Binomial distribution

Binomial distribution gives the probability of exactly k successes in n independent trials each with success probability p. It applies when outcomes are binary, trials are fixed, and p is constant throughout.

ByHoang TruongUpdated June 30, 2026

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An agent closes 30% of calls with a sale. Across 10 calls, the probability of exactly 3 sales follows the binomial formula: C(10,3) × 0.30³ × 0.70⁷ ≈ 0.267, about 27%. The distribution applies whenever outcomes are binary, trials are independent, the number of trials is fixed, and the success probability stays constant.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Variables

$n$: Number of independent trials
$k$: Number of successes (0, 1, 2, …, n)
$p$: Probability of success on each trial
$C (n, k)$: Binomial coefficient: n! ÷ [k! × (n − k)!]

Conditions: binary outcomes, fixed n, constant p, independent trials.

LaTeX

\text{Mean} = n \times p \quad \text{Variance} = n \times p \times (1 - p)

Variables

$M e an$: Expected number of successes
$V a r ian ce$: Spread of the binomial distribution

Check yourself

PracticeCORE

A call-centre manager wants to model the number of calls out of the next 20 that result in a sale. She believes each call independently has a 30% chance of closing, regardless of what happened on previous calls. Under which conditions is the binomial distribution the correct model?

Select an answer to check your understanding.