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 βsuccessβ and βfailureβ) 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. Explore More Resources
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: Explore More Resources
P(X = x) = (n choose x) Β· p^x Β· (1 β p)^(n β x) where
(n choose x) = n! / (x! (n β x)!)
p is the probability of success on a single trial
* x is the number of successes (0 β€ x β€ n) Explore More Resources
Mean, variance, and shape
* Mean (expected value): E[X] = n p
* Variance: Var(X) = n p (1 β p)
* Standard deviation: Ο = sqrt(n p (1 β 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. Explore More Resources
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 β p)^(n β x).
4. Multiply: (n choose x) Γ p^x Γ (1 β p)^(n β 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 Γ (0.5^6) Γ (0.5^14) = 38,760 Γ 0.5^20 β 0.0369 (3.7%) The expected number of heads is E[X] = n p = 20 Γ 0.5 = 10, so 6 heads lies on the left tail of the distribution. Explore More Resources
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 β p)^(n β x).
* Mean = n p and variance = n p (1 β p); shape depends on p (symmetric at p = 0.5).