The Binomial Distribution

By the end of this lesson, you will be able to identify binomial settings, calculate binomial probabilities, and find the mean and standard deviation of a binomial random variable.

The Binomial Setting (BINS)

A random variable X follows a binomial distribution if all four conditions hold:

B — Binary: each trial has exactly two outcomes (success or failure)

I — Independent: trials are independent of each other

N — Number of trials: a fixed number $n$ of trials

S — Same probability: the probability of success $p$ is the same on every trial

If all four hold, we write $X \sim B(n, p)$.

Check the conditions explicitly on the AP exam. Stating "B, I, N, S" and verifying each earns points.

The 10% Condition

Trials are technically only independent when sampling with replacement. However, if the sample is less than 10% of the population, we treat the trials as approximately independent. This is the 10% condition.

Binomial Probability Formula

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

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the number of ways to choose k successes from n trials.

Example: A free throw shooter makes 70% of shots. Find the probability of making exactly 4 out of 5 attempts.

$P(X = 4) = \binom{5}{4}(0.7)^4(0.3)^1 = 5 \cdot 0.2401 \cdot 0.3 = 5 \cdot 0.07203 = 0.3602$

Computing Cumulative Probabilities

P(X ≤ k): sum the binomial formula for x = 0, 1, 2, ..., k. Use calculator functions: binomcdf(n, p, k).

P(X ≥ k) = 1 − P(X ≤ k−1): use the complement rule.

P(X = k): use binompdf(n, p, k) or the formula.

Mean and Standard Deviation of a Binomial

$$\mu_X = np \qquad \sigma_X = \sqrt{np(1-p)}$$

Example: Out of 20 free throw attempts at 70%:

$\mu_X = 20(0.7) = 14$ makes expected on average

$\sigma_X = \sqrt{20(0.7)(0.3)} = \sqrt{4.2} \approx 2.05$

Normal Approximation to the Binomial

When the sample size is large enough, the binomial distribution is approximately normal:

$X \sim B(n,p)$ is approximately $N(\mu, \sigma) = N(np, \sqrt{np(1-p)})$

Condition for approximation: $np \geq 10$ AND $n(1-p) \geq 10$.