written by sweetjazz3 DeucesCracked Contributor
published almost 3 years ago

Odds are a crucial part of poker. But they can often be confusing to beginning and intermediate players. The goal of this article is to discuss the different types of odds that are talked about in poker and clear up common misconceptions people have about odds. It is intended to be written at an elementary level. I hope to do a follow up article in which I discuss the concept of effective odds, which is more subtle and nuanced than what is covered here. The only prerequisite for this article is to know how to compute expected values.

The game of roulette offers a simple example of two different notions of odds that are important in gambling. In roulette, there are 38 numbers on the board (1 – 36, 0 and 00). Your odds of correctly guessing which number the wheel will land on are 1 in 38, or approximately 2.63%. We could also express this as saying you are 37 to 1 against to guess the right number, since there are 37 wrong guesses compared to 1 correct guess. The casino will pay out 35 to 1 on a bet, meaning for every $1 you bet, if you guess correctly, the casino will pay you $36 (for a profit of $35, since $1 is simply the money you bet returned to you). We say that the casino is laying you odds of 35 to 1. The odds the casino is laying you are the odds that would match the probability of guessing the right number if there were only 36 numbers. Since the actual roulette wheel has two more wrong numbers (which is what the 0 and 00 are there for), you are getting poor odds from the casino and your bet will lose you money over the long term. This can easily seen by calculating the expected value of the bet.

One simple use of the term ‘odds’ occurs when we want to figure out how likely we are to make a hand that we are drawing to. In hold em, if we hold A♥ 7♥ and our opponent holds K♥ K♣ and the board after the turn reads 8♥ 6♦ J♥ 4♠. We can ask the question: “What are the odds that we will win this hand?” This is simply another of phrasing the question, “what is the probability that the river card will make us the winning hand”? This is a straightforward calculation – there are 14 cards which will result in us winning the pot, namely the 8 remaining hearts, the 3 remaining aces, and the 3 remaining non-heart fives. And there are a total of 44 cards left in the deck, since we have accounted for 8 of them in the two starting hands and the four community cards. The odds that we will win the hand are 31.8%, the result of dividing 14 by 44 and expressing the quotient as a percent. In order to not confuse this use of odds with the other uses of the term below, I recommend referring to the 31.8% figure calculated here as the probability or likelihood of making your hand instead of the odds of making your hand. This is just a semantic issue, but I find it helpful in keeping the ideas straight in my head.

In the hand as above, let’s imagine that there is $100 in the pot after the flop. On the turn, our opponent bets all-in with his remaining $80 (which we cover). Should we call or fold? The decision relies on a calculation of our immediate odds. They are very easy to calculate, as they are the ratio between the amount we stand to win and the amount we must put at risk. In this case, we stand to win $180 (the $100 in the pot plus our opponent’s $80 bet), while we must risk $80 to call the bet. Thus our immediate odds are 180 to 80; we can also express this as a percent of the final pot that we are contributing to. We are risking $80 to make a final pot of $260, so that is 30.8%, again by dividing 80 by 260 and converting to a percent. In general, if we risk $X to win $Y, we say that we are getting immediate odds of $Y to $X, or expressed as a percentage, X / (X + Y). Recall that you can convert the percentage to a percent by multiplying by 100.

Our decision is now easy. We compare the probability of ending with the winning hand (31.8% as calculated above) with the proportion of the pot that our bet will consist of (30.8%). Whenever our probability of winning the hand is larger than the proportion of the pot that constitutes the additional money we are risking, we should call; when the probability is lower, we should fold. We could also express this less formally by saying that we should call if the odds of making our hand are better than the immediate odds we are being offered, and we should fold otherwise. Please note that this rule only applies if the betting results in a player being all-in; if there are future betting decisions to be made, then we have to consider them as well in our analysis, as we’ll see below.

As an exercise, you should check that if we change the problem so that there is still $100 in the pot after the flop but so that our opponent now bets all-in with his remaining $100 (which we still cover), then we should fold instead of call.

