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Drawing without replacement

When you draw several items without putting them back, the probability changes at each draw. The number of ways to choose k of n, without order, is the binomial coefficient. A hypergeometric probability is the number of favourable choices divided by the number of possible choices. The numerator is the product of the ways of choosing from each group. The order inside the draw does not matter.

\binom{n}{k} = \frac{n!}{k!(n-k)!}

combinations

P = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}

hypergeometric

Symbols

$N$	total number
$K$	number of favourable items
$n$	number drawn

Example

4 red and 6 blue, draw 2. $P(\text{two red}) = \dfrac{\binom{4}{2}}{\binom{10}{2}} = \dfrac{6}{45} \approx 0.1333$ .

Without replacement: combinations. With replacement and independence: a product.

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Part of Mathematics S1: Probability.