## Making Poker Math Easier

### Mathematics, Strategy

October 15, 2007

A while back we did a big example of odds math and discounting outs.  These examples were somewhat complicated because I wanted to give a comprehensive picture of what can be involved in an odds problem.  Indeed, these problems were so complicated that the math really pushed the edge of what a player could be expected to do at the table.  In other words, those articles fell slightly short of my own standards of what constitutes good poker math.  I want to rectify that here and also present the process I use to do so as a template for simplifying other poker math problems. Here’s the original odds problem:

You’re playing 20/40 limit holdem and are dealt Ah7h on the button. Preflop, action is folded to you, and you raise on a semi-steal. A tight player in the small blind re-raises, the big blind folds, and you call. The pot is now \$140 and action is heads-up. The flop comes Qh4d5h. The small blind checks, you bet on the semi-bluff, and the small blind raises. You call (note that raising here is not a bad choice either). Pot is now \$220. Turn is 2c. Villain bets \$40. Do you call?

Principle 1: If you have a hard problem you don’t want to do, see if there’s an easy problem you can do that gives you almost the same results.

The difficulty in solving this odds problem the way we did in the previous two articles really stems from the complexity of discounting the non-nut outs and figuring the implied odds.  So what would happen if we skipped that part and just counted the nut outs and assumed we would just win the pot?  We then have 8 winners and 38 losers.  We’re risking \$40 to win \$220.  A brief examination shows we are winning more than 5:1 money-wise, but less than 5:1 against hitting the nuts, so we call.  This is exactly the same result we got when we discounted the outs and did the implied odds, but was far simpler math-wise.  When you get a math problem you don’t have enough time or mental capacity to do, consider replacing it with a similar but less accurate one you can do.

Principle 2: Make use of best-case and worst-case scenarios

When replacing a complicated problem with a simple one, you have to make assumptions.  For example, we assumed above that none of the non-nut outs were good.  When you make such an assumption, you usually have the choice to be optimistic or pessimistic.  You could assume all the non-nut outs were good, or none of them were.  I chose to be pessimistic for a specific reason.  My guess, ahead of time, was that this was a clear call.  Since I did the math with worst-case assumptions, and it still came out a call, I know that call will actually be MORE profitable than my math showed.

So when you’re pretty sure you should play, do your math with worst-case assumptions.  If it says you should play, you’re good to go.  Similarly, when your guess is you should pass, do you math with best case assumptions and if it says you should pass you know reality is even more dismal and your pass was even more correct.

Principle 3: Memorize key odds thresholds

While the above odds calculation from Principle 1 was pretty simple and definitely doable at the table, there’s an even easier way.  Anyone who’s played much holdem knows you end up with a lot of 8 and 9 out draws to the nuts.  Any time there’s a very common occurrence like this, it’s worthwhile to memorize the odds involved so you don’t have to compute them.  For example, such a draw in holdem is always slightly less than 5:1 against hitting.  With that number in memory, you only need to do math on the bet and pot sizes to see that you’re getting paid more than 5:1 and should definitely call.  Memorizing these commonly used numbers is well worth your time.

Principle 4: Develop intuition

One step beyond memorizing odds is memorizing common scenarios.  For example, you’ll find that in standard structure limit holdem if you have an 8 or 9 out nut straight or flush draw, and there was multiway action or multiple bets on each previous streets, you’ll always have odds to see the next street if it will only cost you one bet.  In practice this means you don’t need to do any math in this situations at all because the solution will always be the same .  As you internalize more of these rules of thumb, you can play correctly in numerous situations without bothering to do any math at the table at all.