AP Statistics · Topic 2.10 · Unit 2

AP Stats 2.10: The Binomial Distribution

By Jude Wallis · Published

Topic 2.10 in the Fall 2026 AP Statistics course covers the binomial distribution. It counts successes in n independent trials, each with success probability p. The probability is (n choose x) times p^x times (1-p)^(n-x), the mean is np, and the standard deviation is the square root of np(1-p).

AP Statistics: Unit 2 (topics 2.10). In the Fall 2026 AP Statistics course, the binomial distribution is Unit 2 Topic 2.10, aligned to skills 3.C, 3.D, 4.B, and 4.D. Its formula, mean np, and standard deviation are on the formula sheet.

What makes a variable binomial

Topic 2.10 studies one important discrete distribution in depth. The objectives are to justify whether a variable is binomial, and to calculate its probabilities, mean, and standard deviation.

A binomial random variable XX counts the number of successes in a fixed number of repeated independent trials, nn, where each trial has only two possible outcomes labeled success and failure. The probability of success is pp and the probability of failure is 1−p1-p, and pp stays the same on every trial. If any of those conditions fails, the variable is not binomial.

The binomial probability formula

The probability that a binomial random variable has exactly xx successes in nn trials, when the success probability is pp, is the binomial probability function:

P(X=x)=(nx)px(1−p)n−x,x=0,1,2,…,nP(X = x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, 2, \ldots, n

The term (nx)\binom{n}{x}, read 'n choose x', counts how many arrangements give xx successes, pxp^x is the probability of those successes, and (1−p)n−x(1-p)^{n-x} is the probability of the remaining failures. You can also estimate binomial probabilities with a simulation. For worked 'at least' and 'at most' problems and a calculator, see the binomial probability guide and the binomial probability calculator.

Mean and standard deviation

Because the trials are identical and independent, the mean and standard deviation have short formulas. If a random variable is binomial, its mean is:

μX=npσX=np(1−p)\mu_X = np \qquad \sigma_X = \sqrt{np(1-p)}

The mean npnp is the long-run average number of successes, and the standard deviation np(1−p)\sqrt{np(1-p)} measures the typical spread around it. Both appear on the AP formula sheet, and both should be interpreted in context with units. Note that a fixed count of trials separates the binomial from the geometric setting explained in binomial vs geometric.

Justifying and interpreting a binomial

Before computing, justify that the setup is binomial by naming all four conditions in context: a fixed number of trials, two outcomes per trial, a constant success probability, and independent trials. If any condition fails, the binomial formulas do not apply, and sampling a small population without replacement is the usual way independence breaks.

After computing, interpret the results in context. A probability answers how likely a specific count of successes is, the mean npnp gives the long-run average number of successes, and the standard deviation np(1−p)\sqrt{np(1-p)} gives the typical spread around that mean. When an exact calculation is awkward, you can also estimate a binomial probability from a simulation, which ties this topic back to estimating probabilities in topic 2.3.

Defective bulbs in a sample of 8

A large batch of light bulbs is 10% defective. You inspect n = 8 bulbs and let X be the number defective, so p = 0.10. Find P(X = 1), the mean, and the standard deviation.

  1. Binomial coefficient: (81)=8\binom{8}{1} = 8.

  2. Powers: p1=0.10p^1 = 0.10 and (0.90)7=0.4782969(0.90)^{7} = 0.4782969.

  3. Multiply the three pieces: P(X=1)=8×0.10×0.4782969=0.3826P(X = 1) = 8 \times 0.10 \times 0.4782969 = 0.3826.

  4. Mean μX=np=8×0.10=0.8\mu_X = np = 8 \times 0.10 = 0.8; standard deviation σX=8×0.10×0.90=0.72=0.8485\sigma_X = \sqrt{8 \times 0.10 \times 0.90} = \sqrt{0.72} = 0.8485.

P(X=1)≈0.3826P(X = 1) \approx 0.3826, the mean is μX=0.8\mu_X = 0.8 defective bulbs, and σX=0.72≈0.849\sigma_X = \sqrt{0.72} \approx 0.849.

Frequently asked questions

What makes a random variable binomial?

Four conditions: each trial has two outcomes (success or failure), the trials are independent, the number of trials n is fixed in advance, and the success probability p is the same on every trial. A binomial random variable then counts the number of successes across those n trials.

What are the mean and standard deviation of a binomial distribution?

The mean is np, the number of trials times the success probability, giving the long-run average number of successes. The standard deviation is the square root of np times (1 minus p). Both are on the AP Statistics formula sheet and should be stated in context with units.