Binomial distribution
Also known as: binomial probability distribution
The binomial distribution gives the probability of getting exactly k successes in n independent trials when each trial has the same probability of success. It is the standard model for counting successes in a fixed number of yes-or-no trials.
A random variable follows a binomial distribution when four conditions hold: the trials are binary (each outcome is a success or a failure), the trials are independent, the number of trials n is fixed in advance, and the probability of success p is the same on every trial. Flipping a coin 20 times and counting heads fits; drawing cards without replacement does not, because p changes from draw to draw.
The probability of exactly k successes is P(X = k) = C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) is the number of ways to choose which k of the n trials are the successes. For example, the probability of exactly 3 heads in 5 fair coin flips is C(5, 3) · (0.5)³ · (0.5)² = 10 · 0.03125 = 0.3125. The center and spread follow simple formulas: the mean is μ = np and the standard deviation is σ = √(np(1 − p)).
Binomial probabilities describe a huge range of situations — defective items in a production batch, patients responding to a treatment, free throws made in a set number of attempts, or voters favoring a candidate in a sample. When n is large and both np and n(1 − p) are at least about 10, the binomial distribution is closely approximated by a normal distribution with the same mean and standard deviation, which is the basis for inference about sample proportions.
The AP Statistics exam expects you to verify the binomial conditions, compute exact and cumulative binomial probabilities, and use the mean and standard deviation formulas — including the connection to the sampling distribution of a sample proportion. The FE Mechanical and FE Civil exams cover the binomial as one of the common probability distributions in their probability and statistics sections, where the reference handbook supplies the formula and you supply the setup.
Key takeaways
- The binomial distribution counts successes in a fixed number of independent trials with a constant success probability.
- P(X = k) = C(n, k) · p^k · (1 − p)^(n − k).
- Mean is np and standard deviation is √(np(1 − p)).
- Sampling without replacement breaks independence, so the binomial model no longer applies exactly.
- For large n with np ≥ 10 and n(1 − p) ≥ 10, the normal distribution is a good approximation.