Ace-Magnets

On a busy Friday afternoon in mid-winter in my local poker room, I folded my hand in a ten-handed $4-8 Hold'Em game. Just on my left, LindaMae raised and everyone except the Big Blind folded. The Flop came [Ah]-[6s]-[7d]. The Big Blind bet $4 and LindaMae mucked her pocket Kings. "Damn Ace-magnets!" she swore quietly. While I waited for the next deal, I wondered whether Kings really do attract Aces the way funerals draw politicians. Given that she held pocket Kings, the probability that one or more Aces would flop is (1-C(4,0)*C(46,3)/ C(50,3)), or .2255. So LindaMae was unlucky on that Flop, because three times out of four no Ace would fall: 77.4% of the time the Flop would be Ace-less. Furthermore, if there were no Ace on the Flop, then the probability of no Aces on the Turn or River is given by C(4,0)*C(43,2)/C(47,2)), or .8353. So, if an Ace hadn't flopped, then five times out of six there wouldn't be one on the Turn or River either. Putting those two events together, starting with pocket Kings, the chance is 64.7% that there won't be an Ace on the table after all the cards are out. Pocket Kings are NOT Ace-magnets; it just seems that way.

(With no Ace on the table, it's a little more likely that one or more of her nine opponents would have been dealt pocket Aces. That chance is .0543, about 1 in 19.)

The table shows the probability that there would be exactly 0 through 4 Aces on the tableau. The table also shows the probability that 0 through 4 Aces would be dealt among the players versus the number on the tableau.

The table shows that for exactly one Ace on the tableau, it's a 21% chance that no Aces were dealt among the nine opponents: so, it's a 79% chance that one or more Aces were dealt. Because most play Ace- Any in low-limit Hold'Em games, when there's exactly one Ace on the tableau, four times out of five somebody will have a Pair of Aces or better.

I wouldn't have criticized her for calling one small bet, because her pot odds were larger than her cards odds: counting the house rake, she had pot odds of $24 for $4, better than the about 1 in 5 odds that an Ace wasn't dealt. If the Turn card didn't improve her hand, then counting the rake and bad-beat drop, her pot odds to see the River card would have been $40 for $8, about equal to 1 in 5. A call there wouldn't have been all that bad either.

LindaMae reasoned differently. The Big Blind was a solid player and because of her early-position, pre- Flop raise, he surely put her on a big Ace or big pockets. She reckoned that: a) he wouldn't have bet into her post-Flop unless he had a good hand, e.g., Two Pairs or a Set; or b) he had a weak Ace and was probing to learn if she had a better Ace. Either way, headsup and having only two outs, her Kings were headed for the muck. I couldn't disagree.