Poker News

Richard Burke

George sat next to me on a sunny Wednesday afternoon in the fall with a question. Had he understood correctly that when there are exactly three trumps on the tableau, the odds are two to one that no one would have been dealt two trumps in a ten-handed Hold’Em game? Yes, I explained that fact in “Three Trumps on the Table,” a column published in Poker Player. What, he asked, are the odds against someone’s having a Straight in a ten-handed game when there are three straight cards on the table.

George had raised pre-Flop with pocket Queens. The Flop, [Qh]-[Js]-[Td], made Top Set for him, and a possible Straight for the enemy. George led the betting. He slowed down when he was raised. Suspecting that the raiser had flopped a Straight, and hoping the dealer would pair the board, he made crying calls on the Turn and River, obtaining this tableau, [Qh]-[Js]-[Td]-[2c]-[3h].

The winner showed [9d]-[8d] and George mucked his Set. Should he have folded sooner, he asked, or what? I told George that it would take me a day or two to reckon it, because a hand with the right two cards from the four Aces, four Kings, four Nines, and four Eights, would make a Straight for the enemy. Three days later, I had made a table of probabilities for anyone having a Straight. 27.3% of the time no one will have any two of the sixteen danger cards in her hand and the probability of a Straight is zero.

The dealer will distribute two danger cards to exactly one opponent 43.1% of the time. While A-K, K-9, and 9-8 would make a Straight, A-A, K-K, 9-9, 8-8, A-9, A-8, and K-8 wouldn’t, so there’s only a 40% chance that two danger cards in a hand would make a Straight. The resultant probability of a Straight for two danger cards in one hand is 0.172.

The dealer will distribute four danger cards to two opponents 23.4% of the time. Some combinations will make a Straight, some won’t. There are 5460 ways, C(16,4)*3!!, to deal four danger cards to those two opponents: excluding Flushes and Straight Flushes, those 5460 possible combinations reduce to 96. I examined each case and determined that when two opponents each had two danger cards, the chance of anyone’s having a Straight was 64.3%. The combined probability for two opponents each having two danger cards is 0.151.

Three or more opponents will each have two danger cards 6.2% of the time. There are 120,120 ways, C(16,6)*5!!, to deal six cards out of sixteen into three hands. Excluding Flushes and Straight Flushes simplifies those ways to 1020 cases. Rather than examine each case, I set the Straight probability to .82. Since the chance of anyone having a Straight must lie between .643 and 1.000, the resultant probability is 0.05 +/-.01. So 63% is the answer, I toldGeorge. When you face a tableau where someone must use both her cards to make a Straight, the odds are about 63 to 37 that no one has it.

Your pot odds were longer than your cards odds for the whole hand, plus you had a redraw for a Full House or Quads, so you played the hand correctly. Thanks, said George, departing for a $4-8 Hold’Em game.

Filed under: Poker News