# A Guide to dbinom, pbinom, qbinom, and rbinom in R The binomial distribution is one of the most commonly used distribution in statistics. This tutorial explains how to work with the binomial distribution in R using the functions dbinom, pbinom, qbinom, and rbinom.

## dbinom

The function dbinom returns the value of the probability density function (pdf) of the binomial distribution given a certain random variable x, number of trials (size) and probability of success on each trial (prob). The syntax for using dbinom is as follows:

dbinom(x, size, prob)

Put simply, dbinom finds the probability of getting a certain number of successes (x) in a certain number of trials (size) where the probability of success on each trial is fixed (prob).

The following examples illustrates how to solve some probability questions using dbinom.

Example 1: Bob makes 60% of his free-throw attempts. If he shoots 12 free throws, what is the probability that he makes exactly 10?

```#find the probability of 10 successes during 12 trials where the probability of
#success on each trial is 0.6
dbinom(x=10, size=12, prob=.6)
#  0.06385228
```

The probability that he makes exactly 10 shots is 0.0639.

Example 2: Sasha flips a fair coin 20 times. What is the probability that the coin lands on heads exactly 7 times?

```#find the probability of 7 successes during 20 trials where the probability of
#success on each trial is 0.5
dbinom(x=7, size=20, prob=.5)
#  0.07392883
```

The probability that the coin lands on heads exactly 7 times is 0.0739.

## pbinom

The function pbinom returns the value of the cumulative density function (cdf) of the binomial distribution given a certain random variable q, number of trials (size) and probability of success on each trial (prob). The syntax for using pbinom is as follows:

pbinom(q, size, prob)

Put simply, pbinom returns the area to the left of a given value q in the binomial distribution. If you’re interested in the area to the right of a given value q, you can simply add the argument lower.tail = FALSE

pbinom(q, size, prob, lower.tail = FALSE)

The following examples illustrates how to solve some probability questions using pbinom.

Example 1: Ando flips a fair coin 5 times. What is the probability that the coin lands on heads more than 2 times?

```#find the probability of more than 2 successes during 5 trials where the
#probability of success on each trial is 0.5
pbinom(2, size=5, prob=.5, lower.tail=FALSE)
#  0.5```

The probability that the coin lands on heads more than 2 times is 0.5.

Example 2:  Suppose Tyler scores a strike on 30% of his attempts when he bowls. If he bowls 10 times, what is the probability that he scores 4 or fewer strikes?

```#find the probability of 4 or fewer successes during 10 trials where the
#probability of success on each trial is 0.3
pbinom(4, size=10, prob=.3)
#  0.8497317```

The probability that he scores 4 or fewer strikes is 0.8497.

## qbinom

The function qbinom returns the value of the inverse cumulative density function (cdf) of the binomial distribution given a certain random variable q, number of trials (size) and probability of success on each trial (prob). The syntax for using qbinom is as follows:

qbinom(q, size, prob)

Put simply, you can use qbinom to find out the pth quantile of the binomial distribution.

The following code illustrates a few examples of qbinom in action:

```#find the 10th quantile of a binomial distribution with 10 trials and prob
#of success on each trial = 0.4
qbinom(.10, size=10, prob=.4)
#  2

#find the 40th quantile of a binomial distribution with 30 trials and prob
#of success on each trial = 0.25
qbinom(.40, size=30, prob=.25)
#  7
```

## rbinom

The function rbinom generates a vector of binomial distributed random variables given a vector length n, number of trials (size) and probability of success on each trial (prob). The syntax for using rbinom is as follows:

rbinom(n, size, prob)

The following code illustrates a few examples of rnorm in action:

```#generate a vector that shows the number of successes of 10 binomial experiments with
#100 trials where the probability of success on each trial is 0.3.
results <- rbinom(10, size=100, prob=.3)
results
#  31 29 28 30 35 30 27 39 30 28

#find mean number of successes in the 10 experiments (compared to expected
#mean of 30)
mean(results)
#  32.8

#generate a vector that shows the number of successes of 1000 binomial experiments
#with 100 trials where the probability of success on each trial is 0.3.
results <- rbinom(1000, size=100, prob=.3)

#find mean number of successes in the 100 experiments (compared to expected
#mean of 30)
mean(results)
#  30.105
```

Notice how the more random variables we create, the closer the mean number of successes is to the expected number of successes.

Note: “Expected number of successes” = n * p where n is the number of trials and p is the probability of success on each trial.