Optimal Boards: Part I

Brian Alspach

Poker Digest, Vol. 5, No. 10, May 3 - 16. 2002

Which two-card player hand has the best chance of being the nuts after all five board cards are dealt? Let's be specific and consider pocket aces. The suits of the aces don't matter, but in order to simplify the discussion, let's assume the player hand in question is the two red aces.

The number of boards is $C(50,5) = 2,118,760$. They break down as follows: straight flushes (36); quads (552); full houses (3,216); flushes (4,122); straights (9,180); trips (46,464); two pairs (102,960); one pair (901,120); and high card (1,051,110).

There are four different possible outcomes relative to the hand being the nuts. The pocket aces, together with the board, may end up giving the player a winning hand that no other player may even tie. In this case, we shall say the board produces the ``absolute nuts.''

The player may end in a situation that A-A cannot lose, but anyone else holding an ace ties the player. We'll say the player has the ``relative nuts'' in this case.

The board may produce a situation for which all players left in the hand tie. We'll say the ``board plays'' in this case.

Finally, all too often, the board produces a situation for which there is a potential for another player to beat the A-A hand. We'll say the A-A is ``vulnerable'' in this case.

We work through the 2,118,760 possible boards and see what they do to the red aces. I don't include all the details, but provide an outline for anyone who wants to have a little fun with this question

Look at the 36 straight flushes first. Two of them are the 9-10-J-Q-K of diamonds or hearts. Our player has the absolute nuts for those two boards. For the two royal flushes in clubs and spades, the board plays. Finally, for the remaining 32 straight flushes, the A-A is vulnerable.

Of the 552 quads, 24 have an ace on board. In this case, the board plays. The remaining 528 quads do not have an ace on board. Thus, any player holding an ace ties for the pot. So A-A is the relative nuts for these 528 boards.

The 3,216 full houses partition into 48 having A-A as the pair, and 3,168 not having an ace on board. Our player holding A-A clearly has the absolute nuts for the 48 full houses whose pair is A-A, but the A-A player hand is vulnerable for all remaining boards because another player might have quads.

Once we reach flushes, things begin to get complicated. Of the 4,122 flushes, 2,554 are in clubs or spades and our A-A hand is vulnerable.

There are 784 heart flushes and the same number of diamond flushes. There are two ways a heart flush on board makes the ace of hearts the absolute nuts. One way is for the board to not allow a straight flush. Then an ace high heart flush is the only winner. On the other hand, if the board allows a straight flush and has the form x-10-J-Q-K of hearts, where x is any of deuce through eight, the ace of hearts again is the absolute nuts.

It is fairly easy to verify there are 48 sets of five ranks chosen from $\{2,3,\ldots,K\}$ (remember the ace is in the player's hand) that do not allow a straight. This means there are 736 sets of five ranks that do allow a straight, but seven of them have the form x-10-J-Q-K.

Thus, 729 heart flushes on board are vulnerable because they allow a straight flush. We have only 55 heart flushes on board for which the ace of hearts is the absolute nuts. So the 4,122 flushes split into 110 giving the A-A hand the absolute nuts, and 4,012 for which the A-A is vulnerable.

There are 7,650 straights whose smallest card has rank ace through eight. All of them make A-A vulnerable. Note that if the straight has four hearts or four diamonds, making red aces look fairly strong, a straight flush is then possible.

There are 1,020 straights with ranks 9-10-J-Q-K. There are interesting possibilities for them relative to red aces. There are six of them that give the red aces a royal flush, that is, the absolute nuts. There are another 414 of them having three or four suited cards, but all of these make A-A vulnerable because of straight flush possibilities.

There are 600 straights with ranks 9-10-J-Q-K not allowing flushes. For these 600 straights, A-A becomes the relative nuts.

The 510 straights with ranks 10-J-Q-K-A also are interesting. For these the board plays unless three or four of the cards are suited. Since only two aces are available, counting those allowing flushes is a little trickier. It turns out that 210 of the straights allow flushes. These make A-A vulnerable except for the four making royal flushes for A-A. For the remaining 300 straights, the board plays.

Therefore, of the 9,180 straights, 10 give A-A the absolute nuts, 600 give A-A the relative nuts, 300 lead to the board playing, and 8,270 lead to A-A being vulnerable.

All 46,464 three-of-a-kind boards make A-A vulnerable.

The 102,960 two-pair boards are interesting. First, 3,168 of these boards have the case aces on board. Thus, the player has quad aces, and this is the absolute nuts unless the board allows a straight flush. This happens only if the other two ranks involved is one of the couples 2-3, 2-4, 2-5, 3-4, 3-5, 4-5, or the corresponding couples of ranks from 10 through king. Counting the boards with these other ranks having two-pair and three suited cards, we find 144 such boards. Subtracting from 3,168 leaves 3,024 two-pair boards with A-A being the absolute nuts. All the other two-pair boards leave A-A vulnerable.

The 901,120 boards with a single pair break down as follows: 14,080 have a pair of aces, 126,720 have a pair with an ace kicker, and 760,320 have no ace on board.

When there is a pair of aces on board, the player has the absolute nuts unless the board allows a straight flush. For this part, I did an exact count and found that of the 14,080 boards with aces on board, 1,244 allow a straight flush and the remaining 12,836 do not.

All of the 126,720 boards with a pair and an ace kicker leave the A-A vulnerable because another player may have quads, and the best our player can do is a full house.

Finally, the 760,320 single pair boards with no ace leave A-A vulnerable unless our player makes a royal flush. Only 24 of the boards allow that. The pair boards then partition into 12,860 giving A-A the absolute nuts, and 888,260 vulnerable boards.

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