The Binomial Distribution

The binomial distribution is simply the count of successes when you repeat the same yes-or-no trial a fixed number of times, like how many of 5 free throws go in. The name comes from the binomial theorem in algebra ("binomial" means "two terms"): these probabilities are exactly the terms you get when you expand $(p + (1 - p))^n$. The formula looks crowded, but it's three pieces multiplied: $\binom nk$ (how many ways to pick which $k$ trials succeed), $p^k$ (those $k$ all succeed) and $(1 - p)^{n - k}$ (the rest all fail). Once you see those three, the formula reads like a sentence.

Conditions (BINS): Binary outcomes (success or failure), Independent trials, a fixed Number of trials, the Same probability $p$ each time.

Why the formula works: take 3 successes in 5 trials. One specific sequence, like $SSFSF$, has probability $p \cdot p \cdot (1 - p) \cdot p \cdot (1 - p) = p^3(1 - p)^2$, by the multiplication rule for independent events. Every other sequence with 3 successes has the same probability, just in a different order. So all that's left is counting them, and that's $\binom53 = 10$. In general there are $\binom nk$ sequences, each with probability $p^k(1 - p)^{n - k}$.

Why $\mu = np$? If you shoot 20 free throws at 70%, you'd expect to make 70% of 20, which is 14. The formula just says that out loud. The spread $\sqrt{np(1 - p)}$ is biggest when $p = 0.5$, where the outcome is least predictable.

"At least" and "at most" questions add several terms, or use the complement: $P(X \ge 1) = 1 - P(X = 0)$.

A common mistake. It's tempting to leave out $\binom nk$ and say the chance of 4 makes in 5 shots is $0.7^4(0.3)$. That's the chance of one particular order, like making the first four and missing the last. The miss could come on any of the 5 shots, so you need the factor $\binom54 = 5$.

Quality control (defects in a batch), clinical trials (patients who respond), polling (people who say yes) and network reliability all use binomial counts.