By now, surely everyone knows that poker is a game of skill as well as a game of chance. And it’s hard to consider yourself a skilled poker player if you don’t understand how probability relates to the game and how you play it.

This post is meant to serve as an introduction to probability in general and to probability in poker. If you’re already an intermediate or advanced poker player, you’re probably familiar with most of these concepts. I remember how complicated I thought these concepts were when I started playing, so I tried to write the post I wish I’d been able to read.

Here’s how to understand probability in poker:

## 1- The 1st Step Is to Understand Probability in General

Probability is a specific branch of math, like algebra or geometry. It concerns itself with the study of how likely events are to happen. Probability is also the word used to describe how likely something is to happen or not happen. You can express an event’s probability in multiple ways, but understand that every event has a probability of between 0 and 1. 0 represents something that will never happen. The closer your probability gets to 0, the less likely it is to happen. 1 represents something that will always happen. The closer your probability gets to 1, the more likely it is to happen.

Most people understand probabilities at an almost intuitive level when they’re presented as percentages. This is how the weather is presented. When we’re told that there’s a 50% chance of rain, we know that it’s just as likely to be raining as not. If we say you have a 20% chance of winning a game, we know that means you’ll win 1 out of 5 times.

You could, though, represent a probability as a fraction. In gambling, that’s how it starts most of the time. The simplest probability formula is this: An event’s probability is the number of ways an event can happen by the total possible number of outcomes.

Here’s an example from poker:

You have a 52-card deck. You have 13 cards in each of 4 suits. If you draw a card at random and want to know the probability that it’s going to be a heart, the probability is 1/4. There are 4 suits, and there are an equal number of cards in each of those suits.

There are 4 cards of each rank, too, so you can also calculate the probability of drawing a card of a specific rank—an ace, for example. That probability is 1/13. Once you know the probability as a fraction, it’s easy to translate it to a percentage. You convert it to a decimal, and that’s done by dividing the numerator by the denominator. Then you move the decimal 2 places to the right.

For 1/13, we’re looking at 1 divided by 13, which is 0.076923077. Moving the decimal 2 places to the right gives me the following percentage: 7.6923077%. Most people round percentages off to the nearest whole number or to 1 or 2 digits after the decimal. Now we’re looking at 8% or 7.7%. An even more useful way of expressing a probability in poker is by expressing it in odds format. This is simply a comparison of the number of ways something can’t happen with the number of ways it can happen.

If the probability of something is 1/13, the odds are 12 to 1. There are 12 ways it can’t happen, and only 1 way it can happen. This becomes useful because you can compare the payout odds on a bet with the odds of winning that bet to estimate whether it’s a good or bad bet.

## 2- Probabilities for Multiple Events

When you’re calculating probabilities for multiple events, you either add those probabilities together or you multiply them by each other. The difference is based on the words “OR” and “AND.” If you want to know the probability that A “or” B will happen, you add the 2 probabilities together.

Here’s an example:

You want to know the probability that you’ll get an ace OR a king from a 52-card deck. That’s 1/13 + 1/13, or 2/13. If you want to know the probability that A “and” B will happen, you multiply the 2 probabilities together.

Here’s an example:

You want to calculate the probability that the 1st card you get will be an ace, and that the 2nd card you get will also be an ace.  That’s 1/13 X 3/51, or 3/661. Most people would reduce that and say it’s roughly 1/221. You’ll notice that the 1st probability used 13 as the denominator and the 2nd used 51 as the denominator. That’s because once you’ve dealt a card out of the deck, there are only 51 cards left.

Most poker probability calculations are going to involved multiple cards which have left the deck. The only cards you subtract from the total are the ones you know the rank and suit of. Just because another player has those cards in the hole doesn’t mean you subtract them from the number in the deck for purposes of calculating the probability.

Here’s an example:

You’re playing Texas holdem with 3 other players. You have 2 hole cards, both of which are of the same suit. The probability of getting another card of the same suit is 11/50. There are 13 cards in each suit, but 2 of them are in your hand. There are 50 cards that are unknown. Those other hearts might be in your opponents’ hands. That’s why they’re including in the 50 cards.

Let’s look at another example:

You’re playing Texas holdem, and you have 2 hearts in the hole. On the flop, 2 more hearts show up, along with a 3rd card of another rank. Now you have 4 hearts, so you just need 1 more to hit your flush. What’s the probability that the next card will make your hand? 13 – 4 = 9. That’s how many hearts are left in the deck. 52 – 5 = 47. That’s how many cards are left in the deck. The probability of getting a heart on the next card is 9/47. Most gamblers would simplify that to about 1/5.

## 3- How Pot Odds Relate to the Odds in Poker

Pot odds are a measure of how much you’ll get paid if you win a bet in poker. It’s the odds offered by the pot. Here’s a simple example:

If there’s \$10 in the pot, and another player bets \$10, you have to call \$10 to play. The pot odds in this situation are the \$10 in the pot plus the \$10 bet you’re calling, or \$20. If you win this bet, you’re getting pot odds of 2 to 1 on your money. If you estimate that your odds of winning this pot are 2 to 1 or less, then this is a profitable (or at least break-even bet). But if the odds of winning are higher than that, this is a bet with a negative expectation.

