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 http://www.twodimes.net/poker/ , a free site for calculating poker odds.

Here's an example based on a hand posted on my site, www.wptfan.com , 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 www.twodimes.com/poker. 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 FullTiltPoker.com.

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