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Binomial distribution

Probability distributions

Japanese school year: Math B

What you learn

This topic covers the binomial distribution, modeling the number of successes in repeated independent trials with a constant success probability. It is widely used in quality control and opinion polling to evaluate proportions. Knowledge of repeated trials and combinations is essential before learning this.

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Key points

This formula calculates the probability of getting exactly kk successes in nn independent trials, each with success chance pp. It is used for events like coin flips or quality inspections.

P(X=k)=nCkpk(1−p)n−kP(X=k) = {}_n\mathrm{C}_k p^k (1-p)^{n-k}

This formula gives the expected number of successes E(X)E(X) in a binomial distribution. You can find it simply by multiplying the total number of trials nn by the single-trial success chance pp.

E(X)=npE(X) = np

This formula calculates the variance V(X)V(X) of the number of successes in a binomial distribution. It is found by multiplying the number of trials nn, the success chance pp, and the failure chance 1−p1-p.

V(X)=np(1−p)V(X) = np(1-p)

This formula gives the standard deviation σ(X)\sigma(X) of the number of successes in a binomial distribution. Taking the square root of the variance np(1−p)np(1-p) measures the typical deviation from the average.

σ(X)=np(1−p)\sigma(X) = \sqrt{np(1-p)}

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