Pot Odds to the Second Best Hand

Calculating basic pot odds is pretty simple once you’ve done it a few times. In a case like this:

Limit Hold’em

Villain: A [img]/images/graemlins/spade.gif[/img] K [img]/images/graemlins/club.gif[/img]
Hero: A [img]/images/graemlins/heart.gif[/img] 7 [img]/images/graemlins/heart.gif[/img]

Flop: K [img]/images/graemlins/heart.gif[/img] 6[img]/images/graemlins/heart.gif[/img] 3[img]/images/graemlins/diamond.gif[/img]

It’s a simple call against a bet given 4:1 pot odds. So how do we figure this calculation out when we might be drawing to the second best hand? A 4 flush has 9 outs on the flop, right? If we know that we have 9 outs but we can reasonably put the villain on a hand with 7 outs to a better hand, do we simply take our 9 and subtract 7 and divide by the remaining cards?

Example:

Villain: A [img]/images/graemlins/spade.gif[/img] K [img]/images/graemlins/club.gif[/img]
Hero: A [img]/images/graemlins/heart.gif[/img] 7 [img]/images/graemlins/heart.gif[/img]

Flop: K [img]/images/graemlins/heart.gif[/img] K [img]/images/graemlins/diamond.gif[/img] 6[img]/images/graemlins/heart.gif[/img]

This, of course, is all assuming that we can reasonably put the villain on a given hand. How do we calculate our pot odds here?

(yes – I know he doesn’t have exactly 7 out to the nuts here. 2A, 1K, 3 6s and any runner runner pair which gives him about 7.5 outs, but we’ll round it..)

[jba]

[ QUOTE ]

Villain: A [img]/images/graemlins/spade.gif[/img] K [img]/images/graemlins/club.gif[/img]
Hero: A [img]/images/graemlins/heart.gif[/img] 7 [img]/images/graemlins/heart.gif[/img]

Flop: K [img]/images/graemlins/heart.gif[/img] K [img]/images/graemlins/diamond.gif[/img] 6[img]/images/graemlins/heart.gif[/img]


[/ QUOTE ]

if you hit your flush on the turn, your opponent will have 9 outs to improve on the river, which is about a 20% chance. In other words, about 20% of the time you hit your flush you lose. Another way to think of this is that only 80% of your outs are good, so you have 7.2 outs. this is the proper way to think of it. the hard numbers here are:

Board: Kh Kd 6h
Dead:

equity (%) win (%) tie (%)
Hand 1: 71.7677 % 71.72% 00.05% { AsKc }
Hand 2: 28.2323 % 28.18% 00.05% { Ah7h }

28% is pretty close to a 7 out draw.

as you pointed out, you are never going to be able to put your opponent that squarely on a hand. what will actually happen in a situation like this is you will put your opponent on a hand range, for example he has trip kings 25% of the time (10 outs) and just two pair the other 75% (4 outs), so he averages about a 5.5 out redraw or 11%, which means you have closer to 8 outs. it is your strength against his range that really matters.

hope this helps

[AaronBrown]

In addition to jba's excellent reply, I would add that counting outs makes the most sense when you're pretty confident that you'll win if you make your hand, and lose otherwise. Otherwise it gets complicated.

There are four kinds of cards in the deck, ones that help no one, ones that help only you, ones that help only him and ones that help both of you. Knowing how many help you (categories 2 and 4 combined) and how many help him (categories 3 and 4 combined), is not enough information. You also need to know how many are in any one single category (that would allow you to deduce all four numbers) plus what happens if neither or both of you improve. If this is after the flop so there are two cards to go, it gets even more complicated.

In a standard out situation, there is only category 2. If you get one of these cards you win, if you don't you lose. Generally, given your information, there's a small chance you'll hit and lose, but you don't bother quantifying it because you don't know enough about his hand (as jba mentioned). You might not hit and have the better hand, but this won't help because you'll fold when he bets.

If you can win when neither player improves, or lose because both players improve, you have to figure all the possibilities. Usually that's too many to compute on the fly and you don't have the information anyway.