Odds & Poker (Pot, Card, Implied, Reverse etc.)
Odds, as applied to poker, are really fairly simple but somehow the poker literature has gotten itself turned around and explained them in a very awkward way. The formula for odds in poker is exactly the same as it is in any other gambling activity. You need to know (or estimate as bet as possible) how much you’re risking, how much you stand to win, how many times you’ll lose, and how many times you win. You then construct the following two ratios:
($ won):($ lost)
(number of times you’ll lose):(number of times you’ll win)
and compare. If the first ratio is bigger than the second, your expectation is positive and you should play. If the first ratio is smaller, you expect to lose money and should pass. If there are multiple valid plays you could make, you need to do the expectation calculation for each one to see which is best. This comparison ensures you don’t forget about opportunity cost.
A real-world example with all the complications is in order:
You’re playing 20/40 limit holdem and are dealt Ah7h on the button. Preflop, action is folded to you, and you raise on a semi-steal. A tight player in the small blind re-raises, the big blind folds, and you call. The pot is now $140 and action is heads-up. The flop comes Qh4d5h. The small blind checks, you bet on the semi-bluff, and the small blind raises. You call (note that raising here is not a bad choice either). Pot is now $220. Turn is 2c. Villain bets $40. Do you call?
Ok, first some observations here. Villain almost certainly has a pair - AA, KK, QQ or possibly slightly smaller. AQ is possible too, but would be unexpected. So clearly we’re behind here. But we also have outs - any heart, any 3, or any A might well win it for us.
Now, let’s work through the odds math piece by piece:
How much will we win? Well, if we win the pot we’re guaranteed the $260 in the pot. But it’s highly unlikely villain will fold the river, and we’ll have a decent chance of even getting in a raise there if we make our draw. So we can expect to win maybe another $60 on average on the river if we’re good for a total of $320. Notice that I’m counting both player’s bets that are already in the pot as part of the win because they’re sunk costs, but I’m only counting villain’s future bets because my future bets are still optional.
How much will we lose if we lose? Remember, money already bet earlier in the hand doesn’t count here - it’s gone. If you’re going to see it again, you have to win it back. So our cost starts with the $40 we’re going to risk by calling. If we miss all our draws, we won’t pay any more. But if we hit the A, and villain has AA or QQ or AQ, we’re going to lose one more bet on the river (assuming we wisely don’t raise if we make a pair of aces). AA, QQ, AQ makes up more than half of villain’s range, but we’ll only hit an A less than 1 time in 10, so this will cost us on average only a couple bucks. Call it $3 for a total risk of $43.
How often will we win vs. lose? There are 46 unseen cards in the deck. Eight hearts make Ah7h the nuts and are sure wins. The 4h is good too unless villain has QQ, so it’s probably 3/4 of an out. The three non-heart treys are at worst a split, but that happens about half the time when villain holds an ace - call them 2 and change outs (more like 9/4 if you wand to get picky). The three aces will win against less than half of villain’s likely holdings - call them one effective out. This means we have effectively 12 outs. The other 34 cards are losers.
Do we call? The odds of winning are 34:12 against. The money is $320:$43. Clearly the money is the bigger ratio (more than 7:1 vs less than 3:1) so we have an easy call.
Opportunity Cost Check: We do have another option - raise. If we think there’s a chance villain will fold, we should probably pump it here. But it’s unlikely villain is folding the strong holdings we put him on and if you run the math with that assumption, then calling gives us the higher expectation (trust me - I’m not going to work it out here). So we call.
Now, I hope that my reasoning there seems simple and direct. There’s actually a lot of complexity buried in this example, and I want to point some things out.
- All the math was done taking future events into consideration when figuring the win and risk. The fancy name for this is “implied odds” which you’ll see mentioned a lot in poker forums and poker books. But really it should just be called common sense because if you don’t take future events into consideration, then obviously your results are going to be skewed.
- In situations where there will be no meaningful future action (you’ll all in, or last to act on the river, or whatever) then you don’t have to worry about said future action (duh). The fancy poker book term for this is “pot odds” because the only money you stand to win is what’s in the pot. The term pot odds is used in a different way by some poker authors, as I’ll explain a future post. But most of the time, it’s used as I’ve described here.
- Our implied odds thinking took into account the fact that if we make a hand less than the nuts and lose, it will cost us money in the future. Poker books call this “reverse implied odds” but really it’s just another case of common sense.
- There was a much easier way to do the math in this example which I avoided because I wanted to make some points explicit. I’ll point out how to do it fast in a future article.
- When considering outs not to the nuts, I “discounted” them depending on how likely I thought they were to be losers. This process requires some skill and discretion, and I’ll go into more detail down the road.
- Some poker books refer to the ratio of wins to losses as “card odds”
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