3  Random Variables

3.1 Horse or Laminator

Today we are talking about random variables. There is a classic scene in the absurdist play Rosencrantz and Guildenstern are Dead by Tom Stoppard in which they improbably flip a coin and it turns up heads 92 times in a row. If you flip a coin, what is the probability that it lands up heads? There is a 0.50 probability, or 50% chance, of flipping heads. There is also a 0.5 probability that the outcome is tails. These two probabilities have to sum to 1 because there is no chance of any other outcome. Put another way, if \(p_h = 0.5\), then \(p_t = 1 - p_h = 0.5\), where \(p_h\) is probability of heads and \(p_t\) is probability of tails.

Now, instead of thinking about flipping a coin as a 50/50 shot at heads or tails, think about it only in terms of succeeding at flipping heads. Each successful attempt you get a ‘1’ for a success, and each time the coin is not heads, we call it a ‘0’ for failure. Put another way, we ask the question ‘Did we flip heads?’ and if the answer is TRUE, and we use a 1 to denote that. This is actually how R encodes TRUE (T = 1) and FALSE (F = 0).

# to leave notes to yourself in R scripts, start with a #
?rbinom # look up the Help file to find out about the arguments to the function

# rbinom wants:
# n, the number of observations / replicates
# size, the number of draws
# p the probability of success

# to simulate a single coin flip, we want to take one replicate (n=1) of 
# one draw (size=1) from a binomial distribution  with a probability of 
# success (heads) of 0.5
rbinom(n=1, size=1, prob=0.5)

[1] 1

# we could keep running this line over and over
# or we could increase n, the number of replicates
# this gives us a vector of 100 observations of a coin flip
rbinom(n=100, size=1, p=c(0.5))

  [1] 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 1
 [38] 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 0
 [75] 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0

# to visualize the outcome, we can create a histogram plotting the results
# using the function hist()
hist(rbinom(n=100, size=1, p=0.5))

# because the function is drawing from a probability distribution, outcomes will
# be a little different every single time. If you need them to be the same, you 
# can set the seed with set.seed(42), or any other number

The outcome we are observing, successfully flipping heads, is a random variable. A random variable is one that is drawn from a probability distribution (not in the colloquial sense of the word random). Our observed response (heads = TRUE) is drawn from a Bernoulli distribution with a probability of success of 0.5. We can write this mathematically as: \[heads \sim Bernoulli(p=0.5)\]

The Bernoulli distribution is a probability distribution that only yields success (1) or failure (0) based on the probability of success \(p\). The probability can vary depending on the process you are modeling, but the outcome will always be 0 or 1.

While we could be very high-brow in this course and simulate the probability of that series of events, instead we will consider the remarkable case of Jo Brand’s performance in a game of ‘Horse or Laminator’ on Taskmaster. In this task, five comedians have to guess correctly whether the taskmaster has selected a card with a horse on it, or a laminator on it. For the sake of this exercise, we will assume that the taskmaster is random in his selections, and that the only thing we need to be concerned with is the probability of a comedian correctly guessing if he has selected a horse or a laminator (i.e., he is not more or less likely to choose a horse or a laminator, and there is no higher chance of a comedian being correct in the case of him selecting one or the other).

Horse or Laminator

There are only two options, so we assume that comedians have a 50/50 shot at being correct, meaning they have a 0.5 probability of successfully guessing if the taskmaster has chosen a horse or a laminator. We can represent ‘success’ with a 1 and ‘failure’ with a 0. If the taskmaster chooses a horse and they select horse, that’s a 1; if he selects a horse and they select a laminator, that’s a 0; if he selects a laminator and they select a laminator, that’s a 1; and if he selects a laminator and they select a horse, that’s a 0. It does not matter what is selected, so long as their answers match.

In R, we can represent this scenario as arising from a binomial distribution. We’ve encountered the function rbinom before when playing around with code, and that is the way to randomly draw from a binomial distribution in R. For a single guess, we will make one draw from a binomial distribution of size 1, and a probability of 0.5.

rbinom(n = 1, size = 1, 0.5)

[1] 1

Note that the value we get from this can only ever be 0 or 1, but, the value returned is random. We don’t always get a 1, and we don’t always get a 0, but every single time we have a 0.5 probability of getting a 1 (i.e., a ‘success’).

