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Probability & Bayes Theorem
In poker, it’s important to be able to estimate the chance or probability that something occurs. This helps you with the balancing act of knowing whether you’re ahead or behind.
Example 1
What is the probability of flipping a fair coin 3x and landing on Tails all three times?
- Each flip has a probability of 50% to become Tails.
- To calculate, do (50%)(50%)(50%) or (1/2)(1/2)(1/2) to get the answer 12.5%
Example 2
What is the chance of being dealt AA pre-flop?
- Since there are only 6 combos of AA out of the 1326 total combos. The chance of being dealt AA or any specific pocket pair every 1 in 221 hands. This represents a 0.45% chance of being dealt AA.
Example 3
What is the chance of our opponent having AA when you assume they have a range of (AA, KK, QQ, JJ, AK)?
- There are 6 combos each of AA, KK, QQ, JJ and 16 combos of AK for a total of 40 combos in this range.
- Of the 40 combos, only 6 combos represent AA.
- To calculate the probability, do (6 / 40) to get 15%
Example 4
From the previous question, if we have AK, what is the chance our opponent still has AA with the range (AA, KK, QQ, JJ, AK)?
- Our AK is blocking some of the combos of AA and AK in our opponents range. There are still 6 combos of QQ, JJ however less combos for the hands containing an A or K. There is 3 combo of AA, 3 combo of KK, and 9 combos of AK. The total combos is (3 + 3 + 6 + 6 + 9) or 27 combos.
- Of the 27 combos, only 3 can be AA.
- To calculate, do (3 / 27) to get 11%
Example 5
This hand is the book (The Math of Holdem by Collin Moshman & Douglas Care).
Suppose we open 3x with KK and the big blind defends.
Which flop is more dangerous for our hand (AAT vs. AT3) when our opponent has one of these three hands (AQ, QJ, JT)?
- On the AAT flop, there are only 8 combos of AQ, 16 combos of QJ, and 12 combos of JT possible. The only hand that out-flopped you is AQ. This means that 8 combos of the (8 + 16 + 12) combos have you beat. You’re beat by only (8 / 36) or 22% on this flop against this range.
- On the AT3 flop, there are 12 combos of AQ, 16 combos of QJ, and 12 combos of JT possible. You’re beat by only (12 / 40) or 30% of the range.
- This means that it’s more likely for you opponent to have an Ace on a flop with only 1 ace, not 2 aces. Each ace acts like a blocker to reduce your combos.
Now, in poker, their actions (fold, check, call, raise) might provide clues for what hand they have. You can use Bayes’ Theorem to approximate the “conditional probability” of an event occurring.
For example, let’s say we have KK, the flop is AT3, and our opponent just check-raised us on the flop. What is the chance that they have AQ and not one of the weaker semi-bluffing hands like QJ or JT? To figure this out, we need to make assumptions and estimations.
AQ — my opponent will check-raise 2/3 of the time on AT3 flop
QJ — my opponent will check-raise 1/3 of the time on AT3 flop
JT — my opponent will check-raise 1/4 of the time on AT3 flop
To calculate, we need to find:
- probability of having AQ
- probability of check-raising with only AQ
- probability of being check-raised by AQ, QJ or JT
The chance of having AQ is 12 combos of 40 combos or (12 / 40) = 30%
The chance of check-raising with AQ is estimated as 2/3 of the time or 67%
The chance of being check-raised is:
- (2/3) of the 12 combos of AQ = 8 combos
- (1/3) of the 16 combos of QJ = 5.3 combos
- (1/4) of the 12 combos of JT = 3 combos
- A total of about ~16 combos will be check-raising of the 40 total combos. This means that you will get check-raised (16 / 40) or 40% of the time.
Use Bayes’ Theorem
= P(check-raising w/ AQ) * P(having AQ) / P(check-raising)
= (8 / 12) * (12 / 40) / (16 / 40) = 50%
In conclusion, using probability in poker will simply make you a better guesser, your gut instinct will be more consistent, and you’ll have a better framework for thinking things out.
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2 Replies
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I love how you always show your work.
Using an equity calculator, KK vs range AQ, QJ, JT
Flop AAT: KK has 75% equity
Flop AT3: KK has 63% equity
I remember in a home game calling down my friend’s bluff. I was in SB with JJ. My friend limped in MP, I raised. He then 3bet and I called. Flop came A 9 2. I called his cbet. I knew he was more likely to open raise with AK and AQ. I thought his limp 3bet line was unlikely with Ax.
When a second A came on the turn, I easily called his turn bet . I knew that the second A meant he was less likely to have an A in this hand. River was a 6 and it went check/check.
He turned over 76 and was surprised that I was able to call him down. He didn’t fully understand the blocker effect of the A on the turn even after I pointed it out to him.
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I must have taken a day off or something, this post is amazing, how did I miss this? Great point about using this way of thinking to “train your gut” to make better intuitive decisions in real time. @ARW do you listen to the Thinking Poker Podcast? They have some good discussions about Bayes and how it applies in hands.
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