Honor Math

Playing limit hold'em, you are in middle position and pick up pocket sevens. A "super rock" who practically never raises before the flop raises in front of you. Since it is quite likely that he has a high pocket pair, you toss away your pocket sevens – no problem.

Even if your opponent is merely a "rock" (as opposed to a "super rock"), for example, and raises before the flop only with Group 1 and Group 2 hands (see the matrix below), he will have a pocket pair higher than your sevens about 40 percent of the time, plus the equity of matching one of his overcards on the flop (about one-third of the time) and your probably choosing to fold your sevens whenever there is an ace, king, or queen on the flop (see the chart below), which occurs about half of the time.

But if your opponent is a loose-aggressive hold'em player who frequently raises before the flop with lesser hands (Group 3 and worse), you want to play – most likely by reraising, in an attempt to get heads up with the raiser.

So, let's say that you reraised before the flop, everyone but the raiser folded, and the flop was two low cards and one high card (an "honor" card, 10 or higher). What is the likelihood that he has a higher pair than yours at this point? It is very important for you to know where you stand.

It depends on which high card is in the flop. If the honor card in the flop is an ace, there are more hands containing an ace than containing a 10 that he would raise with before the flop. And it also depends on whether the raiser typically raises with only the better Group 3 hands or perhaps raises with some or many of the Group 4 hands (as is the case with a late-position raiser).

The following chart gives the percentage likelihood that your pocket sevens are beat after the flop by three types of raisers, either because the raiser has a higher pocket pair or because the honor card in the flop matched one of the two cards in his hand.

Honor Card in the Flop Group 2 Raiser Group 3 Raiser Group 4 Raiser

Ace 69% 66% 45%

King 69% 50% 45%

Queen 64% 43% 38%

Jack 42% 43% 39%

10 35% 33% 39%

Note that against a rock, who raises before the flop only with a Group 2 or better hand, when an ace, king, or queen is in the flop, you are beat approximately two-thirds of the time; hence, you should usually fold his bet.

The following hold'em matrix, which has been revised many times over the past 10 years, puts all hold'em hands into seven groups, 1-7 (1 is best; 7 is worst). Using the standard math notation of row first and column second, in order to get the group number of any hold'em hand, look up the higher card first for two unsuited cards and the lower card first for two suited cards. For example, A-Q (row ace, column queen) is an A-Q of different suits and is in Group 2; but Q-A (row queen, column ace) is for A-Q suited – which is in Group 1.

2 3 4 5 6 7 8 9 10 J Q K A

2 4 6 7 7 7 7 7 7 5 5 5 4 3

3 7 4 5 6 6 7 7 7 5 5 5 4 3

4 7 7 4 4 5 6 6 7 5 5 5 4 3

5 7 7 6 4 4 5 5 6 5 5 5 4 3

6 7 7 7 6 4 4 5 5 5 5 5 4 3

7 7 7 7 6 6 3 4 4 5 5 5 4 3

8 7 7 7 7 6 6 3 4 4 4 5 4 3

9 7 7 7 7 6 6 5 3 3 4 4 4 3

10 7 7 7 7 7 6 5 5 2 2 3 3 3

J 7 7 7 7 7 6 5 5 4 2 3 2 2

Q 7 7 7 7 7 7 6 5 4 4 1 2 1

K 7 7 7 7 7 7 7 6 4 3 2 1 1

A 5 5 5 5 5 5 5 5 4 3 2 1 1

Thus, Group 1 consists of pocket aces, kings, and queens, A-K, and A-Q suited. Group 2 consists of pocket jacks and tens, A-Q unsuited, K-Q, A-J suited, K-J suited, and J-10 suited (but I have reservations about J-10 suited).

The frequency of picking up a Group 1 hand (there are 38 different combinations, 6+6+6+16+4) is a little less than 3 percent of the time (there are 1,326 hold'em hands). There are 42 Group 2 hands – thus, a little more than 3 percent frequency. There are 94 Group 3 hands, about 7 percent frequency. So, you pick up a Group 3 or better hand about 13 percent of the time.