Now let’s assume that each comedian has 10 guesses. That’s not actually what happens on the show, but let’s run with it for a minute. We can adjust our code so that instead of drawing one number from the distribution once, we draw one number ten times.

# take ten draws of size 1 from the distribution
rbinom(n = 10, size=1, prob=0.5)

 [1] 1 1 0 1 0 0 1 1 0 0

# take 1 draw (for one comedian) who gets 10 guesses, and report the number of successes in those 10 attempts
rbinom(n = 1, size=10, prob=0.5)

[1] 5

We could now tally up the number of times a comedian was successful based on this random draw. Over the long-run, this number should balance out to roughly 50/50, but on any given draw it is random. The reason it starts to balance out when we take enough draws is due to the law of large numbers. If we make enough draws from a probability distribution, eventually it will start to resemble the expected value. Let’s try still letting our comedian guess 10 times, but now we will iterate that for 1000 attempts. So in our simulated world, you could think of it like 1000 comedians all get a shot to guess horse or laminator 10 times each, and we want to know the average number of times they were correct. We would expect it to stabilize probably somewhere around five out of 10 times.

# this will give us 1000 simulations of 10 draws from binomial
rbinom(n = 1000, size=10, prob=0.5)

   [1]  3  4  3  7  5  7  5  7  4  5  3  4  4  4  5  4  3  5  4  4  3  7  3  4
  [25]  6  5  5  6  4  1  5  6  5  5  7  8  5  5  6  4  3  5  7  8  5  6  5  5
  [49]  4  2  5  5  5  7  5  1  5  4  5  4  7  5  3  9  5  9  7  6  5  4  7  4
  [73]  4  5  3  4  2  4  5  2  9  8  6  5  3  3  4  5  6  6  2  7  4  4  0  4
  [97]  3  8  6  8  4  4  2  4  6  4  3  5  1  2  4  6  5  5  5  7  5  7  5  8
 [121]  4  8  5  4  2  5  6  5  4  4  7  7  7  5  4  4  4  6  6  5  4  5  7  8
 [145]  7  6  2  6  6  2  4  7  7  3  5  4  3  4  3  5  4  4  6  6  6  3  6  5
 [169]  7  4  7  6  5  6  4  5  5  5  4  4  4  6  5  5  1  5  2  8  6  6  3  3
 [193]  4  5  7  5  4  0  4  2  3  6  7  4  4  5  6  7  4  7  4  5  5  5  2  4
 [217]  4  8  5  6  4  4  5  4  5  6  6  4  3  6  5  6  4  8  3  4  4  4  4  4
 [241]  4  6  4  4  7  1  6  3  5  6  4  3  4  4  5  4  9  5  2  2  4  3  6  5
 [265]  4  5  4  7  3  6  6  5  5  8  4  4  7  2  6  4  6  4  4  5  4  8  3  6
 [289]  3  5  3  5  6  0  5  4  3  8  4  5  7  7  5  4  8  5  4  4  3  5  9  6
 [313]  7  4  5  8  6  7  3  4  8  4  5  7  6  4  5  3  6  4  5  8  7  5  7  7
 [337]  6  4  5  5  9  5  5  4  8  6  3  5  4  3  6  3  8  3  5  6  6  6  6  5
 [361]  3  4  6  5  5  4  2  6  3  6  5  6  5  8  3  7  5  8  6  4  3  4  3  4
 [385]  6  4  1  5  4  4  4  3  4  7  5  7  6  2  7  6  5  7  5  3  5  2  6  5
 [409]  4  6  7  7  5  6  4  8  7  7  6  4  8  7  5  6  5  5  5  4  5  4  5  3
 [433]  3  5  4  5  3  5  5  7  7  9  5  4  2  6  4  7  3  7  5  6  6  5  5  5
 [457]  7  5  7  4  2  3  5  5  6  7  7  1  4  9  8  5  4  4  7  5  8  7  3  4
 [481]  5  4  5  7  5  4  8  3  3  5  9  5  5  5  3  6  5  5  7  6  7  5  4  5
 [505]  6  1  2  2  4  5  4  5  6  7  3  5  5  7  3  5  5  4  3  4  4  4  5  3
 [529]  2  5  5  5  2  4  3  5  3  5  4  6  6  4  7  8  8  7  7  3  6  7  7  5
 [553]  5  6  5  5  5  4  7  8  8  6  7  5  4  2  2  4  3  2  6  4  6  5  7  5
 [577]  6  2  0  4  4  7  5  7  4  5  5  6  7  6  6  3  4  4  5  8  3  5  4  4
 [601]  6  6  4  5  3  5  6  4  4  3  7  6  4  3  5  4  6  1  4  5  3  7  6  7
 [625]  3  3  6  5  5  5  5  7  2  7  3  5  6  5  7  4  6  6  7  7  7  7  4  7
 [649]  3  8  8  6  5  4  4  4  3  5  7  4  3  7  9  4  3  5  3  1  6  3  6  3
 [673]  5  2  4  3  5  8  7  4  6  5  5  7  6  5  4  4  6  6  4  4  7  4  6  4
 [697]  5  5  5  4  5  4  5  6  4  6  5  5  4  7  7  6  7  3  3  7  7  8  7  5
 [721]  6  4  6  7  5  4  4  3  4  8  5  3  5  5  4  5  5  6  4  5  5  5  6  5
 [745]  5  7  5  6  6  6 10  4  3  4  2  6  5  4  3  6  7  4  8  5  6  3  5  7
 [769]  3  4  5  3  4  6  5  5  7  5  5  6  5  5  5  4  5  5  2  4  5  6  5  8
 [793]  4  2  5  8  4  4  5  4  9  1  7  2  5  5  4  3  7  7  4  6  5  6  6  4
 [817]  6  5  5  5  3  6  5  3  6  9  5  7  6  5  1  3  6  4  3  6  5  4  7  7
 [841]  6  6  6  5  6  4  3  6  4  5  3  5  5  5  4  6  7  1  2  7  3  4  5  3
 [865]  4  3  5  6  3  7  3  9  4  2  6  7  5  5  4  3  5  4  2  2  7  6  6  4
 [889]  5  5  3  5  6  4  3  6  5  2  4  7  5  5  6  7  5  6  6  6  5  5  7  4
 [913]  4  5  3  6  5  6  5  3  5  6  3  5  5  8  2  7  7  4  5  3  4  5  1  2
 [937]  4  3  6  4  2  8  3  7  2  7  5  5  5  3  6  3  6  6  7  7  4  4  4  7
 [961]  5  8  9  6  4  3  5  6  6  3  6  8  9  3  5  5  5  6  5  5  8  5  5  5
 [985]  6  4  4  3  4  6  5  5  6  4  6  9  6  4  4  2

