The binomial distribution

What you’ll learn

• Check the conditions for a binomial model.
• Calculate exact and cumulative probabilities.
• Find and interpret the mean and variance.

Understand the idea

A binomial variable counts successes in a fixed number of independent trials, with two outcomes per trial and the same success probability each time. Identify what counts as a success and state $n$ and $p$ before calculating. Sampling without replacement from a small population generally violates constant probability and independence.

The binomial coefficient counts the possible positions of the successes. Multiply it by the probability of any one such arrangement. For phrases such as “at most” and “at least”, translate the wording into an inequality first. For instance, $P(X\geq r)=1-P(X\leq r-1)$.

The mean is a long-run average count rather than a promise about one group. The variance quantifies spread in squared count units. If both the mean and variance are given, their ratio can help determine $p$, and then $n$ follows from the mean.

Key formulae

$$P(X=r)=\binom nr p^r(1-p)^{n-r}$$

$$E(X)=np$$

$$\operatorname{Var}(X)=np(1-p)$$

$$P(X\geq r)=1-P(X\leq r-1)$$

Worked example

Eight independently inspected components each have probability 0.3 of needing adjustment. Find the probability that exactly two need adjustment.

1. Let $X\sim B(8,0.3)$ count components needing adjustment.
2. There are $\binom82=28$ ways to place the two successes.
3. $P(X$ $=2)$ $=28(0.3)^2(0.7)^6$ $=0.29647548$.

Answer: $0.2965$ approximately.