Optimal Boards: Part II

Brian Alspach

Poker Digest, Vol. 5, No. 11, May 17 - 30, 2002

We are in the midst of examining the question: Which two-card player hand has the best chance of being the nuts after all five board cards are dealt? Let me remind you we are using the term ``absolute nuts'' to mean the player has a winning hand that no other player may even tie; ``relative nuts'' to mean the player cannot lose, but other players may tie; ``the board plays'' to mean all players left in the hand tie; and ``the hand is vulnerable'' to mean the board produces a situation for which there is a potential for another player to win.

In Part 1 we restricted ourselves to the player hand A-A and considered all 1,067,650 boards with a pair or better. The tally for the boards considered in Part 1 is as follows: 16,054 boards give the A-A hand the absolute nuts; 1,128 boards give the A-A hand the relative nuts; 326 boards play; and 1,050,142 boards leave A-A vulnerable.

For simplicity recall we are assuming the A-A hand is the red aces. We left the 1,051,110 high card boards to this article because I want to show you a nice trick for deciding how the A-A hand fares for these boards.

A high-card board has five distinct ranks on board (otherwise it would be better than high card) and we refer to this as the rank set. There are $C(13,5) = 1,287$ ways of choosing five ranks from 13, but 10 of the choices correspond to straights. Thus, there are 1,277 possible rank sets for a high card board.

It is easy to write out the rank sets not allowing a straight. There are precisely 80 of them. Of these 80 rank sets, 31 have an ace and 49 do not. Performing the same kinds of considerations we did in Part I, it is easy to determine that each of the 31 rank sets with an ace gives rise to 510 high-card boards, where 210 of the boards allow a flush and 300 boards do not allow a flush. The 300 boards not allowing a flush make the A-A hand the absolute nuts since the player has trip aces, and neither straights nor flushes are possible.

Of the 210 boards allowing a flush, exactly four of them have four suited red cards. Thus, 206 flushing boards make the A-A hand vulnerable, and four of them give A-A the absolute nuts.

For the 49 rank sets not allowing a straight and with no ace, there are 1,020 boards of which 420 allow a flush. From the 420 flushing boards, there are 30 having four suited red cards. So each rank set gives us 30 boards making A-A the absolute nuts, and 990 boards making A-A vulnerable.

The rest of the 1,197 rank sets allow straights. Exactly seven of the rank sets have the form x-10-J-Q-K because $x$ may be any of 2,3,...,8. For each of these seven rank sets, there are 420 boards allowing a flush, and 600 boards not allowing a flush. For the latter 600 boards, A-A is the relative nuts because any player holding an ace has Broadway.

Of the 420 flushing boards, exactly six of them give A-A a royal flush. The remaining 414 flushing boards leave A-A vulnerable.

This leaves 1,190 rank sets. Exactly 462 have an ace and any board for this rank set has a black ace. The only way A-A can be the absolute nuts is if the other four cards are all hearts or all diamonds, and they do not allow a straight flush. There are only four such boards for the rank set anyway. It turns out that 156 of the rank sets do not allow the straight flush when the four non-ace ranks are suited and red. Thus, we get $4\cdot 156 = 624$ boards making A-A the absolute nuts. All the other boards for these 462 rank sets make the A-A vulnerable.

We are down to 728 high card rank sets allowing straights and not containing an ace. This is the ugliest of all the subcases and where we employ the nice trick.

Why is this the ugliest subcase? Each rank set produces 1,020 boards. The only way A-A is the absolute nuts is for the board to contain four suited red cards, and NOT allow a straight flush! Each rank set has 30 boards with four suited red cards. The source of the problem is that for a rank set such as 3-4-5-6-8, all 30 boards with four suited red cards allow a straight flush, whereas, for 3-4-7-10-Q, only 12 of the boards with four suited red cards allow a straight flush.

At first glance then, it seems we must examine all 728 rank sets individually to see how many straight flushes they allow when a red ace has the potential of being the nut flush. This would be tedious, lengthy, and prone to errors.

But wait a second! There is another way. Notice that whenever a set of four ranks allows a straight, such as 3-4-5-6 above, it means a contribution of six straight flushes to the set of five ranks 3-4-5-6-x. This follows because there are two choices for the red suit for 3-4-5-6, and three choices for the suit of x. So all we need to do is count how many times 3-4-5-6 occurs in sets of five ranks and multiply by six. For example, in this case, x can take on any of six ranks because it cannot be A, 2 or 7. This means 3-4-5-6 accounts for 36 boards that allow straight flushes when they are either all diamonds or all hearts.

So we can count potential straight flush boards using sets of four ranks instead of sets of five ranks. This is a much easier problem to handle. There are many fewer sets of four ranks allowing straights, and we always are multiplying by the same number reducing the chance of errors. Doing this leads to 13,494 boards of the $30\cdot 728 = 21,840$ boards allowing a diamond or heart flush that also allow straight flushes. So 21,840 - 13,494 = 8,346 boards give the A-A the absolute nuts, and all other boards allow either a straight flush or a straight beating the best hand A-A can make. (Recall that we removed Broadway hands earlier.)

Adding all the numbers now gives 35,960 boards with A-A the absolute nuts, 5,328 boards with A-A as the relative nuts, 326 boards for which the board plays, and 2,077,146 boards leaving A-A vulnerable.

This means the probability of a board for which A-A cannot lose is precisely .0196, that is, slightly less than one out of 50 boards.

What about pairs of smaller ranks? I may work out the numbers some time in the future, but certainly not now. I suspect most people's intuition leads them to believe pairs of smaller ranks do not fare as well. I took a quick look at pocket kings and believe their numbers compare better than you might think.

For example, with red aces, we saw in Part 1 that for the 4,122 possible flushes on board, 110 boards make A-A the absolute nuts and 4,014 leave them vulnerable. For red kings, 108 flushes on board make them the absolute nuts, while 4,016 leave them vulnerable. They don't give up much here. On the other hand, there are other categories of boards for which pocket kings give up a lot of ground compared to pocket aces. Still, I suspect pocket kings don't do so badly. As you move to smaller pairs, they will diminish more rapidly because one needs stronger restrictions on the board to make them either the absolute or relative nuts.

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