# we should save this to an object so it is easier to work with
ten_attempts <- rbinom(n = 1000, size = 10, prob=0.5)
table(ten_attempts)

ten_attempts
  0   1   2   3   4   5   6   7   8   9  10 
  1  16  39 126 227 235 179 133  30  12   2 

hist(ten_attempts)

Now, what actually happens on Taskmaster is that the first comedian gets five correct in a row (horse, horse, horse, horse, laminator). The second gets one correct. The third gets zero correct, the fourth gets one, and the final comedian, Jo Brand, guesses horse or laminator correct 13 times in a row. Using the rbinom function, work out what the probability of each of these events is. Keep in mind that now instead of getting a certain number of guesses and tallying success across all those, as soon as a comedian guesses incorrectly, they are out. So we are looking only for that number of successful events in a row to calculate the probability.

# recode david's as 1, because otherwise length doesn't work correctly; you can't guess 0 times
# there are many other solutions possible
# you could subset stuff, for example
# or use logical vectors to find true and false, etc.
scores <- c(5, 1, 1, 1, 13)

for(i in scores){
  print(prop.table(table(rbinom(n = 1000000, size = i, prob=0.5)==i)))
}

   FALSE     TRUE 
0.968395 0.031605 

   FALSE     TRUE 
0.499957 0.500043 

  FALSE    TRUE 
0.49931 0.50069 

   FALSE     TRUE 
0.500608 0.499392 

   FALSE     TRUE 
0.999873 0.000127 

## Guillemot chick survival

To move on to a slightly more ecological example than coin flips or horses and laminators, let’s think about guillemots. Guillemots are seabirds that nest on steep cliffsides to escape predators. They spend most of their life on the ocean, however, and as such are very awkward on land and also not great at flying. To get from the cliffside nest to the ocean for the first time, guillemot chicks have to jump and then glide/fly. If they do not make it all the way, they can attempt to run, but are vulnerable to predation. The only option for chicks is to survive or fail - making it 70% of the way to the ocean will result in failure (and, in this case, death). Thus the probability of success does not equate to the actual rate of success - it is the probability per trial (i.e., each attempt).

# let's assume chicks have a 40% chance of making it to the ocean, and there are
# 100 chicks jumping from the cliff hoping to make it to the ocean so we have 
# 100 observations, each with one draw from the Bernoulli distribution
rbinom(n = 100, size=1, p=0.4)

  [1] 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0
 [38] 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1
 [75] 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0

# so far, we have just been running functions and the output is shown in the console
# but we don't have anything in our environment
# can save to an object with the <- assignment operator

survival <- rbinom(n=100, size=1, p=0.4)

# we can then do things with that saved object, like plot it
hist(survival)

3.2 Binomial distribution

3.2.1 Guillemot example

Guillemots typically only lay 1-2 eggs, but for the sake of our example code, let’s pretend that each pair has 3 chicks that are attempting to make it to the ocean. Each individual chick’s attempt could be modeled with a Bernoulli distribution, but if we consider it from the perspective of the breeding pair, the total number of surviving offspring is what matters. We can think of the number of chicks from each nest that make it to the ocean as the sum of their individual attempts for a single nest. This is the Binomial distribution, which is a more generalized version of the Bernoulli distribution (more accurately, the Bernoulli distribution is a special case of the Binomial distribution with one draw) in which each observation has multiple draws.

# for one nest with 3 chicks and a probability of success of 0.4
rbinom(n=1, size=3, p=0.4)

[1] 0

# we could also think about a colony with let's say 100 nests
# how many chicks from each nest will make it?
nest_offspring <- rbinom(n=100, size=3, p=0.4)
hist(nest_offspring)

nest_offspring

  [1] 1 1 1 1 3 1 2 1 1 1 1 2 0 1 2 0 0 3 2 1 0 0 2 0 0 0 2 0 0 2 1 2 1 1 2 1 2
 [38] 1 2 0 0 1 1 2 1 0 2 0 2 0 1 1 1 3 2 3 2 1 2 2 1 1 1 1 1 1 2 1 2 2 2 2 0 2
 [75] 1 2 0 0 1 1 0 2 2 1 0 1 1 0 2 1 3 1 2 1 1 0 2 1 0 2

# what is the probability that all three chicks from a nest make it?
# because we know what we fixed p to, we could multiply the probabilities
0.4^3

[1] 0.064

# what is the probability that none make it?
(1 - 0.4)^3

[1] 0.216

We are able to calculate what the probability of these outcomes are because we fixed the probability of success at 0.4 and then simulated outcomes. In the real world, we rarely know what the probability of success truly is, and instead we are working backwards from our observed outcomes (i.e. our data) to estimate the probability of survival. One of the beautiful things about working with simulated data is that we know the input value, so we can test assumptions, see if we recapture our input, and play around with different model structures and know that anything that is unexpected is most likely a problem with our code or model, not the data.

With discrete distributions like the Binomial, we can convert the frequency of outcomes to a proportion of the total as an estimate of probability. Dividing the frequencies by the total sample size makes it so that the total probability sums to 1.

# table is a function that tallies up all the items in a vector
# a vector is a type of object that is one dimension, i.e. nest_offspring is a 
# vector with length = 100
table(nest_offspring)

nest_offspring
 0  1  2  3 
23 41 31  5 

# proportions will convert the table into proportions instead of counts
# so will prop.table and I often use that function because I learned it first
# there are many different ways to accomplish the same goal in R
survival_prob <- proportions(table(nest_offspring))

# How do these values compare to the calculated probability of all three chicks
# surviving? What about of none surviving?

