JimS's Blog
Ultimate Texas Hold-Em Quads
I'm curious about the odds of something.I was at a California casino playing Ultimate Texas Hold-em. The table was full - 6 players and the dealer - so she dealt 7 hands every five minutes for a total of 84 hands, give or take, over the hour.
In the course of one hour, Quads were dealt 4 times. Twice to me and twice to other players on the table. The trips or better payout is 30-1 and I had 10 bucks on the bonus. The table allows you to bet other players' bonus and I had 5 on each of the other players' bonus at the table. An hour and $1500 bucks later, I got up and went home.
My question is this - what are the odds of quads appearing 4 times in 84 hands?
What are the odds of two of those four times landing on my spot?
Comments
In the case of your hands, you are basically just playing Seven-Card Stud when looking at the individual hands. There are your two cards as well as the five community cards. It sounds like you had twelve total hands deal to you and the result was quads, twice.
https://en.wikipedia.org/wiki/Poker_probability#Frequency_of_7-card_poker_hands
The probability of receiving a four of a kind in a seven-card poker had is roughly 1 in 594. That is roughly a 0.16835% probability on any given hand. This binomial distribution calculator:
http://vassarstats.net/binomialX.html
Puts the probability of seeing exactly two such hands at 0.000183930041 which is 1/0.000183930041 or 1 in 5436.84976398 given twelve total hands.
Before you ask, the cards of the other players and of the dealer are irrelevant because you don't know what they are ahead of time, so they are not removed from the equation. As far as your hand and your final hand only is concerned, you're playing seven-card stud.
The same principle does not apply if you want the overall probability considering all hands. It's much more complex because you have to consider the probability of quads on the board, which would give all players quads. You also have to consider the probability of two cards (in value) appearing in the hands of different players, or the dealer, which then would make a four of a kind using those cards impossible.
The Wizard of Odds site has the probability of quads at .001681, which is likely more exact as the Wikipedia page is probably much more rounded, since we were starting with an assumption of 1 in 594. In other words, it's something less than that which rounds up to 1 in 594, so you can use the binomial distribution calculator the same way with probability .001681 and it won't make a huge difference.
Thanks, #Mission146.