But often knowing our immediate odds is not good enough to analyze the situation completely. We also need to consider our implied odds. Let’s go back to the example hand again. We’ll suppose that the pot after the flop is still $100, but that both players have $200 in their remaining stacks. Our opponent with kings bets $100 on the turn. If that were to put him all-in, we saw in the exercise above that we should fold. But the fact that he has $100 remaining gives us implied odds and that actually turns our decision into a profitable call, under almost any reasonable assumptions on how our opponent plays on the river. Let’s assume that our opponent knows that you have a flush draw, but does not know the rank of your cards. His river strategy will be to fold if a heart comes, but to bet or call all-in on any other card. Of course, we plan to fold any river that does not give us one of our 14 outs that we enumerated earlier. We can do a calculation of our expected value in the hand. There are 30 cards on the river that result in a loss of the pot, so if we call we lose $100 in those cases. There are 8 hearts which will result in us winning the pot, which will generate a profit of $200 (the $100 in the pot on the flop plus the $100 turn bet of our opponent). There are also 6 cards (the aces and the non-heart fives) which will result in us winning the pot and making a profit of $300 (getting the additional $100 left in our opponent’s stack on the river). If we calculate our expected value, we find that we make $9.09 on average by calling, compared with making $0 on average (in fact, $0 every time) if we fold. Thus calling is better than folding, but only because of those times that we can get that extra $100 from our opponent’s stack on the river. As an exercise, you can check that without the extra money left in the stacks on the river (that is, if we were in the situation of the previous exercise), our expected value would be -$4.55, and folding would be better as we said earlier.

One way we could quantify what happened in this hand is as follows. We risked $100 on the turn, but the amount of money we stood to win was not just $200. Instead, those times we won the pot, we won $200 8 out of 14 times and we won $300 6 out of 14 times; in other words, we won $242.86 on average when we won the pot. While our immediate odds were 200 to 100 or 33.3%, our implied odds were 242.86 to 100 or 29.2%. Since the probability of making our hand was higher than our implied odds (expressed as a percent), we were able to make a profitable call.

Our poor opponent with the pocket kings is said to have had reverse implied odds. His bet on the turn was large enough that you did not have the correct immediate odds to call. But because he had more money in his stack and did not know our exact holding, he was unable to fold when we hit some of our outs on the river. In effect, he risked $100 on his turn bet those times when the river came so that either he won the hand or we made our flush, but he risked $200 those times when we made our pair of aces or our inside straight. As we saw in the calculation we just made, he was not really offering us the 2 to 1 immediate odds of his turn bet, but rather the 242.86 to 100 implied odds that result from the way we assume the river will play out on various cards.

Note that it is possible to have both implied odds and reverse implied odds in the same hand. Suppose our opponent had pockets jacks instead of pocket kings and plays the same river strategy except that he will of course bet or call all-in when the flush card pairs the board and gives him a full house. In that case, we have the implied odds of hitting our inside straight, but we have the reverse implied odds of hitting an ace or a board pairing heart (assuming that we play the river with the belief that our opponent has pocket kings, and therefore bet or call all-in when we hit those cards).

Here’s a brief recap of what we covered. The immediate odds are the ratio of the amount of money in the pot that can be won compared to the amount of money we must risk. They can be used to decide whether to call or fold versus an all-in bet by comparing them to the probability that we will win the hand. (Of course, in real games, we don’t know our opponent’s exact holding, but we can do a similar calculation based on the range of hands we believe he holds to estimate how likely we are to win the hand.) But if there is still more betting to come, we must also consider implied odds and reverse implied odds. In general, implied odds allow us to call in some situations when our immediate odds would suggest that we fold. The extra money that we can win if we hit our outs is what can make the call profitable. Reverse implied odds have the opposite effect; if we stand to lose more money on the river even if we hit our outs, then that should make us more inclined to fold. The most common situation in poker is for a pot to be contested between a made hand and a drawing hand; in that case, the drawing hand typically receives implied odds while the made hand typically receives reverse implied odds. But when considering ranges of hands for both players in a pot, often both implied odds and reverse implied odds need to be factored in, since you don’t know for sure whether the cards you think are outs for your hand actually make you a winning hand that can extract a bet from your opponent or a losing hand that will have to pay off a bet to your opponent.