No-limit by the Numbers


I get asked a lot of poker strategy questions, from beginner to advanced. Whenever I get a question about a specific hand, I always use one of a number of free tools to calculate the probability of winning the hand. I've written some of my own software, but my favorite tool is at , a free site for calculating poker odds.


Here's an example based on a hand posted on my site, , by a player with the nickname "Soulcrates." Soulcrates was playing at a small stakes no-limit table online, with the blinds 25 cents and 50 cents. At the start of the hand, Soulcrates had $44. He was dealt A? 10? and raised to $2. Both blinds called. The flop was KdJd2c, giving Soulcrates a royal flush draw. The big blind bet $2, Soulcrates raised $2 more, the next player called, and the big blind (with more chips than Soulcrates) reraised all-in.


Soulcrates asked if he should have called with his last $38. Assuming the third player was folding, if Soulcrates were to call and win, he'd be up to $94 (the $18 in the pot, plus his $38 and his opponent's $38). If Soulcrates won the hand 4 times out of 10, on the average he'd have $37.60 after the hand. In poker, it's the long run that matters, so if Soulcrate's probability of winning was 40% or less, he should fold. So, Soulcrates needed to figure out the probability he'd win the hand.


The first step is to "put" his opponent on a range of hands. Sometimes you can figure out exactly what your opponent must have by the betting or tells.
Most of the time, you're left to guess a little. In this situation, the other player probably has a very strong hand, but there's a chance he's bluffing or semi-bluffing (bluffing, but with a good possibility of winning even if called by a better hand).


The strongest hand Soulcrates could be facing is three kings. Soulcrates has 11 different cards that he will win with -- 8 diamonds (the 2d would give his opponent a full house) and 3 other queens. But if Soulcrates makes his flush or straight, his opponent could still win by making a full house or four of a kind on the last card. I could calculate the probability by hand, but I don't need to.


Instead, head to Enter "K? J? 2?" in the box labeled "Board" and "A? 10?" and "K? K?" under "Hands” and click submit. The result says that A? 10? wins under 34% of the time, less than the 40+% that would make a call the right play. If Soulcrates knew that his opponent had three kings, he should fold. The probabilities for the other possible three of a kinds are the same.


But what if Soulcrates was up against two pair, kings and jacks? Using the poker calculator again, his probability of winning would be 44%. That's enough to make calling correct. Soulcrates could also be against other two pairs, which he'd beat a little less often (42%), or AK (46%). Soulcrates might even already be ahead if he's against an aggressive player, who might be semi-bluffing with something like QT (81%) or Qd9d (82%).


Having calculated the probabilities of winning, we're now left with the subjective part of the answer: guessing the probabilities of what the other player has. I would guess that it's more than twice as likely that the player has two pairs, or AK, or some weaker hand, than it is that he has three of a kind. And I would guess that maybe 5% to 10% of the time, A? 10? is actually ahead. I told Soulcrates that I would have called.


Souldcrates did call, and the other player had KJ, meaning Soulcrates should win 44% of the time. The turn card was the 2?, giving Soulcrates the flush, but the river was a jack, giving the other player a full house. The river card was a tough break, but playing by the numbers, Soulcrates still made the right play.


Andy Bloch
Andy Bloch is a two-time World Poker Tour finalist and game theory expert. He can always be found playing Pot-limit-Omaha on


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