03-16-05, LearnTexasHoldem:

Drawing To Turn Or River Odds

Question: My OLD brain can't understand the 6th paragraph. "For figuring out odds for a draw that can hit on either the turn or the river (keyword either), you need to do something a little more complicated -- you figure out what the odds are of not hitting it and the subtracting that number by 1. Since 5 cards give us our hand, there must be 42 that don't. Then on the river if we haven't hit there must be 41 cards that don't make our hand.

42/47 * 41/46

This gives us .795, now subtract that from 1 to get the percentage of 20%.

*****Flop to River % = 1 - [ ((47 - Outs) / 47) * ((46 - Outs) / 46) ]."

Would you please explain in more detail the calculation? I would really appreciate it.

Thanks so much. Gary.

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Answer: There are two types of draws in texas holdem:

1. When you need to catch both the turn and river for your hand. For example, you have AdKd and the flop is 6d - 2h - 4h. To catch the diamond flush, both turn card AND river card have to be both diamonds.

2. When you need to catch the turn OR river for your hand. For example, you have AdKd and the flop is 6d - 4d - 2h. Notice that on the turn OR river if we catch a diamond card, we hit our flush draw.

To calculate our chances of hitting our draws, we compare the fractions of our outs to the total number of cards. In odds, when both events have to happen, you multiply the fractions. So in the first example when we have to catch a diamond on both the turn and river, the chances are:

10/47 x 9/46

10 diamond cards remaining/47 unseen cards X 9 diamond cards remaining/46 unseen cards

* there are 13 diam. cards total in a deck, and we can see 3 already there. There are also 52 cards total, and we can see 5 of them so only 47 remain unseen.

So the total chance of us hitting our flush draw this way is: 90/2162 or around 4% (not very good).

Now when we can catch our flush draw on the turn OR river -- like in example 2 above -- then we have to use the ugly equation:

Flop to River % = 1 - [ ((47 - Outs) / 47) * ((46 - Outs) / 46) ]

This equation basically says that our chance of catching the draw plus our chance of missing the draw, should equal one; therefore, if we subtract our chance of NOT catching from one, we should have our chance OF catching. Let's try it out:

We have AdKd and the flop is 6d - 5d - 4h. We have 9 outs (cards that will make our flush). Plug those 9 outs into our equation above:

((47 - 9)/47) * ((46 - 9)/46)

38/47 * 37/46

1406/2162

= ~65% of the time we will miss it, which means that 35% of the time we will hit a diamond on the turn or river card.

As we stated above, when both events have to occur we multiply the fractions together. In odds books it then says if an event can happen on either chance, then you add the fractions together. What would happen if we did that?

(our outs/total cards) + (our outs/total cards)

9/47 + 9/46

We get something closer to 38% or 39% of the time. Is this inconsistent with our other equation? No. Our equation above calculated the chance of us not catching a flush card on either the turn or river. When we subtract our chance of not catching the card by the turn or river from 1, it gave us the chance of catching on the turn OR river, but it didn't take into account catching on BOTH the turn AND river. Notice that when we add our 4% to our 35%, we get something right around where we should be according the to last equation: 39%.

This is why if you have a flush draw against another player who doesn't have one of those cards, you are about a 3:2 underdog (60/40 --- out of 100 times, he will win 60, you will win 40).

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