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Binomial Probability Formula

Binomial Probability Formula

A probability formula for Bernoulli trials. The probability of achieving exactly k successes in n trials is shown below.

Formula:

P(k successes in n trials) = (n choose k) times p^k times q^(n−k)

n = number of trials
k = number of successes
n – k = number of failures
p = probability of success in one trial
q = 1 – p = probability of failure in one trial
Example:

You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting exactly 7 questions correct?

n = 10
k = 7
n – k = 3
p = 0.25 = probability of guessing the correct answer on a question
q = 0.75 = probability of guessing the wrong answer on a question

P(7 correct out of 10) = C(10,7) × (0.25)^7 × (0.75)^3 ≈ 0.0031

See also

Binomial coefficients