The binomial distribution

Covers AS91586

When to use the binomial distribution

The binomial distribution models the number of successes in a fixed number of trials. Use it when all four conditions hold:
- a fixed number of trials, $n$ ;
- each trial has just two outcomes (success or failure);
- a constant probability of success, $p$ , on every trial;
- the trials are independent.

The binomial formula

The probability of exactly $k$ successes in $n$ trials is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{\,n-k}$

$\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$ counts the number of ways to arrange the $k$ successes.
The mean (expected number of successes) is $E(X) = np$ .

Worked ExampleA binomial probability

A fair coin is tossed 8 times. Find the probability of getting exactly 3 heads.

Step 1 — Identify $n$ , $p$ and $k$

$n = 8, \quad p = 0.5 \text{ (heads)}, \quad k = 3$

Step 2 — Substitute into the formula

$P(X = 3) = \binom{8}{3}(0.5)^3(0.5)^{5}$

Step 3 — Evaluate

$\binom{8}{3} = 56$ , and $(0.5)^3(0.5)^5 = (0.5)^8 = \dfrac{1}{256}$ :

$P(X = 3) = 56 \times \frac{1}{256} = \frac{56}{256} \approx 0.219$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

For a binomial with $n = 20$ and $p = 0.3$ , find the mean (expected number of successes).

Merit

A multiple-choice test has 10 questions, each with probability 0.25 of a correct guess. Find the probability of guessing exactly 2 correct.

Excellence

A machine produces items with a 5% defect rate. In a sample of 20, find the probability that at least one is defective.

The binomial distribution

Covers AS91586

When to use the binomial distribution

The binomial distribution models the number of successes in a fixed number of trials. Use it when all four conditions hold:
- a fixed number of trials, $n$ ;
- each trial has just two outcomes (success or failure);
- a constant probability of success, $p$ , on every trial;
- the trials are independent.

The binomial formula

The probability of exactly $k$ successes in $n$ trials is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{\,n-k}$

$\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$ counts the number of ways to arrange the $k$ successes.
The mean (expected number of successes) is $E(X) = np$ .

Worked ExampleA binomial probability

A fair coin is tossed 8 times. Find the probability of getting exactly 3 heads.

Step 1 — Identify $n$ , $p$ and $k$

$n = 8, \quad p = 0.5 \text{ (heads)}, \quad k = 3$

Step 2 — Substitute into the formula

$P(X = 3) = \binom{8}{3}(0.5)^3(0.5)^{5}$

Step 3 — Evaluate

$\binom{8}{3} = 56$ , and $(0.5)^3(0.5)^5 = (0.5)^8 = \dfrac{1}{256}$ :

$P(X = 3) = 56 \times \frac{1}{256} = \frac{56}{256} \approx 0.219$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

For a binomial with $n = 20$ and $p = 0.3$ , find the mean (expected number of successes).

Merit

A multiple-choice test has 10 questions, each with probability 0.25 of a correct guess. Find the probability of guessing exactly 2 correct.

Excellence

A machine produces items with a 5% defect rate. In a sample of 20, find the probability that at least one is defective.

The binomial distribution

When to use the binomial distribution

The binomial formula

Step 1 — Identify nnn, ppp and kkk

Step 2 — Substitute into the formula

Step 3 — Evaluate

Test yourself

Practice by grade

The binomial distribution

When to use the binomial distribution

The binomial formula

Step 1 — Identify nnn, ppp and kkk

Step 2 — Substitute into the formula

Step 3 — Evaluate

Test yourself

Practice by grade

Step 1 — Identify $n$ , $p$ and $k$

Step 1 — Identify $n$ , $p$ and $k$