Calculating binomial probabilities P(X = r)

The binomial probability formula P(X = r) = nCr · p^r · (1−p)^(n−r), why it has that shape (choose which r of the n trials succeed, then multiply the success and failure probabilities), how to evaluate it including the special cases r = 0 and r = n, and how it matches the binompdf function on a calculator.

Statistics · Statistical distributions

Calculating binomial probabilities P(X = r)

Why it works

For

X \sim B(n, p)

, the probability of getting exactly

r

successes is

P(X = r) = \binom{n}{r} p^{r} (1-p)^{n-r}, \qquad r = 0, 1, 2, \dots, n,

where

\binom{n}{r} = {}^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}

is the binomial coefficient. Writing

q = 1 - p

for the probability of failure, this is

\binom{n}{r} p^{r} q^{\,n-r}

.

Why this works. Picture one particular way to get

r

successes — say the first

r

trials succeed and the rest fail. That single sequence has probability

\underbrace{p \times \dots \times p}_{r \text{ successes}} \times \underbrace{q \times \dots \times q}_{n-r \text{ failures}} = p^{r} q^{\,n-r},

because the trials are independent. But the

r

successes could fall in any of the positions, and the number of ways to choose which

r

of the

n

trials succeed is

\binom{n}{r}

. Each of those arrangements has the same probability

p^{r} q^{\,n-r}

and they are mutually exclusive, so we add them — that is, multiply by

\binom{n}{r}

.

The ends.

P(X = 0) = q^{n}

(every trial a failure) and

P(X = n) = p^{n}

(every trial a success), since

\binom{n}{0} = \binom{n}{n} = 1

.

On a calculator. This single-value probability is binompdf(n, p, r). Use it directly for "exactly", and keep full accuracy until the final rounding.