The binomial distribution

By the end of this lesson

By the end you can decide whether a situation fits the binomial model, write it as $X \sim B(n, p)$ , and calculate $P(X = r)$ with the binomial formula.

A shortcut for repeated identical trials

Many experiments are the same trial repeated a fixed number of times: one archer takes ten shots, twenty components are each tested, a coin is tossed eight times.

If each trial has just two possible results, we usually call them success and failure. Success is bookkeeping, not a compliment: in a quality check, a defective component can be the "success" if that is what you are counting.

When the trials are identical and independent, counting successes has a standard formula, and it saves writing out a huge distribution table.

Definition

The binomial distribution

If $X$ is the number of successes in $n$ independent trials, each with the same probability $p$ of success, then $X$ has a binomial distribution, written

$X \sim B(n, p)$

Its probability function is

$P(X = r) = {}^{n}C_{r}\, p^{r}(1-p)^{\,n-r} \qquad \text{for } r = 0, 1, \dots, n$

Here:

$r$ is the number of successes you want;
$1 - p$ is the probability of failure on a single trial, often written $q$ ;
${}^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}$ is the number of ways to choose $r$ items from $n$ (from topic 5.2), where $n! = n \times (n-1) \times \cdots \times 1$ .

Where the formula comes from

Look at one specific order of successes and failures, say S S F with $n = 3$ .

Independence lets us multiply: the probability of that order is $p \times p \times (1-p)$ , that is $p^2(1-p)^{1}$ . In general, any particular sequence with $r$ successes has probability $p^{r}(1-p)^{n-r}$ .

But we do not care about the order. There are $^{n}C_{r}$ different orders that place $r$ successes among $n$ trials, and combinations are exactly the tool for choosing positions.

Adding those equally probable sequences gives the formula. So the two factors do distinct jobs:

$p^{r}(1-p)^{n-r}$ is the probability of one arrangement.
$^{n}C_{r}$ counts how many arrangements there are.

B(10, 0.3)

: the tallest bar is at

3

successes, the same as

np = 10 \times 0.3 = 3

(you will meet

np

as the mean in the next lesson). Bars far from

3

are tiny:

10

successes out of

10

is almost impossible.

Worked example: Example 1: exactly three successes

A biased coin lands heads with probability $0.3$ . It is tossed $10$ times, independently. Find the probability of exactly $3$ heads.

Have a go first?

Check 1 — using the formula

CalculateQuestion 1 · Apply

Multiple choiceQuestion 2 · Recall

Answer the 2 questions above to continue.