Your deck
Probability
A hypergeometric calculator answers the “what are the chances I draw it?” question for any deck. Five cards from a shuffled 52 and you want an ace: there are four in the pack, so the chance of at least one is 34.1%. Want two? It drops to 4.2%. The free calculator above does this for any deck, any number of hits and any hand size, and shows the full spread from zero hits upwards.
How the hypergeometric calculator works
It’s for draws without replacement. Once a card is out, it’s out, so every draw changes the odds of the next one. That’s why you can’t just multiply 4/52 five times.
P(exactly k) = C(K, k) × C(N − K, n − k) ÷ C(N, n)
N is the deck, K the cards that count as hits, n the cards you draw and k the hits you want. C(a, b) counts the ways to choose b things from a. “At least” adds up every k from your number to the top. The calculator works in logarithms, so a 1,000-card deck doesn’t break it.
Where it’s useful
- Card games. The chance of a pair in your first two cards, of a flush card turning up, of finding the one card your combo needs. There are presets for Magic: The Gathering and Yu-Gi-Oh! with the right hand sizes built in.
- Raffles and draws. Ten winning tickets among 500 and you hold 20: that’s N = 500, K = 10, n = 20.
- Quality checks. Sample 20 items from a box of 200 with 6 faulty ones and see how likely you are to catch at least one (47%).
The expected number of hits is simply n × K ÷ N. Handy as a sense check: if you expect 0.4 hits, don’t be surprised when you get none most of the time.
For lottery-style questions, the combination calculator counts the lines, and the probability calculator handles repeated independent events like dice and spins.
To shuffle and deal a real deck for practice, use the card shuffler.