Implied Odds in Holdem

Real Texas Holdem is played with hands facedown, though. If you have pocket jacks and all you know from your opponent’s reraise is that he has a "good" poker hand, but not necessarily pocket aces, then your situation is weaker.

For instance, we said before that about one-ninth of the time you’ll flop a jack without him flopping an ace. But about one-one-hundredth of the time, you’ll both flop a set, and if you don’t know he has pocket aces, you’ll almost certainly get stacked. Now your EV for a $65 reraise is -$4.80.

$4.80 = (0.88)(- $65) + (0.11)($520) + (0.0l)(- $480)

The remote threat of getting stacked if you both flop a set costs you almost an extra $5 a poker hand. That means that your opponent with secret pocket aces doesn’t have to raise as much to keep your call from being profitable.

Also, if your opponent sometimes will reraise preflop with smaller pairs or unpaired hands like ace-king (as almost everyone will), then your implied odds situation is much weaker still. Sometimes when you hit your set, you won’t get your opponent’s stack. Meanwhile, when you don’t hit your set, the raiser will get you to lay down what turns out to be the best hand. In this situation, the most typical in real Texas Holdem play, the raiser needs to raise less still to avoid offering you too high implied odds.

Also in this situation, the player with aces need not reraise so much because he has to balance out the threat of giving a pair the right implied odds with the risk of scaring out hands like ace-king, ace-queen, or king-queen.

So, putting you in the raiser’s shoes for a moment, the more your opponent can narrow down the range of poker hands you might have, the larger of a percentage of your total stack you must bet immediately to avoid rewarding your opponent’s ability to take advantage of his knowledge.

Again, when the stacks are deep, knowing when to get away from your poker hand will allow you to make smaller bets. Keep reading hands.

The more your opponents know about the exact nature of your hand in Holdem, the more you have to bet immediately to avoid offering them too high implied odds.

Here’s another example. Say you are playing $5-$ 10 blind no limit Texas Holdem with $2,000 stacks. Someone raises to $30 in early position, and you call on the button with 6♦ 5♦. Both blinds fold ($75 in the pot and $1,970 left to bet). The flop comes J♣ 7♣ 3♠, giving you a gutshot. Your opponent bets $100. At that point, he tells you (he’s not lying) that he has a set of jacks. Furthermore, he promises to bet all the rest of his money on the turn no matter what comes.

If you call and miss your straight, you lose $100. If you call and make your straight, you still have to dodge a board pair on the river to win. Ten cards out of forty-four pair the board, so 23 percent of the time (10/44) you’ll lose $1,970, and 77 percent of the time (34/44) you’ll win $1,970 + $75 = $2,045. So your expected win if you catch your straight on the turn is $1,132.50.

$1,132.50 = (34/44)($2,045) – (10/44)($1,970)

Thus, your Holdem implied odds are $l,132.50-to-$100 or about 11.3-to-1. Your chance to catch your straight is 4/45 or about 10.25-to-1. Since your implied odds are greater than your chance to make the straight, you should call.

Let’s say that the preflop action and flop cards are the same, but now your opponent tells you (again, he’s not lying) that he either has a set of jacks or A♣ K♣ (for a big flush draw). And again, he promises to bet all-in on the turn no matter what happens.

First, there are three ways to have a set of jacks (J♥ J♦, J♠ J♦, J♠ J♥), while only one way to have A♣ K♣. So there is a 75 percent chance he has the set and only a 25 percent chance of the flush draw.

Knowing that, you have to fold to the all-in turn bet if you miss, even if you catch a pair, since even though you are now ahead of the flush draw hand, he’s three times more likely to have the set and have you still drawing to the gutshot.

So again, you lose $100 if you call and miss your straight. But if you call and catch your straight, the math changes.

If you call and catch the 4♦, 4♥, or 4♠, then you are the favorite over both the set and flush draw. The set has a 23 percent chance to outdraw you (10/44), and the flush draw has a 20 percent chance (9/44), so weighting the jacks at three times the likelihood, he has on average about a 22 percent chance to outdraw you.

0.22 = (1/44)[(10)(3)+(9)(1)/4]

If you catch the 4♣, then 75 percent of the time, you are ahead and will be outdrawn 23 percent of the time (10/44). The other 25 percent of the time, you are drawing dead, the equivalent of being outdrawn 100 percent of the time. So he has about a 42 percent chance to outdraw you.

0.42 = (1/44)[(10)(3)+(44)(1)/4]

That is, three times out of four, you’ll have a 22 percent chance (9.75/44) of being outdrawn. And one time out of four, you’ll have a 42 percent chance (18.5/44). Together, this represents an aggregate 27 percent chance to be outdrawn.

0.27 = (1/44)[(9,75)(3)+(18,5)(1)/4]

Note that your chance to be outdrawn when your opponent always had the jacks was 23 percent. So things are now worse for you. How much worse exactly? Well, now your expected win is approximately $970.

$970 = (32,06/44)($2.065)-(11.94/44)($1,970)

Thus, your implied odds are $970-to-$100 or 9.7-to-l. Your chance to catch the straight in Texas Holdem poker is still 10.25-to-l, so now you can’t call profitably. Adding a little uncertainty about your opponent’s poker hand turned a profitable call into an unprofitable one.

Put another way, your opponent made a mistake by betting only $100 when he told you he had exactly jacks. He offered you too high implied odds. With some uncertainty about his holding, though, his $100 bet is enough to make you fold.

While you will rarely play a Texas Holdem poker game where you will know your opponents’ hands with such certainty as in this example, the general principle applies: The more accurately your opponents can read your poker hand, the more you have to bet in proportion to your stack to prevent them from calling profitably. The more accurately you can read your opponents’ possible holdings, the larger the bets you can call to try to bust (or bluff) them.

Also recommended:

1. Implied and Effective Odds Explained
2. The Importance of Implied Odds in Texas Holdem
3. Quantifying the Value of Information
4. Another Opportunity to Gather Information
5. Paying for Information in Texas Holdem