Poker Probability - Poker Probabilities Poker is a game that involves cards. Of course, poker isn’t just about your own cards. It’s also about reading your opponents and anticipating their responses to all possible lines of play you can employ. But before we can get to those higher planes of poker thinking, we need to be rock solid with our fundamentals. (Like poker probability, for example, which is the subject of this article.)

What’s the probability that I’ll hit a flush with two cards to come?
Given that my opponent has AA, KK, QQ, JJ, or AK, what’s the probability that I’m beaten given that I have AQ on a Q93 flop?
If I’m down to 7 big blinds in a tournament, what’s the probability that the 4 players behind me will fold given that I have 98s and that they will call my all-in with 66+ or AK-AT?

Beneath everything, poker is a game of probabilities.

Poker Probability Defined

Simply put, probability is the math of random events. A random event is anything with more than one outcome where the outcome can’t be predicted with complete certainty.

Random Events and Possible Outcomes

Here are some examples of random events, along with their corresponding sets of outcomes:

Flipping a fair coin {1, 2}
Rolling a six-sided die {1, 2, 3, 4, 5, 6}
Being dealt a hand in hold’em {AA, AKs, AKo, AQs, AQo, AJs, AJo, ATs, ATo, A9s, A9o, A8s, A8o, A7s, A7o, A6s, A6o, A5s, A5o, A4s, A4o, A3s, A3o, A2s, A2o, KK, KQs, KQo, KJs, KJo, KTs, KTo, K9s, K9o, K8s, K8o, K7s, K7o, K6s, K6o, K5s, K5o, K4s, K4o, K3s, K3o, K2s, K2o, QQ, QJs, QJo, QTs, QTo, Q9s, Q9o, Q8s, Q8o, Q7s, Q7o, Q6s, Q6o, Q5s, Q5o, Q4s, Q4o, Q3s, Q3o, Q2s, Q2o, JJ, JTs, JTo, J9s, J9o, J8s, J8o, J7s, J7o, J6s, J6o, J5s, J5o, J4s, J4o, J3s, J3o, J2s, J2o, TT, T9s, T9o, T8s, T8o, T7s, T7o, T6s, T6o, T5s, T5o, T4s, T4o, T3s, T3o, T2s, T2o, 99, 98s, 98o, 97s, 97o, 96s, 96o, 95s, 95o, 94s, 94o, 93s, 93o, 92s, 92o, 88, 87s, 87o, 86s, 86o, 85s, 85o, 84s, 84o, 83s, 83o, 82s, 82o, 77, 76s, 76o, 75s, 75o, 74s, 74o, 73s, 73o, 72s, 72o, 66, 65s, 65o, 64s, 64o, 63s, 63o, 62s, 62o, 55, 54s, 54o, 53s, 53o, 52s, 52o, 44, 43s, 43o, 42s, 42o, 33, 32s, 32o, 22}

Given a random event, the probability that an outcome of interest will occur is given by the following expression:

P(Outcome of Interest) = Number of Ways an Outcome of Interest Can Occur/Total Number of Ways Possible Outcomes Can Occur

Example

Let’s consider some probabilities involving rolling a six-sided die. The probability of rolling a 2 is because there’s 1 outcome that results in a 2 and 6 total possible outcomes. The probability of rolling a number less than 5 is because there are 4 rolls less than 5 (1, 2, 3, and 5), and 6 total possible rolls.

Fancy Counting: Permutations and Combinations

Permutations

The dice examples I just gave are all straightforward. Trickier probability problems usually involve counting things that aren’t always easy to count. Before we start talking about probabilities having to do with cards, we need to talk about some advanced counting tools.

A permutation is a specific ordering of elements in a set. Let’s say a set has three elements: A♠, K♣, and Q♦. The permutations of this set are {A♠,K♣,Q♦}, {A♠,Q♦,K♣}, {K♣,A♠,Q♦}, {K♣,Q♦,A♠}, {Q♦,A♠,K♣}, and {Q♦,K♣,A♠}. To find the number of permutations that exist for objects in a set containing N unique objects, use the following formula (W(K) represents the number of ways that you can choose the Kth object):

# of Permutations = W(1)W(2)W(3)…W(N-2)W(N-1)W(N)

Poker Probability Example

Let’s follow this procedure for the set containing A♠, K♣, and Q♦. There are three ways to choose the first element (A♠, K♣, or Q♦). After the first element is chosen, only two elements remain, and after the second element is chosen, the last one is completely determined since we’re choosing from a set that only has three objects. Therefore, there are (3)(2)(1) = 6 permutations of the set {A♠,K♣,Q♦}.