# With the values calculated from the simulated data (i.e. not the parameter
# value that we fixed), what is the probability that *at least* one chick 
# from a nest survives?
0.47 + 0.27 + 0.06 # note: numbers might be different because we didn't set.seed()

[1] 0.8

In general with coding, you want to avoid hard coding like this where you put in fixed values. It is much better to use code that is flexible if the input data changes (e.g. we are using a randomly generated dataset, so it will change every time), or if you want to change some parameter across a lot of different parts of the code. For example, instead of repeatedly typing p=0.4, we could have created an object in our environment with the probability of success such as p.surv <- 0.4 and then specifying p=p.surv throughout the script, which would make it really easy to change the simulations for a new scenario (e.g. if the guillemot chicks were given little hang-gliders, we might increase p.surv and would only have to type it out once as p.surv <- 0.8.

Subsetting vectors in R is a really useful tool when you want to apply a function to only part of a vector, inspect part of an object, etc. We use the square brackets [] for subsetting, and within them specify which elements to return. The elements to return can be a numeric vector (e.g. c(1,2,4)) to return the first, second, and fourth elements, or a logical vector indicating if an element should or should not be returned (e.g. c(TRUE TRUE FALSE TRUE)). If subsetting a range of consecutive elements, the : operator can also be used (e.g. 1:4 is the same as c(1,2,3,4)).

# To avoid hard coding our estimated percent, we can use sum() on a subset of
# the table of proportions to estimate the probability at least one survives?
sum(survival_prob[2:4])

[1] 0.77

# what is the probability at least two survive?
sum(survival_prob[3:4])

[1] 0.36

# side note: if you're not sure which indices to subset, it can help to look at
# your object; you can easily do this in RStudio by highlighting just the bit of
# code with your object name and using Ctrl + Enter (or Cmd + Enter) to print
# just that object to the console, i.e. if you highlight part of a line, you will
# only run the highlighted bit, not the entire line

3.2.2 Coral example

With the guillemots, we were assuming that each pair had three potential offspring surviving. What happens if we vastly increase the number of draws from the binomial distribution? Take corals as an example. Some broadcast spawning corals have mass synchronized spawning events where individual corals on a reef all release bundles of sperm and eggs at the same time. The reproductive success of each individual coral can still be modeled as a binomial distribution, i.e. how many of those released sperm and eggs will actually encounter a bundle of the same species and be a successful mating attempt is a random variable drawn from a Binomial distribution with a number of draws \(N\) equal to the number of released sperm and eggs and a probability of success \(p\) for each of those. \[ offspring \sim Binomial(N, p)\].

Note: the \(N\) for denoting the number of draws from the Binomial is distinct from the n = argument in the rbinom() function. Don’t let this confuse you! \(N\) equates to the size = argument, while n = is the number of observations of the random variable, i.e. the total number of individual corals on the reef.

# let's assume now the coral reef has 100,00 individual corals (I have no 
# clue if this is an accurate number, but let's roll with it)
# let's also assume each individual coral releases 10,000 sperm and eggs
# and the probability for each of those resulting in a successful mating event 
# 0.0002 (i.e. 0.002%)

gametes <- rbinom(n=100000, size=10000, p=0.0002)
hist(gametes, breaks = 100)

# what is the probability of reproductive failure?
prop.table(table(gametes))

gametes
      0       1       2       3       4       5       6       7       8       9 
0.13420 0.27083 0.26959 0.18136 0.09046 0.03714 0.01177 0.00354 0.00086 0.00020 
     10      12 
0.00004 0.00001 

# what's the probability of more than 5 offspring?
prop.table(table(gametes>5))

  FALSE    TRUE 
0.98358 0.01642 

# now let's increase the number of individual corals to 1 million, and also increase
# the probability of success to 0.001 

gametes2 <- rbinom(n=1000000, size=10000, p=0.001)
hist(gametes2, breaks=100)

