Bernoulli and Binomial Distributions

Some processes have only two outcomes, such as a user clicking an ad or not. These are modelled with the Bernoulli and binomial distributions.

Bernoulli Distribution

A Bernoulli trial is a single attempt with two outcomes, success (1) or failure (0). If the probability of success is p, then the mean is p, and the variance is p(1 − p).

Binomial Distribution

A binomial random variable counts the successes in n repeated Bernoulli trials.

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

• n: the number of trials.
• k: the number of successes you are asking about.
• p: the probability of success on each trial.
• C(n, k): the number of different ways to place k successes among n trials, calculated as n! / (k!(n − k)!).
• pᵏ × (1 − p)ⁿ⁻ᵏ: the probability of any one specific arrangement of k successes and n − k failures.

The mean is np, and the variance is np(1 − p). The formula applies only when the number of trials is fixed, the trials are independent, and p stays the same on every trial.

Example: A marketing email is sent to 5 people, and each clicks with probability 0.2. Find the probability that exactly 2 click.

1. C(5, 2) = 5! / (2! × 3!) = 10
2. 0.2² = 0.04 and 0.8³ = 0.512
3. P(X = 2) = 10 × 0.04 × 0.512 = 0.2048

The expected number of clicks is 5 × 0.2 = 1, with a variance of 5 × 0.2 × 0.8 = 0.8.