## Could Have One of Several Draws

In the last example, we knew exactly what our opponent's draw was. In practice, you rarely will. You may know that he's likely to be drawing, but you won't know whether he has a straight draw, a flush draw, bottom pair, etc.

Say you again have AVA4 on the turn in a \$100 pot with \$400 behind. But now the board is You are fairly sure your opponent has a draw, but you don't know whether it's a diamond draw or a straight draw. It could even be a backdoor club draw with a hand like

Whatever draw he has, he's likely to have about eight or nine outs (though big combination straight, flush, and pair draws are also possible). So he's still likely to be approximately 4-to-l against to make his hand.

Unfortunately, your opponent won't make it easy on you and tell you which draw he has. If any of the "obvious" draws comes in, that is any diamond, king, queen, eight, or seven, he may bluff even if the card didn't complete his hand.

Now you can't just fold on the river if a diamond comes and your opponent bets. Depending on exactly how much he bets and how often he bluffs, you may still fold, or you may call. But either way, you lose money: if you call, sometimes you'll be paying him off, and if you fold, sometimes you'll be getting bluffed out.

Since your opponent can now sometimes make money from you on the river, his implied odds are significantly better than the pot odds. A bet offering slightly worse than his pot odds doesn't cut it anymore. You have to bet a larger amount to prevent him from calling profitably.

If your opponent could hold one of several draws, bet a larger amount than you would if you knew which draw he had.

Don't Bet Too Much

Once you observe the basic rule and bet more than your opponent can call profitably, you should now root for him to call. That's because calling would be a mistake (if your opponent knew what you had), and you want your opponents to make mistakes even if they sometimes draw out and it costs you the pot.

While moving all-in anytime you know you have the best hand might prevent your opponent from calling profitably, it's still a dumb thing to do. Huge bets will blow your opponents out of the hand and force them to play correctly. According to the Fundamental Theorem of Poker, you should avoid plays that force your opponents to play correctly. Put them to a decision; let them make mistakes.

Bet more than your opponents can call profitably, but don't bet so much that you blow your opponents off their hands. Bet an amount that entices them to make a bad call.

How Big Do You

Want Their Mistake to Be?

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 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 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 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 opponent will draw out and win the extra dollar).

On the other hand, if your 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 opponent is a 3-to-1 dog. Ignore future betting for the moment. If your opponent calls, on average it will be break-even for him.

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

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

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

So when you bet \$50 more than break-even, 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, 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 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 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:

\$3 5 = (\$ 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 opponent's potential mistake times the chance your opponent will make the mistake._

### Expectation and Multiple Possible Hands

In the previous example, you maximized your expectation against a single, known hand. If your 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 opponent to draw profitably with the strongest of his possible draws.

Put another way, if your 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 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 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 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 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 opponent's mistake of calling with the 4-to-l draw is \$100.

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

So you gain \$100 when your 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.

Even though you made a mistake by allowing your opponent to draw correctly sometimes, your 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.

• 33.33 - (0.75)(\$100) + (0.25)(- \$167)
• Choose your bet size to maximize . your overall expectation» even if that I 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 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.

Don't Take Away Their Rope

In the first example of the section, you had AVA# on a Q4742<£4± board, the pot was \$100, you and your opponent each had \$400 behind, and your opponent was on a diamond flush draw. Our conclusion was that you should have bet at least \$40 because he was 3.9-to-l to make his draw, so you should have offered him no better than 3.5-to-l pot odds.

Let's reconsider the same example, except now you hold You have top set instead of an overpair, and the and 24 make your opponent's flush, but give you a full house. Your opponent now has seven outs instead of nine, so he's 5.3-to-l to beat you (7/44). By our earlier reasoning, you should offer no better than 5-to-1 pot odds, so you should bet at least \$25 (offering \$125-to-\$25).

But our earlier reasoning doesn't hold anymore! Why not? Because if the 44 or 24 comes, not only do you not lose, but you stand to win your opponent's remaining \$400 on the river. Let's compare two expectations: one where you make a big bet, forcing your opponent to fold, and another where you check, allowing him to draw for free.

If you bet a lot, forcing your opponent to fold, you'll win the \$100 pot every time. So your expectation is \$100.

If you check, then you win \$ 100 whenever no diamond comes (35/44), win nothing when a non-pairing diamond comes (7/44), and win \$500 (\$100 plus \$400) when the 44 or 24 comes (2/44). Your expectation if you check is \$102.28.

Because your opponent will occasionally make a second-best hand and get stacked, you'd prefer that he draw for free than that he fold. The lower bound of your betting range isn't \$25 — it's \$0.

Obviously, you'd rather bet and have your opponent call than check. But you should bet an amount that you're fairly sure your opponent will call, even if that's less than \$25 (although in this case it wouldn't be).

If your opponent could catch his draw, but still be second-best, tend to bet an amount you're fairly sure he'll call. Don't miss a chance to stack him by blowing him out too early._