# now what is the probability of reproductive failure?
prop.table(table(gametes2==0))

  FALSE    TRUE 
0.99996 0.00004 

# and what is the probability of more than 15 offspring?
prop.table(table(gametes2>15))

   FALSE     TRUE 
0.951553 0.048447 

# let's visualize the data as a histogram again, but add a vertical bar using 
# the abline() function at 15; note there are other plotting options to make 
# your plots slightly more aesthetic
hist(gametes2, 
     breaks=100,
     border=F, # i just don't like borders on bars
     col="turquoise4", 
     main="", # removes the 'title'
     xlab="Number of successful mating attempts", # changes x axis label
     las=1 # rotates the axis labels the right way
     )
abline(v=15, 
       lwd=2, # width of line
       col="tomato2")

3.3 Probability mass

In the example above, you can see that there is very low probability of being in the tails of the distribution. So far, we have only been thinking about discrete distributions where the outcome will fall into a ‘bin’ of some kind. With the Bernoulli distribution, the outcome could only be 0 or 1. With the Binomial distribution, the outcome could only be an integer, including zero, and can only be positive. There are no numbers in between; you either make it to the ocean, or you don’t, and you never end up with 1.5 surviving offspring.

Discrete distributions have a probability mass function, which you can think of as the amount of probability associated with each bar in the histogram (including potential ‘bars’ that are never observed and their probability mass is 0).

We can visualize the probability mass associated with different outcomes using the dbinom function.

dbinom(c(0, 1), 1, 0.5)

[1] 0.5 0.5

# what if we tried for a non-possible outcome?

check_me <- seq(0, 1, by=0.001)
plot(dbinom(check_me, 1, 0.5), type="l")

