No-limit by the Numbers by Andy Bloch

I get asked a lot of poker strategy questions, from beginner to advanced. Some are easy, but some involve the kind of math I can’t always do off the top of my head. When that happens, I rely on one of a number of free tools to calculate the probability of winning the hand.

Here’s an example based on a hand posted on a website I run:

Our hero was playing at a small stakes No-Limit table online, with $.25-$.50 blinds. At the start of the hand, he had $44. He was dealt Ad-Td and raised to $2. Both blinds called. The flop was Kd-Jd-2c, giving our hero a royal flush draw. The big blind bet $2, hero raised $2 more, the next player called, and the big blind (with more chips than our hero) re-raised all-in.

Should our hero call with his last $38? Let’s assume the third player will fold. If our hero 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 he wins the hand four times out of 10, on the average he’d have $37.60 after the hand ($94 multiplied by four, and divided by 10). In poker, it’s the long run that matters, so he should only call if his probability of winning is greater than 40%. Now he needs 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 even semi-bluffing.

The strongest hand our hero could be facing is three kings. He has 11 outs to win the pot – every diamond but the 2d, and three queens. But even if our hero makes his flush or straight, his opponent could still win by making a full house or quads on the last card. I could calculate the probability by hand, but I don’t need to.

Instead, I head to the Internet and one of the many free poker odds calculators, such as the one at Enter "Kd Jd 2c" in the box labeled "Board" and "Ad Td" and "Ks Kc" under "Hands", and click submit. The result says that Ad-Td wins under 34% of the time – less than the 40+% that would make a call the right play. If our hero knows 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 he’s 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. Our hero might also be against other two pairs, which he’d beat a little less often (42%), or A-K (46%). He might even already be ahead if he’s against an aggressive player who would semi-bluff with something like Q-T (81%) or Qd-9d (82%).

Having calculated the probabilities of winning, our hero is 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 pair, or A-K, or even some weaker hand than that he has three of a kind. And I would guess that maybe 5% to 10% of the time, Ad-Td is actually ahead. I told our hero that, based on the numbers, I would have called.

Our hero did call, and the other player had K-J, giving our hero a 44% chance of winning the hand. The turn card was the 2d, but the river was a jack and our hero’s flush lost to a full house. The river card was a tough break, but playing by the numbers, he still made the right play.

It’s good to know the numbers, but it’s equally important to know how to get them. And if you use the available tools whenever you aren’t sure, you’ll start to remember them when they come up at the table. In poker, every tool in your toolbox brings you one step closer to mastery of the game.

Andy Bloch