01-13-04, LearnTexasHoldem:

How to calculate poker odds...

Question:

I read your article and I'll be working those suggestions this week in my first ever NL tournament. I was wondering if you could help out with another item. I see all these hand odds/probablities charts and have found brief summeries on how to calculate poker odds, for example the chance of getting an ace with your first card would be 1 in 52 or 52:1.

Assuming you didn't get an ace for a first card, the odds of getting it on your second increase to 51:1. This is pretty straight forward but with the sheer weight of all the statistics out there, remembering them by wrote is impracticle so I'd like to know how to calculate them myself in given situations.

For instance if I have two suited cards in my hand, I'd like to be able to calculate the poker odds of drawing a complete flush by the flop, by the turn and by the river. What if I get 3 to the flush on the flop, what then, what about 4 to the flush on the flop and so on. This situation is probably more simplistic than others but the concept is how to calculate odds not only on the next card but on a series of cards. I guess this is pretty much a math question but I can't really find any documentation on how to set these equations up.

Thanks!
Cassie

Hi there Cassie,

You may want to check out this page: Pot Odds and Implied Odds That should help you figure out odds but I'll go over a couple more examples here to make sure you get the right idea.

To answer your first question, in most games you will already know most of the odds going into the game. You won't try to figure them all out in your head. The main calculation you'll do is figuring out what your opponent has and if a draw is worth it based on the pot size, your hand and what you think he has.

Here is how you figure them out:

Let's say you have Jh Th. The flop comes back 4h - 5h - Tc That gives you a really strong hand since you have top pair and the flush draw. To figure out what the odds are of the flush getting there you calculate the chances of it getting there on the turn and then the river and then putting them together.

Let's do that now. There are 13 cards of each suit in the deck, you can see that you have 4 of the hearts already so that leaves 9 remaining hearts to make your flush. You can also see 5 total cards (two in your hand and 3 on the flop). That leaves 47 unseen cards out of the 52. That means that you have a 9/47 of catching the flush on the turn.

We do the same thing on the river. There are now only 46 unseen cards but still 9 remaining hearts so the chance is now 9/46. If you are just calculating the chance of one card hitting the you can just do a simple division to get a percentage. For example, the chance of hitting your draw just on the turn card would be 9/47 or 19%.

Things get slightly more tricky when you have to figure out if you can hit the hand on either card (turn or river). To do this we flip the fractions to cards that WON'T help us and then we multiply them together. That will give us a percentage for NOT hitting our hand which we can use to figure out the percentage for hitting it.

So, for the turn card we said we had 9 outs from 47 unseen cards. That means that 38 must not help us then (47-9). That gives us 38/47. For the river we do the same thing (46-9): 37/46.

Now convert these to decimals and multiply them together. .80 * .80 = .64 or 64% of the time we won't hit it which must mean that 36% of the time we will.

Summary

9 cards give us our hand. There are 47 unseen cards on the turn and 46 unseen cards on the river.
38 cards on the turn don't give us our hand 38/47
37 cards on the river don't give us our hand 37/46

Convert these to decimals and multiply = .80 * .80 = .64
.64 is 64% of the time we won't catch, so subract that from 100 to get how often we do = 36%

The 2/4 rule

You can also use the 2/4 rule which gives a rough estimation. Just times the outs you have by 2 if you'll just see the turn card or by 4 if you see the river.

Catching AA pre-flop

Now let's do another example of catching AA preflop like you mentioned above, how often does it happen? There are four Aces in the deck so your chance of catching one of the first card is 4/52. One the second card there are only 3 aces remaining and 51 unseen cards remaining. So it is 4/52 and 3/51 but unlike the example above we don't have to flip them, we multiply them since you have to do BOTH. So that lowers our chance signifcantly. For multiplying just do it like this: 4 x 3 = 12 and 52 x 51 = 2652. The total is then 12/2652 which is the same as 1/221. So that means you'll get aces about once out of every 220 hands.

Catching a set on turn or river

Last example... Let's say you have JJ and your opponent has AA. The flop is K - 4 - 2. To catch a J on the turn card your chances are 2/47 (2 Jacks remaining in the deck out of 47 unseen cards). That means your percentage to catch it on the turn is a little over 4%. To catch it on the river is slightly higher but still about 4%. Let's use the 2/4 rule to figure the chance of hitting on the river. Four times our two outs gives us 8%. That's really bad and why if you have a big pair cracked by a guy chasing a smaller one then it really hurts. Remember too that he has the same chance to catch his other A as you do your J (besides already being a big favorite).