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.001000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.002000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.003000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.004000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.005000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.006000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.007000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.008000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.009000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.010000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.011000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.012000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.013000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.014000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.015000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.016000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.017000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.018000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.019000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.020000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.021000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.022000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.023000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.024000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.025000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.026000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.027000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.028000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.029000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.030000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.031000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.032000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.033000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.034000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.035000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.036000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.037000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.038000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.039000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.040000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.041000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.042000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.043000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.044000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.045000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.046000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.047000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.048000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.049000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.050000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.051000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.052000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.053000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.054000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.055000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.056000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.057000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.058000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.059000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.060000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.061000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.062000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.063000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.064000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.065000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.066000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.067000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.068000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.069000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.070000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.071000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.072000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.073000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.074000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.075000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.076000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.077000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.078000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.079000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.080000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.081000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.082000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.083000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.084000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.085000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.086000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.087000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.088000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.089000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.090000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.091000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.092000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.093000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.094000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.095000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.096000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.097000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.098000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.099000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.100000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.101000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.102000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.103000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.104000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.105000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.106000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.107000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.108000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.109000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.110000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.111000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.112000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.113000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.114000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.115000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.116000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.117000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.118000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.119000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.120000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.121000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.122000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.123000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.124000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.125000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.126000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.127000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.128000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.129000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.130000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.131000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.132000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.133000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.134000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.135000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.136000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.137000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.138000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.139000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.140000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.141000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.142000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.143000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.144000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.145000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.146000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.147000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.148000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.149000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.150000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.151000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.152000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.153000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.154000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.155000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.156000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.157000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.158000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.159000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.160000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.161000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.162000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.163000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.164000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.165000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.166000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.167000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.168000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.169000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.170000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.171000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.172000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.173000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.174000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.175000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.176000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.177000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.178000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.179000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.180000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.181000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.182000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.183000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.184000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.185000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.186000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.187000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.188000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.189000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.190000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.191000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.192000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.193000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.194000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.195000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.196000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.197000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.198000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.199000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.200000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.201000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.202000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.203000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.204000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.205000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.206000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.207000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.208000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.209000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.210000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.211000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.212000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.213000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.214000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.215000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.216000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.217000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.218000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.219000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.220000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.221000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.222000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.223000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.224000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.225000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.226000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.227000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.228000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.229000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.230000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.231000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.232000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.233000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.234000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.235000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.236000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.237000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.238000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.239000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.240000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.241000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.242000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.243000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.244000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.245000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.246000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.247000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.248000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.249000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.250000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.251000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.252000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.253000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.254000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.255000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.256000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.257000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.258000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.259000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.260000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.261000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.262000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.263000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.264000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.265000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.266000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.267000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.268000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.269000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.270000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.271000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.272000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.273000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.274000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.275000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.276000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.277000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.278000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.279000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.280000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.281000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.282000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.283000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.284000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.285000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.286000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.287000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.288000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.289000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.290000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.291000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.292000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.293000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.294000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.295000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.296000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.297000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.298000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.299000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.300000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.301000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.302000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.303000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.304000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.305000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.306000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.307000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.308000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.309000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.310000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.311000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.312000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.313000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.314000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.315000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.316000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.317000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.318000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.319000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.320000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.321000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.322000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.323000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.324000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.325000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.326000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.327000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.328000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.329000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.330000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.331000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.332000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.333000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.334000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.335000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.336000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.337000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.338000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.339000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