Poker Math
Calculating Probability Percentages
Ok, behind all of the charts on the site is some kind of formula. And if you aren't scared by a number two lead pencil, you can follow along on this page to get an understanding of the math behind the charts.
If you have a four flush, and are waiting on the turn card to be dealt, you can use the Probability of Catching an Out Chart to see that there is a 19.1 % chance of landing your flush on the next card.
How do we get that number?
Take the number of cards that can help you (divided by) the number of unseen cards in the deck, basically.
(# of outs) / ( # of unseen cards)
How many cards can help? There are 13 cards of each suit. We have four of them between our hand and the board cards. 13 minus 4 gives us 9 remaining "outs".
How many unseen cards are there? Well, we start with 52 cards in the deck. Two are dealt to you, leaving 50 unseen cards. Three are placed face up on the board during the flop, so we're down to 47 unseen cards (52 - 2 = 50, 50 - 3 = 47).
So, the simple equation is 9/47, or .1914, rounded down to 19.1%.
You can use this formula for a lot of situations. For example, you have a gutshot straight draw and are waiting on the river card.
A gutshot straight can be filled in with four different cards (say you have A,2,4,5 and need any of the remaining 3's in the deck to complete your straight).
If you are waiting on the river card, there are 46 unseen cards in the deck (52 minus your 2 hole cards, minus 4 cards on the board).
That gives us 4/46, which is .0869. Rounded to 8.7%.
While this isn't math you want to do at the poker table during a live hand, it isn't terribly complicated when you pull out the calculator or paper and pencil.
I Have a Confession to Make
That's the simple-but-accurate version of the formula. The full formula is . . .
1 - [(# of unseen cards - # of outs) / (# of unseen cards)]
Of course, if you were to plug in our numbers, we'd get the same results. For the gutshot straight . . .
1 - [46 unseen cards - 4 outs] / 46 unseen cards
1 - [42 / 46] which is . .
1 - [.913] which is . . .
.087 the same results as with our simple version of the fomula, 8.7%. So, why bother telling you this? Because you need it to answer the next question . . .
What about landing your out on either the Turn OR the River?
We use need full formula to calculate this one. But first let me caution you, look at the probability of landing your out at each point you pay into the pot. So, if you have to pay for the turn card, and then you have to pay for the river card, look at the probability of catching your out on the turn, then look again at your probability of catching an out on the river.
When you pay up front for both cards (say you're going all-in), then it's the right time to look at catching your out with either card. After all, you paid for both of them already.
The formula we use for this situation is:
1 - [((# of unseen cards - # of outs) / (# of unseen cards)) * ((# of unseen cards - # of outs) / (# of unseen cards))]
Example 1:
We're sitting on a flush draw after the flop. What are the odds of landing that flush on either the turn or the river?
There are 9 outs on a flush draw. There are 47 unseen cards on the turn, and 46 on the river, so we plug in some numbers to get:
1 - [((47 unseen cards - 9 outs)/(47 unseen cards)) * ((46 unseen cards - 9 outs)/(46 unseen cards)] after subtraction we have
1 - [(38/47) * (37/46)] so we do the division
1 - [(.8085) * (.8043)] and then we multiply
1 - [.6502]
1 - .6502 equals .3498 or 35%
Well, those are the formulas behind the numbers in the charts. Honestly, I recommend just printing out the charts and referring to them when you need the information. Soon enough you'll have the percentages for common situations memorized.
But, for those with curious minds who want to know how things work behind the scenes, now you know.
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