The dead man's hand: aces and eights and other named hands
Aces and eights is plain two pair: 1,584 of 2,598,960 five-card hands, 1 in 1,641. Hickok's 1876 hand, its disputed fifth card, and other named hands.
Trips, boats and wheels: poker's hand nicknames explained
Trips is three of a kind (54,912 hands), a boat is a full house (3,744), a wheel is A-2-3-4-5 (1,020 plus 4 steel wheels). Each nickname, counted.
Pat hands: how often five cards need no exchange
A pat hand uses all five cards: straight or better. Exactly 19,716 of 2,598,960 deals qualify, 0.76%, or about one hand in 132.
Break-even odds: what a 3-to-1 payout needs you to win
A payout of N-to-1 breaks even when you win 1 time in N+1. At 3-to-1 that is 25%, which a flush draw (19.15%) misses and an improving pair (28.71%) clears.
Card removal: how the cards you can see change your odds
Every visible card leaves the unseen pile. A flush draw of 9 in 47 (19.15%) falls to 17.39% if another spade shows, and rises to 19.57% for any other card.
Five-card draw: where the oldest form of poker came from
Early poker used 20 cards and four players. What is documented about the road to five-card draw, and which dates stay uncertain.
Why no suit outranks another in poker
Poker ranks hands by card rank, never by suit. Each suit has 13 cards and 1,287 five-card combinations, so none is rarer and equal hands split.
Three of a kind: exchange one card or two?
Three of a kind drawing two cards makes a full house 66 in 1,081 and quads 46 in 1,081. Keeping a kicker and drawing one gives 3 in 47 and 1 in 47.
Two pair on the draw: how often it becomes a full house
Hold two pair and exchange the fifth card: 4 of 47 unseen cards make a full house, 8.51%, about 10.75 to 1 against. One pair drawing three manages 1.02%.
Drawing one card to a flush: nine outs from 47 cards
Hold four cards of one suit and replace the fifth: 9 of the 47 unseen cards finish the flush, a 19.15% chance, or about one attempt in five, and 80.85% of the time you miss.
Is the ace high or low? The ace in five-card poker
In five-card poker the ace is high, except in the wheel A-2-3-4-5, the lowest straight. Q-K-A-2-3 is no straight, and there are exactly 10 straight ranks.
Breaking ties in five-card poker: high cards and split pots
Hands in the same category are settled card by card, highest first, and suits never count. There are 7,462 distinct strengths; exact ties split the pot.
The odds of improving a pair when you exchange cards
Hold a pair and exchange the other three cards, and you improve 28.7% of the time: two pair 16.0%, trips 11.4%, a full house 1.0%, four of a kind 0.28%.
Why a flush beats a straight: the rankings follow the odds
In five-card poker there are 5,108 flushes and 10,200 straights, so the flush is about twice as rare. Each rung of the hand ranking is the next rarest.
How rare each five-card poker hand is
There are 2,598,960 five-card hands. Half of them are high card, 42% hold one pair, and a royal flush is 4 of them: one deal in 649,740.