We’ve limited your bet sizes to a range: Bet more than they can call profitably, but bet less than what would almost certainly blow them off their poker hand. Now we need to figure out what the right size is within that range.

You want to choose the size that will maximize your expectation. Roughly speaking, your expectation is equal to the approximate value of the mistake times the chance that they’ll make the mistake.

By "value of the mistake" we mean how much money, on average, your holdem opponent loses to you by making the mistake. Say your opponent can break even by calling a $100 bet (and profit by calling a bet smaller than $100). If you bet $101, then your holdem opponent is making a mistake by calling, but it’s a tiny mistake. The value of that mistake is less than $1 (less than because sometimes your holdem opponent will draw out and win the extra dollar).

On the other poker hand, if your holdem opponent calls a $1,000 bet, then he’s made a huge mistake. Let’s do a little math to get a feel for exactly how big these mistakes are.

Say you bet $100 into a $200 pot, and your poker opponent is a 3-to-1 dog. Ignore future betting for the moment. If your holdem opponent calls, on average it will be break-even for him.

$0 = (1/4)($300) + (3/4)(- $100)

Now say you bet $150, and your holdem opponent calls. On average, your opponent expects to lose $25 on a call.

$25 = (1/4)($350)+ (3/4)(-$150)

If you bet $200, and your holdem opponent calls, on average he will lose $50.

$50 = (1/4)($400) + (3/4)(- $200)

If you bet $600, and your holdem opponent calls, on average he will lose $250.

$250 = (1/4)($800) + (3/4)(- $600)

So when you bet $50 more than break-even in Texas Holdem, he loses $25. When you bet $100 more, he loses $50. When you bet $500 more, he loses $250.

In general, the value of your opponent’s mistake will be proportional to the excess amount you bet beyond the break-even point.

This is an important concept of Texas Holdem betting, so we’ll repeat it. Your opponent’s expected loss (and, thus, your gain) is proportional to the excess amount you bet (and he called), beyond what would have been break-even, not the total size of the bet. If $500 is a break-even amount, then you double your profit by getting $600 called versus $550. (A conclusion worth noting is that $600 will almost always be better than $550 in this scenario, as it offers double the profit potential. Your opponent would have to call $600 less than half as often as $550 to make the smaller bet better, and in practice, that will almost never happen.)

The value of your opponent’s mistake is only half of the expectation equation. To get your total expectation, you have to multiply the value of the mistake by the chance your opponent will make the mistake. Again, a big all-in bet may offer your opponent the opportunity to make a huge mistake, but if your opponent will never be dumb enough to call, then you don’t gain anything.

Say you are fairly sure your poker opponent has a flush draw, and a $100 bet will be break-even for her. You are choosing between three bet sizes: $150, $200, and $500.

You think that your poker opponent will call the $ 150 bet about 70 percent of the time, the $200 bet about 40 percent of the time, and the $500 bet 5 percent of the time. To find the best bet, you have to multiply the size of the mistake by the chance your opponent will make it:

$35 = ($ 150-$100)(0.70)

$40 = ($200-$100)(0.40)

$20 = ($500-$100)(0.05)

The best bet is the $200 bet. It doesn’t get called the most often, but it has the highest expectation.

Bet the amount that maximizes your expectation: the value of your holdem opponent’s potential mistake times the chance your opponent will make the mistake.

Good luck.

Also recommended:

1. Expectation and Multiple Possible Hands
2. Don’t Take Away Their Rope
3. You Might Not Have the Best Poker Hand
4. When Bigger Bluff Sizes Will Fold More Hands
5. The Importance of Implied Odds in Texas Holdem