O Binomial Distribution

The Binomial Distribution is a distribution of probabilities where there is a known number of trials, a known number of desired outcomes (often referred to as “successes”), and a known and stable probability of success. We are most often interested in the probability of a certain outcome or the probability of any of several outcomes for these distributions.

The two R functions of most use to us in dealing with binomial distributions are “dbinom” and “pbinom” functions

“dbinom” finds the probability of a particular outcome for a given set of circumstances. For the following examples we will assume that we are rolling a regular six sided die 20 times and that we have defined a success as a result of three on the die. For example, we might roll a six sided die twenty times and want to know the probability that we will see exactly 5 results of three. The number of trials is 20, the number of “successes” is 5, and the probability of a success is 1/6.

We should code this into R as follows: dbinom(x=5, size=20, prob=1/6)

dbinom(x = 5, size = 20, prob = 1/6)

## [1] 0.1294103

More often we are interested in knowing the probability of several different events. Obviously one way to find this would be to calculate each individual probability and add them up.

If, for example, we want to know the probability of rolling 4 or fewer threes (p(x <= 4)), we could use this code:

dbinom(x=0:4, size = 20, prob = 1/6)

## [1] 0.02608405 0.10433621 0.19823881 0.23788657 0.20220358

The colon tells R to begin with zero and calculate the probability for every value up to and including 4.

We could also get R to do the addition for us with the following code:

sum( dbinom(x=0:4, size = 20, prob = 1/6))

## [1] 0.7687492

Alternatively, we could use the “pbinom” command:

pbinom(q=4, size = 20, prob = 1/6, lower.tail = T)

## [1] 0.7687492

“lower.tail” tells R that we want 4 or fewer threes. If we set “lower.tail = F” we would get everything from 5 to 20 inclusive. Note that if we evaluate the upper tail that the value of “q” is not included in the calculations.

pbinom(q=4, size = 20, prob = 1/6, lower.tail = F)

## [1] 0.2312508

Either way works, you just need to find the one that seems to make the most sense to you and stick with it. One advantage of dbinom is that you can find the probability associated with a section in the middle of the distribution. If, for example, we wanted to know the probability of 5 to 9 threes we could use the following code:

dbinom(x=5:9, size=20, prob=1/6)

## [1] 0.129410292 0.064705146 0.025882058 0.008411669 0.002243112

or, if we want R to do the addition for us:

sum(dbinom(x=5:9, size=20, prob=1/6))

## [1] 0.2306523