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The binomial distribution

The binomial distribution counts the number of successes in a fixed number of independent trials, with the same probability each time. The expectation is the number of trials times the probability. The standard deviation is the square root of the expectation times the probability of failure. The probability of exactly k successes uses the binomial coefficient, because the order of the successes does not matter. Without a fixed number of trials the model is not binomial.

Try it yourself

E(X) = np

expectation

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

standard deviation

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

point probability

Symbols

$n$	number of trials
$p$	probability of success
$k$	number of successes

Example

$n = 10$ and $p = 0.2$ . $E(X) = 2$ and $\sigma = \sqrt{1.6} \approx 1.265$ .

$P(X = 2) = \binom{10}{2} \cdot 0.2^2 \cdot 0.8^8 \approx 0.302$ .

The same p, independent trials, fixed n, and we count successes.

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Part of Mathematics S2: Probability distributions.