Combinations: The Math Behind your Opponents Range of Hands
The key to all games of poker is putting your opponents on a range of hands, calculating how likely those range of hands are, then comparing to the pot odds, and deciding what to do.
Putting your opponents on a range of hands is the Art of Hand Reading
Calculating how likely that range is to occur is the Art of Combinations
(And the Art of Combinations is really the Math of Combinatorics)
Since it's more fun to do math in a real hand, we'll do some combinations related to my hand today:
I'm dealt T T J 5 in a PLO8 game
A player in EP min-raises, 3 others call and I call in the BB getting 9:1 on my money
** Dealing Flop ** T, 7, 8
5 players and checked around. I have top set, but chances are very good someone has some sort of flush
** Dealing Turn ** K
5 players all check around. Weaker flushes are likely to call me down as I have a loose table image.
** Dealing River ** 8
Good, now I have an overboat. I bet hoping a flush will call me.
Hero bets [$15].
Villain raises [$40], and has $45 left in his stack.
So, what do we do here? First, we put our opponent on a range of hands. We'll divide those into hands that beat us, and hands that don't.
Hands he could play that beat us: 88xx, KKxx, for quads or a better fullhouse.
Hands he could play that we're ahead of: 77xx, 8Kxx, 8Txx, 78xx, for worse full houses.
Then of course there are some bluffing types that think I'm trying to steal here, that could have a wider range of hands. It's possible that some weaker flushes might play this way: they didn't want to bet for fear of bigger flushes, but now spring to life. But on the whole, those bluffing/weak flush hands are fairly unlikely.
So, how often are the hands that beat us dealt out?
For 88xx, it's C(2,2) or 1 way we're beat. (See the Math of Combinatorics if this doesn't make much sense)
For KKxx, it's C(3,2) or 3 ways we're beat.
That's a total of just 4 possible hands dealt that beat us.
Now, how often are the hands that beat us dealt out?
For 77xx, it's the same as KK or 3 ways
For 8K, it's C(3,1) * C(2,1) or 6 ways. (3 choose 1 for the K's, then 2 choose 1 for the 8's)
For 8T, it's C (1,1) * C(2,1) or 2 ways. (since I have two of the T's there's just T left)
For 78, same as 8K, 6 ways
Another way to think of the grouping of 8K,9T,87 hands is to say, "I can take the 8 and match it with each of the unseen K,T,7 cards, of which there are 9. I can then repeat that with the 8"
That's a total of 17 ways
So, if I'm sure my opponent has a full house here, then of the 21 possible ways he could have the boat, I am ahead 17/21th of the time, or I'm winning 80% of the time here.
However, we need to make some adjustments: This is just based on the likelihood of certain hands being dealt. Now we need to adjust for the probability of those hands BEING PLAYED. In practice, it's more likely that opponents will play hands with KK in them than hands with 77, 8K, 8T, 78 in them. So we will arbitrarily adjust the KK probability upward. I'll guess that the chances of KK getting played are about 2x as likely as the others. Is that right? Who knows, but its seems like a reasonable guess. So basically, I'll double count the ways KK is dealt to adjust for this.
So now I'm beat 7 ways, and ahead 17 ways.
Now, if I'm ahead the vast majority of the time here, should I reraise? Not necessarily!
I need to look at the number of times a worse hand will call my reraise. After all, a better hand is always going to call, but if a worse hand will always fold, then there is no point in raising. Instead I would just call.
In this case, my opponent was a moderate player. If I reraised, I figured he would fold his 77xx, and 78xx hands, but call with the 8T and 8K hands.
So if I reraise, I lose the 7 times (adjusted) he has me beat with his 88,KK, but win the 8 times I'm ahead of his 8K, 8T.
So reraising is slightly more profitable, so I reraise!
Sadly, Villain showed 8 K K 7 for a bigger full-house, and I lost. But it was the right decision, and that's all you can control in poker.