Draw Exactly 2 Aces in 5 Cards

Model guide

Why this probability model fits

This is a hypergeometric probability problem because the cards are drawn without replacement from a fixed deck, so each draw changes the composition of the remaining deck. If you draw an ace first there would be one less to draw the second time. On every sequential draw there is one less card to draw from, changing the odds.

Setup: The deck is a fixed population of 52 cards, the 4 aces are the success states, and the 5 drawn cards form the sample.

Replacement: Without replacement is the key phrase. Once one card is drawn, it cannot appear again and the next draw comes from a smaller remaining deck.

Independence: The draws are not independent because each card drawn changes what remains in the deck.

Worked solution

3.993%

The probability of exactly 2 successes is 3.993%.

1. Identify this as without-replacement sampling from a fixed population.
2. Use P(X = k) = [C(K, k) C(N-K, n-k)] / C(N, n).
3. Set N = 52, K = 4, n = 5, and target successes = 2.
4. Count the favorable samples and divide by all 5-draw samples from the population.
5. The final probability is 2,162/54,145, which is 3.993%.