Section 2.4 - The Binomial Distribution—Bernoulli Trials

A Bernoulli trial produces an outcome which satisfies a criterion with probability $p$, and so fails to satisfy the criterion with compliment probability $q = 1 - p$. What is the probability that $n$ successive, independent Bernoulli trials results in exactly $k$ satisfactory outcomes?

The compound outcome is an arrangement of $k$ satisfactory outcomes and $n - k$ unsatisfactory outcomes. Because the Bernoulli-trial outcomes are independent, the outcome probabilities are multiplied to get the combined-outcome probability, and because multiplication is commutative, any individual compound outcome has probability $p^kq^{n-k}$. There are $C(n, k)$ such compound outcomes, and the probability that $n$ successive, independent Bernoulli trials results in exactly $0 \leq k \leq n$ satisfactory outcomes when satisfactory outcomes have probability $p$ is $$b(k; n, p) = C(n, k)p^kq^{n - k}$$ The compound-outcome sample space is not uniform. Even if $p = 1/2$,^?that is, $p = q$. $C(n, k)$ varies with $k$; the variation becomes more complex when $p\neq q$.