Binomial distribution

Definition

A binomial distribution models the probability of obtaining a given number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes (commonly called “success” and “failure”) and the probability of success p is constant across trials.

When to use it

Use the binomial distribution when:
* You have a fixed number of trials n.
* Each trial has only two possible outcomes.
* Trials are independent.
* The probability of success p is the same for every trial.

Examples include coin flips, pass/fail test outcomes, quality-control defect checks, clinical trial results, and binary survey responses.

Probability formula

The probability of observing exactly x successes in n trials is:

P(X = x) = (n choose x) · p^x · (1 − p)^(n − x)

where
* (n choose x) = n! / (x! (n − x)!)
* p is the probability of success on a single trial
* x is the number of successes (0 ≤ x ≤ n)

Mean, variance, and shape

• Mean (expected value): E[X] = n p
• Variance: Var(X) = n p (1 − p)
• Standard deviation: σ = sqrt(n p (1 − p))

Shape properties:
* Symmetric when p = 0.5 (e.g., fair coin flips).
* Skewed left (longer tail on the left) when p > 0.5.
* Skewed right (longer tail on the right) when p < 0.5.

Relation to Bernoulli trials

A Bernoulli trial is a single binary trial with success probability p. The binomial distribution is the sum of n independent, identically distributed Bernoulli trials. The Bernoulli distribution is the special case of the binomial when n = 1.

How to compute (step-by-step)

1. Identify n (number of trials), x (number of successes), and p (success probability per trial).
2. Compute the combination (n choose x).
3. Compute p^x and (1 − p)^(n − x).
4. Multiply: (n choose x) × p^x × (1 − p)^(n − x).

Example

What is the probability of exactly 6 heads in 20 fair coin flips?
* n = 20, x = 6, p = 0.5
20 choose 6 = 38,760
P(X = 6) = 38,760 × (0.5^6) × (0.5^14) = 38,760 × 0.5^20 ≈ 0.0369 (3.7%)

The expected number of heads is E[X] = n p = 20 × 0.5 = 10, so 6 heads lies on the left tail of the distribution.

Applications

• Social sciences: modeling dichotomous outcomes (e.g., yes/no, party votes).
• Quality control: defects per batch.
• Clinical trials: number of responders to treatment.
• Finance and insurance: estimating counts of defaults, claim occurrences, or other binary events for pricing and reserve planning.
• Surveys and polling: probability distributions of categorical responses.

Key takeaways

• The binomial distribution describes the probability of a given number of successes in a fixed number of independent binary trials with constant success probability.
• Use P(X = x) = (n choose x) p^x (1 − p)^(n − x).
• Mean = n p and variance = n p (1 − p); shape depends on p (symmetric at p = 0.5).