.340000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.341000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.342000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.343000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.344000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.345000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.346000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.347000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.348000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.349000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.350000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.351000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.352000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.353000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.354000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.355000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.356000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.357000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.358000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.359000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.360000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.361000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.362000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.363000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.364000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.365000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.366000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.367000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.368000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.369000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.370000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.371000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.372000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.373000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.374000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.375000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.376000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.377000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.378000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.379000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.380000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.381000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.382000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.383000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.384000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.385000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.386000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.387000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.388000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.389000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.390000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.391000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.392000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.393000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.394000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.395000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.396000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.397000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.398000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.399000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.400000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.401000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.402000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.403000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.404000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.405000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.406000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.407000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.408000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.409000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.410000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.411000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.412000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.413000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.414000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.415000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.416000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.417000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.418000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.419000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.420000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.421000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.422000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.423000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.424000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.425000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.426000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.427000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.428000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.429000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.430000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.431000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.432000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.433000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.434000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.435000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.436000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.437000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.438000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.439000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.440000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.441000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.442000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.443000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.444000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.445000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.446000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.447000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.448000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.449000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.450000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.451000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.452000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.453000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.454000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.455000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.456000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.457000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.458000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.459000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.460000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.461000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.462000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.463000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.464000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.465000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.466000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.467000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.468000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.469000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.470000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.471000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.472000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.473000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.474000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.475000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.476000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.477000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.478000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.479000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.480000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.481000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.482000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.483000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.484000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.485000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.486000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.487000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.488000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.489000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.490000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.491000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.492000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.493000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.494000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.495000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.496000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.497000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.498000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.499000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.500000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.501000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.502000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.503000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.504000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.505000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.506000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.507000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.508000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.509000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.510000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.511000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.512000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.513000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.514000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.515000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.516000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.517000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.518000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.519000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.520000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.521000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.522000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.523000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.524000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.525000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.526000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.527000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.528000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.529000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.530000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.531000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.532000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.533000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.534000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.535000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.536000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.537000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.538000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.539000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.540000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.541000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.542000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.543000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.544000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.545000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.546000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.547000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.548000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.549000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.550000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.551000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.552000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.553000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.554000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.555000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.556000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.557000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.558000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.559000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.560000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.561000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.562000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.563000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.564000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.565000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.566000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.567000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.568000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.569000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.570000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.571000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.572000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.573000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.574000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.575000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.576000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.577000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.578000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.579000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.580000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.581000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.582000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.583000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.584000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.585000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.586000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.587000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.588000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.589000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.590000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.591000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.592000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.593000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.594000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.595000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.596000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.597000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.598000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.599000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.600000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.601000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.602000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.603000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.604000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.605000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.606000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.607000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.608000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.609000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.610000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.611000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.612000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.613000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.614000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.615000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.616000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.617000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.618000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.619000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.620000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.621000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.622000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.623000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.624000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.625000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.626000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.627000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.628000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.629000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.630000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.631000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.632000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.633000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.634000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.635000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.636000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.637000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.638000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.639000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.640000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.641000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.642000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.643000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.644000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.645000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.646000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.647000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.648000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.649000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.650000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.651000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.652000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.653000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.654000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.655000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.656000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.657000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.658000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.659000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.660000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.661000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.662000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.663000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.664000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.665000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.666000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.667000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.668000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.669000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.670000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.671000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.672000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.673000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.674000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.675000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.676000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.677000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.678000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.679000