Formulas & tables/Mathematics/Probability/Bernoulli scheme

Bernoulli scheme

Pk=(nk)pk(1−p)n−kP_k=\binom{n}{k}p^k(1-p)^{n-k}

Result unit: — ($\langle 0,1\rangle$)

What it describes

The probability of exactly k successes in n independent trials (Bernoulli scheme).

Symbols

nn
number of trials
kk
number of successes
pp
probability of success

Where it comes from

A model of n independent trials with two outcomes (success/failure). We compute the probability of exactly k successes, accounting for all their arrangements (binomial coefficient).

When to use it

Repeated independent trials (coin tosses, quality control).

Example

3 coin tosses. Find the probability of exactly 2 heads.
Substitutionn=3n=3, k=2k=2, p=0.5p=0.5.
Result(32)⋅0.52⋅0.5=38\binom{3}{2}\cdot0.5^2\cdot0.5=\tfrac38.

Related formulas

Combinations (binomial coefficient) Conditional probability

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