Poker News

Richard Burke

LindaMae sashayed into my local poker room on a Wednesday afternoon in late summer, signed up for a few games, sat next to me and asked if I had figured out the answer to her question. I had. LindaMae was curious about a hand she played last week. In a ten-handed $4-8 Hold’Em game, she raised with [Ad]-[Kd] in late position. Three-handed they saw this Flop: [Kh]-[Tc]-[5h]. It was checked to her and she bet. The Turn was the [4h]. It was checked to her, and she bet. George raised! She called. The River was the [Qc], making the tableau, [Kh]-[Tc]-[5h]-[4h]-[Qc].

George bet. She thought for a bit and finally made a crying call. George showed down Aa-Ts. LindaMae had almost folded, she said, because George had strongly represented a Flush. With three trumps on the table, she asked, what was the chance that anyone had a Flush.

#Prob.?0.66611?1.31400?2.04440?3.00206?The situation arises often, I told her, because the tableau will have exactly three cards of the same suit about 28% of the time. When you are suited and there are exactly three cards of another suit on the board after all the cards are out, the table shows the probability that zero to three of your opponents each will have been dealt two trumps in a ten-handed Hold’Em game. The table shows that about 31% of the time, one person will have been dealt two trumps, etc. (The odds against four and five opponents each having been dealt two trumps are huge, 38 thousand to 1 and 25 million to 1, respectively. All these results are from a mathematical proof, not a simulation.) The odds are about two to one, a 66.6% chance, that nobody has a Flush.

In practice, your odds that no one will have a Flush will be a tad better than two to one, because many players will muck marginal hands like 7a-3a and 6a-2a pre-Flop, especially if there’s been a raise. On the other hand, with three trumps on the table the chance is almost one-third that someone does have a Flush, and she can’t ignore that possibility. Of course if there were four or more trumps on the table, then she’d likely be toast, I told her, because low-limit players often hang in there with one high trump when there are two trumps on the Flop. If there are four or more trumps on the table, then she should figure onmucking.

Her bets on the Flop and Turn were correct, because she had Top Pair, Top Kicker, and she didn’t want to give anyone another card for free. From his point of view, George also bet correctly because with the Ace of trumps and middle Pair, he had ten+ outs. (If his Ace were live, then he would have had thirteen outs.) His chance of winning the pot was one in four or better, his pot odds were five for one or better, and it was possible that he could win by guile, if he could convince LindaMae to lay down her hand. That’s why he check-raised the Turn and bet out on the River.

It’s rarely easy, I told her, but she had played the hand correctly. Having just been called to a $2-4 Hold’Em game, she nodded, blew me a kiss, and swung into battle.

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