On a chilly Friday in mid-winter Fred ranted about pocket rockets. "I'll never raise with Aces again," he swore. "I should just muck 'em in the first place, because they always get cracked!"
On the button in a $2-4 Hold'Em game, Fred raised with Aa-Ad. Eight called. The Flop was [Jc]-[Js]-[9c]. After they all checked, Fred bet. Three called. The Turn card was the [3h]. Fred bet; the Big Blind raised; Fred called. The River was a blank; the Big Blind bet; Fred called. Fred mucked his Aces after the Big Blind tabled [Kh]-[Jh].
It's about 50-50 to have a pair or better on the tableau after all the cards are out, I told Fred. And the higher the pair's rank the more likely it is that someone will crack your Aces with Trips or better. Even with an unpaired tableau, someone could have a Straight, a Flush, even a Straight Flush. Fred said he knew that!
Without resorting to simulation, I told Fred to consider a special tableau such that no ranks are duplicated and neither a Flush nor a Straight are possible, something like [Kh]-[Qd]-[8s]- [7s]-[3c]. These are special cases, I told Fred, happening less than 37% of the time. For special cases like that, the enemy can crack your Aces only with Two Pairs or a Set. I had done the math and developed the table below.
The table shows for these special cases where only Two Pairs or a Set could beat A-A, that with one opponent, Aces will win nine times out of ten. With ten opponents, Aces will win one time in three. Generally, each additional opponent will lower your chance of winning about 7%.
The next graphic shows that your expectation for these special tableaux increases with each opponent up to seven, and then decreases slightly. Expectation is your profit on these hands, I told Fred, and it would always be positive in these special cases. Even though your chance of winning decreases, the additional chips from having more players more than makes up for your diminished chances. These findings suggest some pre- Flop tactics when you have pocket Aces. In early position you would raise in order to induce players with marginal hands to fold, thus raising your win probability. In late position and in the blinds, raising probably won't cause many to fold: you would raise in those positions to build a bigger pot, hoping that your Aces will prevail. In middle position, you would raise to do both. Limping is the WORST. That lets the opponents with marginal hands see the Flop cheaply, and the ensuing pot will be smaller. I told Fred his pre-Flop choice with pocket Aces is either to raise and re-raise in any position, or to muck. And, why would he ever muck the best starting hand in Hold'Em? Because the tableau will often be suited, paired, and/or sequenced, the general win probabilities for Aces will be lower than those shown. You can't know in advance that the tableau won't be benign, so the principle applies, I told him, raise with A-A, anytime, anywhere you can!
As he headed toward his Hold'Em table, I told him just to ensure the tableau isn't paired, sequenced, or suited. He heard my little joke, but I'm not sure he heeded it.