One of the key skills in poker is estimating your odds of winning and comparing it to the payout odds for the bet. If you can do this well, you can profit from poker consistently. It’s just a matter of putting money into the pot when you have a positive expectation and folding when you have a negative expectation.

## 4- Outs in Poker and How They Relate to Probabilities

One of the ways to estimate your odds of winning a hand is by calculating your “outs.” An out, in poker, is a card that will improve your hand and make it the winner. The tricky part is accounting for what cards your opponents might be holding.

Here’s an example:

You’re playing heads up with another player. You have a pair of jacks in the hole, but there’s a queen on the flop. You’re pretty sure your opponent either has a queen in the hole or a pair of aces or kings. You’re dominated. But if you catch another jack on the turn or the river, you’ve got him beat. Since you already have 2 of the jacks, there are only 2 jacks left in the deck. This means you have 2 outs.

A quick and easy way of approximating the odds of getting your outs is by multiplying them by 2% for both the turn and for the river.  In this case, you’re looking at 2 X 2 X 2, or 6%. That’s about the same as 16 to 1. If there’s \$40 in the pot, and it costs you \$10 to call a bet, you’re only getting 4 to 1 on your money. In the long run, you’re going to lose money in this situation.

On the other hand, if there’s \$400 in the pot, and it costs you \$20 to call a bet, you’re getting 20 to 1 on your money. You’ll profit in the long run from this bet. In some cases, you’ll also deal with implied odds. This takes into account future bets into the pot.

In other cases, you’ll also deal with partial outs. These are outs that might make you a winner of the hand some of the time. Or sometimes you’ll have an out that will improve your hand, but it might also improve your opponent’s hand enough so that he wins.

## 5- Probability and Bluffing

Another way probability can be useful in poker is when deciding whether to bluff. It’s especially useful when determining whether a bluff against multiple players is worthwhile. It’s also illustrative of the fact that you don’t need to win with a tactic most of the time to be profitable. Let’s say you’re heads-up with another player, and there’s \$20 in the pot already. You don’t have anything, but you suspect maybe he doesn’t either. You think there’s a 60% chance he’ll fold if you bet \$20 here.

If you’re right in your estimate, then you’ll win \$20 60% of the time, which is worth \$12 in expected value. You’ll also lose \$20 (or at least have to continue to play with it), 40% of the time. That’s worth \$8 in negative expected value. Since \$12 is more than -\$8, this is a profitable bluff.

If you reversed it, though, and estimated that the other player would only fold to a bluff 40% of the time, you’re making a negative expectation play. Bluffing multiple opponents makes it even harder to succeed. If you’re in a pot with 3 other players, and you want to raise, hoping they’ll fold, your probabilities need to be multiplied. Let’s say there’s a 50% chance that each of them will fold.

You want to calculate the probability that player 1 AND player 2 AND player 3 fold. To calculate that, you multiply: (50% X 50% X 50% = 12.5% ). You need roughly 8 to 1 pot odds to make that a break-even bluff. There’d have to be a lot of dead money in that pot.

If you semi-bluff, which is when you bet with a hand that might improve, you get to add the probability of making your hand to those odds. If you have a 12.5% chance of winning when the other players fold, AND a 12.5% chance of hitting a hand that will win, you have a 25% chance of winning the pot. You only need 4 to 1 from the pot to make this semi-bluff worthwhile. Of course, if you’re lousy at estimating these probabilities, you’ll probably not profit at all.  That’s one of the things that makes poker such a great game.

## 6- Probability in Video Poker

I hesitate to even mention video poker in this post, as it’s not really a poker game. But it does use the poker hand rankings and the probabilities of a 52-card deck. If you understand probability in poker well enough, it can inform your video poker strategy very well.

Video poker is like a slot machine game with payouts based on combinations that come from poker hand rankings instead of just arbitrary symbols. When you look at probabilities related to video poker, you’re most interested in the game’s payback percentage. This is a function of the probability of getting each hand that pays out multiplied by the probability of winding up with that hand.

For example:

In a Jacks or Better game, you have a probability of about 21.5% of winning 1 unit on a pair of jacks or higher. That adds 21.5% to the payback percentage for the game. You also have a roughly 12.9% of getting 2 pair. That pays off at 2 for 1, so that adds (12.9% X 2) or 25.8% to the payback percentage for the game. As the hands get harder to hit, the payoffs for those hands go up. When you add them all together, you have the payback percentage for the game.

Most video poker games have better payback percentages than slot machines, but only if you know how to make close to the right decisions. Understanding the odds of catching various cards versus keeping various other cards, and taking into account the payouts for those hands, is crucial to getting the best possible payback percentage.

Conclusion

This post was meant to serve as an introduction to probability in poker. You can and should delve deeper into the subject. Any good book on poker will cover pot odds versus payoff odds. It will also cover calculating outs.

The great thing about poker, of course, is that it’s a game of incomplete information, so success is tied to learning how to account for the potential range of hands your opponent might have, too.

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