The Significance of Specificity in Probability: Analyzing "Quads" in Poker
The difference in calculations regarding extremely specific events in statistics can be massive, as seen in this example using a poker game I witnessed the other day
The other day, I was sitting down casually watching a basketball game before I heard a friend playing poker on the table behind me yell, “holy cow! What is the probability of that!”. After hearing the buzz word “probability” I had to see what just went down: On back to back hands of Texas-hold ‘em poker, my friend and the player to his right got “quads”. “Quads”, featured above, is when there are 4 of one card between your two and the five cards on the board. This can be 4 Aces, twos, sixes, fives, etc. Now, getting just one hand of quads is extremely rare with a staggering 1 out of 1,042 or .096% of all hands that are dealt getting quads after all of the cards are revealed. My friend’s logic was that since the chances of getting quads was .096%, the probability of two players in the game of poker getting quads on back to back hands would be 1/1,042 x 1/1,042 which equals .00009216% or a 1 out of 1,085,069 chance. Now, I am not writing this article to say that this player is wrong. He is correct in saying that the probability of one specific person at the table and then one other specific player at the table both getting quads on back to back hands is calculated to be 1 out of 1,085,069 chance. However, the true probability of this event happening at a poker table is a lot more likely than this.
First, it’s important to recognize how we can find the probability of getting quads. Below, I added a tree diagram of the probability of getting quads.
In order to find these probabilities, I separated the odds of getting dealt a pair as your first two cards and the odds of getting dealt a hand without a pair, then accounted for each way to get quads. I could go more in depth on this process, but unfortunately it would take a whole different blog post to fully explain.
Generally, when people calculate the odds of two events happening, they consider just those events specifically at that specific time. Here, those events are one specific person getting quads, then one specific person getting quads the hand after. What we should be asking is…
What are the chances that, throughout the entire poker session, there is an instance where one person gets quads, and then the next hand someone also gets quads?
This is the question we will be trying to answer throughout this post. Answering it will address the fact that we would ask “what are the odds of getting quads on back to back hands”, but, unlike my friend did, this new question will allow us to say this at any time throughout the entire session of poker if, on back to back hands, anyone at the table can get quads each time, not just one specific person on each hand.
We will answer this question by first learning about our assumptions for our calculations to be accurate, then find the probability for anyone at the table to get quads, find the probability that it happens on back to back hands, and then find the probability that this could happen at any point in a standard game of poker, not just any two specific hands.
Let’s first look at some of the things we must assume for our calculations to work.
Assumptions:
To do this we will need to assume three things that may not typically happen during a poker game.
All eight players are dealt into every hand (Therefore 8 hands are dealt to everybody for each hand of poker)
All five cards are shown (the flop, turn and river) at the end of the hand
We include folded cards to count as someone’s hand whether or not they get quads. Players in poker fold cards when they do not want to play with them anymore
Probability of Anyone at the Table Getting Quads On One Specific Hand
We have already established that that probability of getting quads is .00096. Normally, when people play poker, they play with eight players. To find the probability that anyone at a table with eight players can get quads, we can find the probability a specific person gets quads, and multiply that number by eight to account for everyone at the table. Therefore, this number will show at least one person at the table getting quads.
Therefore, .77% (1 out of every 130 hands) on average, any hand with eight poker players will have at least one player get quads.
Probability of Anyone at the Table Getting Quads, and Anyone Getting Quads the Following Hand
If the probability of any person getting quads on one specific hand is .0077, we can find the probability of this happening twice by multiplying the probability by itself.
Therefore, the probability of two people getting quads on back to back hands is .00005929 or 1 out of every 16,866 instances of two consecutive hands, on average.
Probability of This Happening at All During the Entire Session of Poker
After talking with some friends, it seems as though they play poker for about 3 hours on average, playing about one hand every three minutes. This would mean they play about 60 hands an hour, and, although it may sound strange, this means there are 59 instances of their being back to back hands in that time frame. In order to find out the chances of anyone at the table getting quads one or more times in this period, we can create what’s called a “binomial function”. This is where we find the true probability of an event happening once, .00005929 in this case, and then find the probability that if this event happened a certain amount of times, 59 times in this case, assuming the events are independent. In this case, the events of having back to back hands that have are quads are independent of one another. We can expedite this long mathematical process by plugging this information into a calculator.
Therefore, the probability of exactly one situation where there are back to back hands of poker that feature a player getting quads are approximately .00349 or about 1 in every 287 three hour sessions of poker, on average.
My point with this is not to say that the probability of any two people getting quads on back to back hands at any point throughout a poker session is not low, a 1 in 287 chance is still unlikely! However, the point of this is more so to identify that there are many times in probability where certain datapoints are extrapolated from a sample in order to create a more enticing probability, as my friend in the introduction may have done. It’s always important to recognize how specific or general someone’s events are when they are calculating the probability of something, and whether or not they, quite literally, factor those specificities into the equation as well.
Thank you all for reading this. I thought this one was pretty fun to make as I see this type of statistical extrapolation in my every day life. Again, feel free to email me at haydenblair7@gmail.com with any questions about this, or advice with statistics and probability, I’d genuinely love to learn more and hear some opinions.