I enjoy playing online poker for the competitive aspects of this beautiful game. When you play as many hands as I do, only a few tend to catch your attention and below I present one for poker flush odds that made me stop and have to do the math behind it.
As you can see from the picture at the top of this article, the setup is the following:
- Four players in the hand (a total of 8 known cards)
- 5 Diamonds, for a flush, on the community board
- At showdown, all players are playing the board and all have the community flush as their hand i.e. no player had a Diamond in their hand
What are the odds that no player would have a Diamond in their hand AND that all five cards on the community board would be a Diamond?
It’s tempting to do this with a quick script in Julia or Python but we can just as easily solve it with probabilistic statistics. We will need to calculate a couple of components for this:
Poker Flush Odds: 8 Cards, No Diamonds
We know that a standard poker deck has 52 cards, 13 of which are Diamonds. We need to deal out 8 cards, in a row, that are not Diamonds. For the first card, we know that this calculation looks like 39/52
We can see from the calculations above that at 8.18%, the probability of 4 players being dealt no Diamonds is small, but not rare. Of course, this is only the first part of the equation that we need.
Poker Flush Odds: 5 Cards, All Diamonds
At this point we are left with a deck of 44 cards that are relatively “rich” in Diamonds. Now we need to deal out 5 cards that are all Diamonds. We know that the probability of the first card being a Diamond is 13/44 and the remaining calculations are as follows:
Now we are left with a relatively small probability of 0.12% that five Diamonds would be dealt out on a community board.
Poker Flush Odds: Putting It All Together
Up to this point, we have treated the four players with no Diamonds and five Diamonds being dealt out as independent events. And that’s OK, but we know that we need to find the probability of both events happening, also known as the intersection of the two events, in order to get the probability of this hand playing out in this exact manner. That is shown below:
That’s a very small number indeed! If we take the inverse of this number, we get 10,187, making this hand scenario about a 1 in 10,200 hand occurrence!
A poker hand will play out in this manner about 1 in 10,200 times!
By going through these exercises, not only do I get to sharpen my probabilistic statistics mind, but it also helps me to improve my poker game. I hope it does the same for you and moves your game forward!