Poker Odds

A flush draw has nine outs, and by the river those nine outs produce 34.97%, not 19.15% counted twice. The two numbers belong to different questions, and the pot-odds formula is where a single figure has to be chosen between them. Where the nine outs come from Thirteen cards of one suit exist. Two are in the hand and two are on the board, which leaves nine unseen cards of that suit. That is the whole source of the count: it is fixed by the suit’s size and the four cards already known, so a flush draw on the flop always starts at nine outs. ...

A gutshot straight draw has four outs; an open-ended straight draw has eight. By the river those are 16.47% and 31.45%, roughly double, and both sit in the same poker odds chart rather than in separate tables. Why one rank produces four cards A gutshot is a draw completed by a single rank. Four cards of that rank are still unseen, so the out count is four. The shape matters because the count is fixed by the number of completing ranks, not by which cards a player holds or by where the gap sits inside the sequence. One rank always means the same four cards. ...

A pocket pair flops a set or better 11.76% of the time, or 1 in 8.5. That figure comes from counting flops, not from counting individual cards, and the hand rankings chart is where the resulting hand sits in the order. The two cards that matter After a pocket pair is dealt, 50 cards remain unseen. Two of them match the rank in hand. Those two are the only cards that can produce a set on the flop, so the entire question is whether at least one of them appears among the three flop cards. ...

The Rule of 4 and 2 multiplies the out count by four on the flop and by two on the turn. It is a linear approximation, and it always lands high: 9 outs gives 36% against an exact 34.97%, while 15 outs gives 60% against an exact 54.12%. The poker odds chart is where the exact numbers live. What the shortcut is actually computing The rule treats each out as one independent chance on each card still to be dealt. Two cards remain on the flop, so every out is counted across two streets; one card remains on the turn, so it is counted once. That is where the four and the two come from. It is a count of shares, not a probability calculation, and shares add cleanly while probabilities do not. ...

Three of a kind outranks two pair because it is rarer: 54,912 five-card combinations against 123,552. The hand rankings chart places them at ranks 7 and 8, and the probability column is the reason for that order. Rarity sets the order Poker hand rankings are an ordering by how often each hand occurs among all five-card combinations. A hand that appears less often beats a hand that appears more often. The table lists 2.112845% for three of a kind and 4.753902% for two pair, so the rarer hand is the one with the smaller share of all five-card hands. ...

A poker odds chart maps unseen cards (outs) to the mathematical probability of completing a draw on the turn or river. In Texas Hold’em, after the flop is dealt, a player sees 5 cards (2 hole cards in hand plus 3 flop cards on the board). From the player’s information set, 47 cards are unknown after the flop, assuming no additional cards have been exposed; some of those unknown cards sit in opponents’ hands rather than in the deck. The probability of hitting a specific out on the turn is calculated as outs divided by 47, while the cumulative probability of hitting an out by the river is calculated using the formula 1 - (1 - outs / 47) x (1 - outs / 46). ...