by Richard G. Burke
In our last column, responding to Larry Duplessis’s question, we found that ace-ten led or tied at a ten-handed table 62.7 percent of the time prior to the flop. In this column we find ace-ten improving as the number of players decrease from ten to two.
Each deck has twenty Broadway cards: four each aces, kings, queens, jacks, and tens. Given that you hold ace-ten, we have 153 doubletons possible from 18 remaining Broadway cards, C(18,2). If the dealer happened to pitch one or more of those danger doubletons to an opponent then, as the table shows, 60 of them dominate ace-ten, and ace-ten leads or ties the other 93. (If you have three or fewer outs if you both whiffed the flop, then we call your hand “dominated” by having a win probability of 30 percent or less.) However, your ace-ten leads over the 84 doubletons of K-Q, K-J, K-T, Q-J, Q-T, and J-T; your ace-ten ties with the other 9 possible ace-tens.
When you play at a shorthanded table, the chance that the dealer pitches exactly one danger doubleton generally decreases as the number of players decrease, as shown.
Even when an opponent has a Broadway doubleton, we still have a 93/153 probability, about a 60 percent chance, that our ace-ten leads or ties her before the flop. We arrived at the table below by multiplying the probability of the number of danger cards dealt times the probability that they lay as doubletons in enemy hands times the probability that your ace-ten led or tied them, and added.
We saw in the previous column that ace-ten leads or ties 62.7 percent of the time against nine opponents. We see from the graphic that the lead or tie probability increases monotonically as the number of other players decrease. Heads-up, ace-ten leads or ties 95 percent of the time.
We deliberately avoided considering suited vs. unsuited aceten. We prefer suited, particularly in limit hold’em. We used PokerStoveTM to perform a Monte Carlo analysis of ace-ten: the suited version against a random hand prevailed 64.6 percent of the time; ace-ten unsuited prevailed 62.8 percent of the time. Suited ace-ten looks less than 2 percent better with all the cards out.
Of course you already knew that as the number of players decrease, the number of playable hands increase. Now you know how much better.
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