How many permutations of hole cards exist in hold’em?

52 possibilities exist for the 1^st card, and 51 possibilities exist for the 2^nd card. Therefore, there are (52)(51) = 2,652 permutations of hole cards in hold’em.

The biggest thing to remember about permutations is that order matters. For example, J♦T♦ and T♦J♦ count as different permutations even though they are effectively the same hand. Permutations are important because you can use them to represent “the number of ways” when calculating poker probabilities.

Poker Probability Example

Since I just mentioned JTs, let’s calculate the probability of getting dealt JTs. There are 8 possibilities for the first card (4 jacks and 4 tens). Given that the first card is a jack or a ten, there’s only one possibility for the second card (if the first card is a jack, then it’s the ten of the same suit; and if the first card is a ten, then it’s the jack of the same suit). The number of JTs permutations is therefore (8)(1) = 8. To find P(JTs), we divide 8 (the number of JTs permutations), by 2,652 (the total number of permutations for two hole cards):

Combinations

Permutations can be useful for some poker calculations, even in cases where card order doesn’t matter. However, if you’re always dealing in permutations, then you’ll be dealing with numbers that are larger than you need to deal with.

And if you’re trying to do some quick calculations in your head at the tables, you prefer to keep the numbers as small as possible. In the interest of simplifying our mental math, it’s now time to introduce combinations. A combination is an unordered sampling of objects. When dealing with permutations, J♦T♦ and T♦J♦ are different. But when dealing with combinations, J♦T♦ and T♦J♦ are the same.

Avoid Double Counting

The key to finding the number of combinations is to avoid double counting. One foolproof way of calculating the number of combinations corresponding to a specific outcome is to take the total number of permutations and then to divide by the permutations of the number of elements comprising a permutation. That’s a bit of a mouthful, so let’s make that more transparent with an example.

How many combinations of hole cards are there?

We already know that 2,652 permutations exist. A set of hole cards contains 2 hole cards, meaning that there are (2)(1) = 2 permutations of hole cards. Therefore, there are combinations of hole cards.

Now, what about the number of AK combinations? We can say that there are 32 AK permutations and then divide by 2 to get 16. But we can also say that each ace can match with each king, meaning that there are (4)(4) = 16 combinations.

Using Combinatons to Calculate Poker Probabilities

We can use combinations to calculate probabilities much in the same way that we can use permutations. Suppose you’re down to six big blinds in a tournament. Action folds to you, and you have 23o in the small blind. The big blind has you covered, and you know that if you go all-in, the big blind will call you with [AA-55, AK-A9]. If you push all-in, what’s the probability that your opponent will fold?

You know your hole cards, so the big blind has two random cards from the remaining 50 cards (it doesn’t matter that other people were in the hand…as long as you don’t know what they had, your knowledge of the big blind’s hand remains unaffected). The number of 2-card combinations from a deck of 50 cards is . The number of combinations for a specific pocket pair, say AA, is provided that you don’t hold any cards from the pocket pair. You only hold a 2 and a 3, meaning that there are 6 combinations each of AA-55 in your opponent’s calling distribution and (4)(4) = 16 combinations each of AK-A9 in your opponent’s calling distribution. In total, your opponent’s calling distribution consists of (10)(6) + (5)(16) = 140 combinations. This means that your opponent’s folding distribution consists of 1,225 – 140 = 1,085 combinations. If you push all-in with your 23o against this opponent, the probability that you’ll win uncontested is .

Be Flexible and Independent

This article won’t make you an expert on poker probability. But it has more than enough basic knowledge to get you going. By granting yourself the power to apply mathematical analysis to the reads you put on your opponents, you’re putting yourself in the position to explore any line of play that you wish. And an intelligent, independent thinker at the poker tables is a very dangerous player.

Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus).