Poker hands and card puzzles
A five-card hand is one of C(52, 5) = 2,598,960 equally likely hands. Deal thousands of them and the frequencies line up with the counts — which is exactly why a full house beats a flush.
From the slides: Counting and Probability and what happens when you learn something.
Poker hands
Press Deal one hand.
Hands dealt
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Puzzles with aces
Shuffle a deck thoroughly and look at it five different ways. Each row is a puzzle from the slides.
Press Shuffle one deck to see a deck turned over up to the first ace and the card after it.
Decks shuffled
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What to notice
- The ranking of poker hands is a ranking by probability: the rarer hand wins. A straight flush turns up about once in 65,000 hands, so even 100,000 hands will show only one or two.
- The card after the first ace is equally likely to be the ace of spades or the two of clubs: 1 in 52 each. The competing arguments (“the ace of spades may already be used”, “the two of clubs may already have gone”) cancel exactly.
- The second card is an ace with probability 1/13, exactly like the first. An unseen card cannot change your information.
- Only about one deal in ten puts exactly one ace in each of the four 13-card piles.