## The model

Suppose we’re interested in counting the number of successes in $$n$$ independent trials of a success/failure random experiment, where the probability of success is $$p \in (0,1)$$. Some real world examples of this are:

• counting the number of heads in $$n$$ coin tosses (here $$p = 1/2$$ for a fair coin)
• counting the number of times 3 comes up in $$n$$ die rolls ($$p = 1/6$$)
• counting the number of defective components in a manufacturing process that produces $$n$$ components in a day ($$p = 1/1000$$ in an earlier example of this type)

## The distribution

Each trial of the experiment can be modeled as independent Bernoulli random variables $$X_1,\ldots,X_n$$ where $P(X_i = 1) = p, \quad P(X_i = 0) = 1-p$ for each $$i = 1,\ldots,n$$. Then the number of succeses is $$X = X_1 + \cdots + X_n$$.

The random variable $$X$$ is called a binomial random variable. Its distribution is given by $P(X = k) = \binom{n}{k} p^k(1-p)^{n-k}, \quad k=0,\ldots,n.$ We write $X \sim \mathrm{Bin}(n,p)$ to denote a binomial random variable with parameters $$n$$ and $$p$$.

## Visualization

The following gives a visualization of the probabilities associated with $$X \sim \mathrm{Bin}(n,p)$$ random variable with some specific choices for $$n$$ and $$p$$. Along the $$x$$-axis are the possible values $$X$$ can take, and along the $$y$$-axis are the associated probabilities.

n <- 20
p <- 1/3
barplot(dbinom(0:n,n,p), names = 0:n)

Here is another with different choices for $$n$$ and $$p$$.

n <- 8
p <- 1/2
barplot(dbinom(0:n,n,p), names = 0:n)

## Computation

We can use the R function pbinom to do computation of probabilities like $P(X \leq m) = P(X = 0) + P(X = 1) + \cdots P(X = m)$ which would otherwise be tedious.

For example, suppose we flip a biased coin, with head probability 1/3, twenty times. The following command lets us compute $$P(X \leq 10)$$ the probability of getting at most 10 heads.

n <- 20
p <- 1/3
pbinom(10, 20, 1/3)
## [1] 0.9623634