Binomial Distribution

The probability distribution of the number of successes in a fixed number of independent yes/no trials.

Binomial distribution — B(n=10, p=0.50) | mean=5.00, σ=1.58

n (trials) = 10p (prob.) = 0.50

Definition

The binomial distribution counts the number of successes in a fixed number of independent yes/no trials, where each trial has the same probability of success.

Three ingredients:

$n$ — the number of trials
$p$ — the probability of success on each trial
$k$ — the number of successes we're counting

We write $X \sim \text{Binomial}(n, p)$ .

Flipping coins

Flip a fair coin 5 times. How likely is exactly 3 heads?

Each flip is independent, $p = 0.5$ , $n = 5$ , $k = 3$ .

There are several ways to get 3 heads: HHHTT, HTHHT, THHHT, … — in fact, $\binom{5}{3} = 10$ different arrangements.

Each arrangement has probability $(0.5)^3 \times (0.5)^2 = 0.03125$ .

So $P(X = 3) = 10 \times 0.03125 = 0.3125$ , about 31%.

Try it

A basketball player makes 70% of free throws. They take 4 free throws. What is the probability of making exactly 2?

Solution

$n = 4$ , $p = 0.7$ , $k = 2$ .

$\binom{4}{2} = 6$ arrangements. Each has probability $(0.7)^2 \times (0.3)^2 = 0.49 \times 0.09 = 0.0441$ .

$P(X = 2) = 6 \times 0.0441 = 0.2646$ , about 26%.

Related concepts

Statistics· Probability

ProbabilityA number between 0 and 1 measuring how likely an event is to occur.

Statistics· Distributions

Normal DistributionThe symmetric bell-shaped distribution that arises naturally from sums of many small random effects.

Statistics· Probability

Conditional ProbabilityThe probability of an event given that another event has already occurred.