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SigmaCalc
The distribution behind card hands, lottery draws, and quality-control sampling from a fixed batch.
Binomial probability assumes each draw doesn't affect the odds of the next (like flipping a coin). Hypergeometric probability is for sampling without replacement — like dealing cards — where each draw changes the remaining pool.
The probability of drawing exactly 2 aces in a 5-card poker hand is a hypergeometric problem: population size 52, successes in population 4 (the aces), sample size 5, and you're solving for k = 2.
P(X = k)
0.274
P(X=k) = C(K,k)×C(N−K,n−k) ÷ C(N,n)
What you entered
Ways to choose the successes from the population
C(13, 2)= 78Ways to choose the failures from the rest of the population
C(39, 3)= 9139Total ways to draw the sample
C(52, 5)= 2598960P(X = k)
(78 × 9139) ÷ 2598960= 0.2743Result
P(X = k): 0.274
Drawing 5 items without replacement from a population of 52 (13 successes), there's a 27.43% chance of getting exactly 2 successes — this is the distribution behind card-hand and lottery-style problems.