In the previous example, you maximized your expectation against a single, known poker hand. If your holdem opponent can have one of several draws, you should maximize your expectation against the range as a whole. Sometimes doing this will mean allowing your holdem opponent to draw profitably with the strongest of his possible draws.

Put another way, if your holdem opponent can have a 4 out draw, an 8 out draw, or a 15 out draw, the bet size that maximizes your expectation might allow the 15 out draw to draw profitably if your opponent will call incorrectly those times he has the 4 or 8 out draws.

Say you think your holdem opponent has one of two draws: one that’s 4-to-l to come in and one that’s 2-to-l. You think your poker opponent will have the 4-to-l draw 75 percent of the time and the 2-to-l draw 25 percent of the time.

Again, for simplicity, assume that there will be no betting on the river (we’ll adjust for river betting at the end). The pot is $1,000.

The break-even point for the 2-to-l draw is a $1,000 bet ($2,000-to-$ 1,000). The break-even point for the 4-to-l draw is a $333 bet ($l, 333-to-$333).

You’re considering two bet sizes: $ 1,500 and $500. If you bet $ 1,500, you’re fairly sure your poker opponent will fold either draw (and be correct to do so). If you bet $500, you’re fairly sure your opponent will call with both draws (correctly with the 2-to-1 draw, but incorrectly with the 4-to-l).

If you bet $1,500, you will win the pot and no more. We’ll call this the "baseline" and assign it a value of $0. You don’t win anything from your holdem opponent’s mistakes, but you don’t lose anything by giving away a profitable call either.

If you bet $500, then you gain because the 4-to-l draw calls incorrectly, but you lose because the 2-to-l draw calls correctly. The value of your poker opponent’s mistake of calling with the 4-to-l draw is $100.

$100 = (0.20)($1,500) + (0.80)(-$500)

The value of your mistake by allowing your holdem opponent to call with the 2-to-l draw is $167.

$167 = (1/3)($l,500) + (2/3)(-$500)

So you gain $100 when your Texas Holdem opponent calls incorrectly with the 4-to-l draw, and you lose $167 when he calls correctly with the 2-to-l draw. But he has the 4-to-l draw three times more often (75 percent versus 25 percent), so your total gain against the baseline is $33.33.

$33.33 = (0.75)($100) + (0.25)(-$167)

Even though you made a mistake by allowing your opponent to draw correctly sometimes, your poker opponent made a bigger mistake by drawing incorrectly the rest of the time. Overall, in this case, you maximize your expectation with the smaller bet.

Choose your bet size to maximize your overall expectation, even if that sometimes means that your opponent can draw correctly against you.

We ignored possible river betting in our analysis. In reality, the fact that your opponent can have one of several draws will mean that his implied odds are greater than his pot odds. Thus, according to the rule from earlier, you should bet a larger amount than you would if you knew your opponent’s poker hand. So you might want to bet significantly more than $500 to ensure that his calls with the 4-to-l draw are still significant mistakes.

Also recommended:

1. When Bigger Bluff Sizes Will Fold More Hands
2. How Big Do You Want Their Mistake to Be?
3. You Might Not Have the Best Poker Hand
4. Choosing Starting Hands in Seven-Card Stud Hi/lo
5. Your Opponent Thinks He May Have the Best Hand