Card Probability Calculator

In short Drawing five cards from a standard deck, the chance of getting at least one ace is 34.12%, and the chance of exactly one is 29.95%. This is the hypergeometric distribution: sampling WITHOUT replacement, where every card you take changes what is left. That dependence is the whole point. If you put each card back before drawing again, the draws become independent and the answer changes — exactly one ace becomes 27.92% rather than 29.95%. The page shows both columns side by side, because the difference between them is the difference between two genuinely different questions, and using the binomial for a without-replacement problem is one of the most common modelling mistakes there is. One thing that does NOT change is the average. The expected number of aces in five cards is 0.3846 either way; removing cards alters the spread of the outcome, not its mean, which surprises people who expect dependence to shift everything. The same engine runs the lottery page, because "how many of my six numbers are among the six drawn" is the same question as "how many of my five cards are aces". Set the deck size and the number of matching cards for any drawing problem at all — raffle tickets, quality sampling from a batch, or a card game with a non-standard deck.