Gary Dee smiled politely and took a break from his ten-handed $4-8 Hold'Em game after his Top Two Pairs lost to a Straight. He was, he explained, going to take a walk outside and think pleasant thoughts. Upon his return he headed in my direction and told me that he had held A-Q; the dealer flopped A-Q-T and then put blanks on the Turn and River. After some spirited betting, his remaining opponent tabled K-J and pulled a nice pot. "What were the chances that anyone had a Straight with that tableau?" he asked.
Sequences on the tableau that have two gaps can make a Straight only one way. For examples, A-Q-T-x-x, A-J-T-x-x, 5-3-A-x-x, and 5-4-A-x-x can make Straights only with K-J, K-Q, 4-2, and 3-2.
The tableau may have a Pair or it may be suited such that the Straight may not win, but we'll ignore that. We'll also ignore the possibility that someone mucked her 4-2 start and then watched the dealer place 5-3-Ax- x on the table. We will concern ourselves only with the event that anyone was dealt two cards that would make a Straight after all the cards are on the table.
For a one-way Straight, the starting hands must be a zero-gap connector like K-Q that would make a Straight with A-J-T-x-x, or one-gap connectors like K-J or 5-3 that would make a Straight with A-Q-T-x-x or 6-4-2-x-x.
The 18 cards in enemy hands can include zero to eight of the cards that could make a Straight. Given that you have two cards that don't affect those odds, the probabilities are shown in the table.
Nine times out of ten, two or more of the eight cards that could make a Straight will have been dealt into enemy hands. Although it's quite likely that two or more danger cards are in enemy hands, they might not make a Straight: for example, the cards might be scattered as singletons among the nine players. A Straight is possible only when one or more enemies have two of the eight cards in their starting hands.
The next table shows the probabilities that 0 to 4 enemies have doubletons that might make a Straight. Our first result is that in a ten-handed game, the chance that one or more of the nine enemies have any two cards from the two ranks that might make a Straight is 21.9%.
Just because they have two cards from the two ranks that could make a Straight doesn't automatically mean that they have one. The doubletons K-K and J-J wouldn't make a Straight with a two-gap sequence like A-Q-T-x-x; only the doubleton K-J would. Of the 28 ways, C(8,2), to deal two cards from two ranks, there are 16 ways to deal one from each rank. Multiplying obtains the probability that anyone who was dealt a doubleton would make a Straight. Similar computations for 2, 3, and 4 doubletons result in a probability of .12842, once in about 7.8 times.
Gary Dee was a bit unlucky, but that's low-limit Hold'Em in any poker room, real or virtual, in the world.
