Advanced Probability for Poker Players
By Tony Guerrera
The first article in this two-part series featured on Poker Helper gave you the basics that will help you make informed decisions at the tables. This article will go into more advanced material that you’ll be able to use when doing analysis away from the table. Doing analysis away from the table is just as important as putting in playing time because when you’re away from the table, you can think long and carefully about situations that you might not have been able to think clearly about in the heat of battle. Having done quality analysis away from the table, you can then return to the tables as a more knowledgeable and profitable player.
Mutual Exclusivity and Complements
Suppose that we hold a set of unpaired hole cards: AK for example. What’s the probability of flopping at least one ace or king? One way of expressing this probability is as follows:
Note that a “GC” refers to a Good Card, meaning an ace or a king. The reason we can add the probabilities P(1 GC), P(2 GCs), and P(3 GCs) is that the events “1 Good Card,” “2 Good Cards,” and “3 Good Cards” are mutually exclusive, meaning that there’s no overlap between them. It’s impossible to have a flop that satisfies more than one condition at once.
Even though we know how to find P(1 GC), P(2 GCs), and P(3 GCs), mutual exclusivity gives us a big shortcut. The entire set of outcomes for flops, with respect to the number of Good Cards, is {0, 1, 2, 3}. This means that (in general, the probabilities of the entire set of outcomes must add up to 1). This means that . Flopping no Good Cards is said to be the complement of flopping at least one Good Card. To find the probability of flopping at least one good card, we can do the following:
- Find the probability of the complement (flopping no Good Cards).
- Subtract the probability of the complement from 1 to get the probability of flopping at least one Good Card
Given that you hold AK, there are combinations of flops without an ace or a kind. There are total combinations of flops given that you hold AK. Therefore, the probability of flopping no aces or kings is , and the probability of flopping at least one ace or king is about .
Joint Probabilities and Conditional Probabilities
A joint probability is the probability of an event that’s composed of multiple outcomes. For example, getting dealt AK is a joint event because it involves being dealt an ace and being dealt a king–being dealt an ace and being dealt a king are considered to be individual outcomes that must happen for one to hold AK. Joint probabilities can be found using permutations or combinations, but another useful method exists.
To find joint probabilities, you can take the probability of the first event and multiply it by the probability of the second event. If there’s a third event, you’d multiply by that probability as well, and you’d keep on multiplying until you’ve multiplied every probability of every individual outcome comprising the joint event. Sometimes it’s easier to use permutations or combinations, but other times, this method will get you your answer much faster.
Let’s consider the probability of being dealt pocket aces. The probability that the first card is an ace is . If one card is drawn, and it’s an ace, then there are 51 cards left in the deck for the second draw–3 which are aces. The probability that the second card is also an ace is . The probability of being dealt pocket aces is therefore
Note that the probability of the first card being an ace is not the same as the probability of the second card being an ace because the deck is not identical on both draws. The second probability, , is called a conditional probability because it’s the probability of the second card being an ace, given that the first card drawn was an ace.
If we refer to the outcome of the first card being an ace as A and the outcome of the second card being an ace as B, the probability of B given that A occurred is written symbolically as ; the symbol “|” means “given.” The probabilities of A and B are said to be dependent because the probability of B changes depending on whether A happened. In general, the joint probability of an event comprising dependent outcomes is , and if P(B|A) is simply equal to P(B), that means that B is independent of A.
Probabilities are Based Off Of How Much We Know
Ultimately, probabilities are a function of our knowledge. Any probability calculation we do is dependent on what we know about a situation. Let’s find the probability of seeing a flop containing three aces. The way this situation is worded, we must assume that the flop is being drawn from a deck of 52 cards. The number of combinations of flops containing three aces is . The total number of combinations of flops is . Therefore, the probability of seeing a flop containing three aces is .
Now, let’s find the probability of seeing a flop containing three aces given that we hold AK. The information we have changes the numbers that go into the calculation. First, there are only three aces left in the deck, meaning that there’s only one combination of flops that contains three aces. Second, holding AK reduces the number of cards in the deck to 50, meaning that there are combinations of flops. Therefore, given that we hold AK, the probability of seeing a flop containing three aces is instead of .
When calculating probabilities, it’s very important to keep track of whether events are independent of each other. You also need to be aware of your state of knowledge concerning the situation you’re calculating probabilities for. Whenever you write down a probability, check to see if it makes sense according to what’s physically happening. For example, if you draw an ace from a deck of cards, there’s no possible way that the probability of the second card being an ace is also because after drawing the first ace, there are only 3 aces remaining in a 51-card deck. By constantly checking yourself along the way, you should minimize the likelihood of you making a mistake when calculating probabilities, and maximize the validity of the information you take with you to the tables!
Tony Guerrera is the author of Killer Poker By The Numbers and co-author of Killer Poker Shorthanded (with John Vorhaus)
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