Draw At Least 1 Ace 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.

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

34.1158%

The probability of at least 1 success is 34.1158%.

1. Identify this as without-replacement sampling from a fixed population.
2. Use P(X in range) = sum of [C(K, k) C(N-K, n-k)] / C(N, n) terms.
3. Set N = 52, K = 4, n = 5, and target successes = 1.
4. Count the favorable samples and divide by all 5-draw samples from the population.
5. The final probability is 18,472/54,145, which is 34.1158%.