Poker News

Richard Burke

Fred and I were playing low-limit hold’em nine-handed on a fine Wednesday afternoon in late spring in our local poker room. In this particular hand, we watched the dealer flop these cards, 4d Tc 4h. First to act, Fred bet and two others called. The dealer turned the 3s and Fred bet again. One called; the other folded.

The dealer put out the last card of the deal, the 8h, for a board of 4d Tc 4h 3s 8h. Fred bet again. His opponent raised. Fred called and then mucked his hand after his opponent showed down As 4s.

Musing about that deal, Fred told me he was surprised when his opponent showed the case four. Holding 5c 4c, he had figured the odds against flopping two fours when the other four had been dealt among his eight opponents were about 400-to-1 against. (It’s 391-to-1.) “Was that wrong?” he asked. I told him he erred by using the before-the-flop odds after the flop. However unlikely it was for you to flop trips, it happened. The right question then became, given that you flopped trips, what was the probability that anyone else was dealt the case card? In other words, what was the chance that the last four in the deck was among the 16 cards dealt to his eight opponents? There were 47 unknown cards, so there were 31 cards in the stock and burn. The odds that the case four wasn’t dealt among his opponents were 31-to-16. So there was a 34 percent chance that someone was dealt the case card.

Whether his opponent kept the case card or mucked it, depended on the game’s loose-tightness, the limits, the card’s rank, its kicker, and the individual players’ styles. If the case card isin the “playing zone” of eights and higher, then it’s more likely that someone will keep it. Seven and smaller ranks gravitate to the muck, unless they’re suited and/or connected.

Fred and I were playing in a loose, low-limit game, and the player who won the deal, a regular, rarely folded before the flop anyway. (In my book, Flop, I named this player type “Seymour Floppes,” because he will always see the flop no matter what cards he holds, and regardless of pre-flop raises.) In that game, keeping A-4 suited was typical. After his opponent had smooth-called twice, I told Fred, he should have check-called on the river, saving himself one big bet.

Pondering, Fred asked how often anyone would flop trips. Before you look at your cards, I told him, the chance that the flop will contain exactly two cards of the same rank is 17 percent. The chance that anyone will have been dealt one or more of that same rank is 60.5 percent, for a combined chance of about 10 percent.

Fred asked me how looking at his hole cards could change the flop. I told him that it couldn’t, the flop was set when the dealer cut the cards. Knowing his hole cards changes what we can predict about the flop. With an unpaired starting hand, he asked how often he flop would trips.About 1 time in 74, I answered, as given by 2*C(3,2)*C(44,1)/C(50,3). Fred thought that someone had trips much more often than that when the flop was paired. Yes, because there are eight opponents, it’s eight times more likely that one of them will have trips. That’s why it seems rare for you and commonplace for them.

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