Hand ranges and combinations
Having worked through these two examples, you may have noticed something particularly unrealistic in them. In general, we will not know our opponentâ€™s hole cards. Rather, we will assign a hand range to our opponent. To do this, we begin by assuming that any two card combination of hole cards is as likely to be dealt to our opponent. We then begin to rule out hands based on his actions. In examples 1 and 2, if our opponent is very tight, we might be able to narrow our opponentâ€™s range down to AA through TT, AK and AQs based on his early position raise. Of course, in practice, we may not be able to definitively pin down a hand range, and weâ€™ll come back to this point later, but let us ignore this issue for the time being. We might assume that our opponent will bet his entire range on the flop, but that when he bets the turn after we call the flop, that he only has AA, AK, or AQs. If that is the case, how do we decide how to evaluate our options? The basic idea is to compute the EV of each option for each of our opponentâ€™s possible holdings and then compute a weighted average of these individual EVs. The reason that we use a weighted average is that some hands are more likely to be dealt than others. A hand combination is the number of ways that a particular hand can be dealt. For example, there are 6 ways to be dealt pocket 4s. This can be computed either by enumerating all the possible suit combinations or by realizing that there are 4 possible choices for the first suit and 3 choices for the second suit, but this double counts the combinations since the order of the suits does not matter, so there are 4 Ã— 3 / 2 = 6 total combinations. Similarly, there are 16 ways to be dealt K7: 4 possible suits for the king times 4 possible suits for the seven. There will be less combinations possible if some of the cards are accounted for. If you hold Ks Js and the flop comes Kh 7d 4d, then there is only possible combination of pocket kings for your opponent, since he must hold exactly Kd Kc to have pocket kings given what you know.
We will repeat Example 2 and compute the EV of calling a turn bet, assuming now that our opponentâ€™s hand range is AA, AK or AQs. Let us first determine how many combinations of each hand are possible. Since one ace is accounted for on the flop, there are only 3 combinations of pocket aces possible for our opponent (As Ah, As Ad, Ah Ad). Similarly, there are 12 combinations of AK possible and 3 combinations of AQs possible. Next, observe that our EV calculation in Example 2 applies to each of the AK combinations, not just Ad Ks. Additionally, the EV calculation will be the same for AQs. Thus for 15 combinations, a turn call will have an EV of 0.274 bets. However, we must do a separate calculation for the 3 combinations of AA. You did this in Exercise 1 and should have found that the EV of the turn call in that case was -1.364 bets. The EV of a turn call versus this hand range is therefore equal to [(number of combinations of AK and AQs) x (EV of a turn call versus AK or AQs) + (number of combinations of AA) x (EV of a turn call versus AA) ] / (total number of combinations) = [(15) x (0.274) + (3) x (-1.364)] / 18 = 0.001 bets. This is about as close to a neutral EV decision as one can come up with. Intuitively, this should make some sense. You are very likely to be up against a hand where you have a positive EV turn call (15 out of the 18 combinations), but the 3 combinations where you are drawing dead has a much higher negative EV. In this particular calculation, those two factors balance out almost exactly evenly.
Repeat Example 3 under the assumptions that our opponent (a) has a preflop range of AA through 88, AK, AQ, AJs, KQs and his range for betting the flop and turn is AA, 99, AK, AQ, AJs, (b) bets his entire original preflop range (AA â€“ TT, AK, AQs) on flop, turn and river and always calls a river raise.
To finish up, let us return briefly to the fact that hand ranges often are not known precisely. We can handle this uncertainty by assigning weighted hand ranges. For example, suppose that we think an opponent could only have AA or AK in a certain situation. We are certain that if he had AA, then he would take the line he has taken. But if he had AK, then we think that there is only a 30% chance that he would play his hand this way. This could an estimate based on our uncertainty (meaning that he either will or will not play AK this way 100% of the time, and the 30% represents a quantification of our certainty that he will play his hand this way) or it could be based on a detailed read that our opponent mixes up his play (so that 30% of the time he has AK he will play it this way, and he will take a different line the other 70% of the time he has AK). While we would still count 6 combinations of AA, we weight the 16 combinations of AK by the 0.30 weighting factor, so we only count 4.8 combinations. (Donâ€™t worry that the combinations are not whole numbers; the math still works the same.) Given these assumptions, the likelihood that our opponent has AA is 6 / (6 + 4.8) = 0.56 and similarly there is a 44% chance he has AK. If there was no weighting, he would have been much more likely to have AK than AA, but since he doesnâ€™t always AK this way, the likelihood that he has aces goes up.