Mathematics.

probability distributions

Binomial Distribution

Statistics25 minDifficulty3 out of 10

You should know: discrete probability distributions, expectation

Overview

The binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability p of success (a Bernoulli trial). It is characterized by two parameters: n (number of trials) and p (probability of success on each trial). Examples include the number of heads in 10 coin flips, or the number of defective items in a batch of 50 when each item independently has a 2% chance of being defective. The distribution's mean is np and its variance is np(1-p), reflecting how spread out the possible success counts are.

Intuition

The binomial coefficient C(n,k) counts how many different orderings of k successes and n−k failures are possible (e.g., HHT, HTH, THH all give 2 heads in 3 flips), and each such ordering has the same probability p^k(1-p)^(n-k). Multiplying the count of arrangements by the probability of any one arrangement gives the total probability of exactly k successes — this is why the formula combines a combinatorial term with a probability term.

Formal Definition

Definition

If X counts the number of successes in n independent Bernoulli(p) trials, X ~ Binomial(n, p), with probability mass function:

P(X=k)=(nk)pk(1−p)n−k,k=0,1,…,nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad k = 0, 1, \ldots, n
Binomial PMF
E[X]=npE[X] = np
Mean
Var⁡(X)=np(1−p)\operatorname{Var}(X) = np(1-p)
Variance

Worked Examples

  1. Here n = 4, p = 0.5, k = 2. Compute the binomial coefficient.

    (42)=4!2!2!=6\binom{4}{2} = \frac{4!}{2!2!} = 6

  2. Apply the PMF formula.

    P(X=2)=6⋅(0.5)2(0.5)2=6⋅0.0625=0.375P(X=2) = 6 \cdot (0.5)^2 (0.5)^2 = 6 \cdot 0.0625 = 0.375

Answer: P(X = 2) = 0.375.

Practice Problems

Difficulty 3/10

A fair coin is flipped 5 times. Find P(exactly 3 heads).

Difficulty 3/10

For X ~ Binomial(n=20, p=0.3), find the mean and standard deviation.

Difficulty 5/10

A multiple-choice quiz has 4 questions, each with 4 options (only 1 correct). If a student guesses randomly on every question, find the probability of getting exactly 1 question correct.

Quiz

Which of the following is required for a random variable to follow a binomial distribution?
For X ~ Binomial(n, p), the mean is:
The binomial coefficient C(n,k) in the PMF counts:

Summary

  • The binomial distribution models the count of successes in n independent trials with constant success probability p.
  • Its PMF is P(X=k) = C(n,k) p^k (1-p)^(n-k), with mean np and variance np(1-p).
  • It generalizes the single Bernoulli trial to repeated, independent trials — the foundation for many discrete probability models.

References

  1. WebsiteWikipedia — Binomial distribution