SL 4.8—Binomial distribution

The binomial model counts successes in fixed independent trials.

It requires a fixed n, two outcomes per trial, constant success probability p and independence; then P(X=k)=nCk pᵏ(1−p)ⁿ⁻ᵏ.

Example

For five independent shots with p=.4, P(X=2)=10(.4)²(.6)³≈.3456.

Test the four assumptions before using the formula or technology.

Repeated trials from a changing population or with different p are not binomial just because they count successes.

If $X\sim B(n,p)$, then $E(X)=np$ and $Var(X)=np(1-p)$, so $SD(X)=\sqrt{np(1-p)}$. For the five-shot example, the mean number of successes is $5(0.4)=2$ and the variance is $5(0.4)(0.6)=1.2$. Technology may calculate cumulative or interval probabilities, but the fixed-$n$, two-outcome, constant-$p$ and independence assumptions must be checked first.