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.680000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.681000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.682000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.683000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.684000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.685000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.686000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.687000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.688000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.689000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.690000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.691000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.692000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.693000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.694000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.695000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.696000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.697000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.698000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.699000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.700000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.701000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.702000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.703000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.704000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.705000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.706000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.707000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.708000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.709000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.710000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.711000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.712000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.713000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.714000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.715000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.716000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.717000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.718000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.719000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.720000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.721000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.722000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.723000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.724000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.725000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.726000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.727000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.728000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.729000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.730000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.731000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.732000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.733000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.734000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.735000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.736000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.737000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.738000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.739000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.740000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.741000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.742000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.743000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.744000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.745000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.746000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.747000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.748000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.749000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.750000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.751000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.752000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.753000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.754000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.755000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.756000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.757000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.758000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.759000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.760000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.761000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.762000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.763000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.764000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.765000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.766000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.767000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.768000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.769000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.770000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.771000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.772000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.773000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.774000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.775000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.776000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.777000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.778000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.779000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.780000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.781000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.782000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.783000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.784000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.785000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.786000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.787000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.788000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.789000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.790000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.791000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.792000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.793000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.794000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.795000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.796000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.797000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.798000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.799000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.800000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.801000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.802000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.803000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.804000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.805000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.806000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.807000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.808000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.809000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.810000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.811000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.812000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.813000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.814000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.815000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.816000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.817000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.818000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.819000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.820000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.821000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.822000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.823000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.824000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.825000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.826000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.827000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.828000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.829000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.830000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.831000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.832000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.833000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.834000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.835000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.836000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.837000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.838000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.839000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.840000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.841000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.842000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.843000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.844000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.845000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.846000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.847000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.848000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.849000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.850000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.851000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.852000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.853000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.854000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.855000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.856000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.857000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.858000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.859000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.860000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.861000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.862000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.863000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.864000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.865000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.866000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.867000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.868000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.869000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.870000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.871000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.872000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.873000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.874000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.875000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.876000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.877000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.878000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.879000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.880000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.881000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.882000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.883000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.884000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.885000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.886000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.887000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.888000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.889000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.890000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.891000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.892000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.893000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.894000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.895000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.896000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.897000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.898000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.899000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.900000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.901000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.902000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.903000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.904000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.905000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.906000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.907000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.908000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.909000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.910000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.911000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.912000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.913000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.914000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.915000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.916000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.917000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.918000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.919000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.920000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.921000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.922000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.923000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.924000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.925000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.926000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.927000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.928000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.929000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.930000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.931000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.932000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.933000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.934000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.935000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.936000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.937000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.938000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.939000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.940000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.941000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.942000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.943000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.944000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.945000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.946000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.947000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.948000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.949000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.950000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.951000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.952000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.953000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.954000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.955000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.956000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.957000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.958000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.959000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.960000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.961000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.962000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.963000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.964000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.965000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.966000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.967000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.968000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.969000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.970000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.971000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.972000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.973000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.974000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.975000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.976000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.977000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.978000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.979000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.980000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.981000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.982000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.983000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.984000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.985000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.986000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.987000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.988000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.989000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.990000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.991000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.992000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.993000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.994000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.995000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.996000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.997000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.998000

Warning in dbinom(check_me, 1, 0.5): non-integer x = 0.999000

# why do we get little spikes?
plot(dbinom(seq(0, 10, 0.5), 10, 0.5), type="l")

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 0.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 1.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 2.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 3.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 4.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 5.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 6.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 7.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 8.500000

Warning in dbinom(seq(0, 10, 0.5), 10, 0.5): non-integer x = 9.500000

xvals <- seq(0, 10, by=1)
plot(dbinom(xvals, 10, 0.5) ~ xvals, type="l")

barplot(dbinom(xvals, 10, 0.5), names.arg = xvals)

# how closely does that match when we draw from the distribution?
# at low sample size?
hist(rbinom(n = 12, size=10, prob=0.5))

# at high sample size?
hist(rbinom(n = 1000, size=10, prob=0.5))

# what is the probability of getting at least 8?
sum(dbinom(8:10, 10, 0.5))

[1] 0.0546875

# what if we did the same calculation with simulated data?
sum(rbinom(100, 10, 0.5)>=8)/100

[1] 0.05