Peeking at your hole cards, you see a beautiful red Ace. Now you hope for another Ace in the hole… Pocket Aces – Aces-in-the-hole – A-A!
Probability and Odds
With four Aces in the 52-card deck, you have one chance out of 13 that the first hole card will be an Ace. Now, with three Aces left among the remaining 51 cards, you can expect a second Ace just one out of 17 times. So the chance of being dealt pocket Aces is: 1/13 x 1/17 = 1/221.
The probability is simply 1/221 = 0.0045 (expressed as a decimal), or 0.45% (expressed as a percentage).
On average, in the long run, you will be dealt pocket Aces only one out of 221 hands. You miss the other 220 times. Thus the odds against being dealt A-A are 220-to-1.
Since A-A is so rare, it would be a shame to waste it by not building a pot for yourself. But be careful; you can lose with Aces in the hole. (Some casinos offer a bonus during certain hours of the day if your pocket Aces are “cracked.”)
Applying Probability Law
Four hundred years ago, Galileo, the famed Italian physicist, astronomer and mathematician, wrote about gambling with dice, leading to the science of probability – the relative frequency “in the long run” of selected outcomes.
A-A in the hole – pocket Aces – is the best possible starting hand. But what is the probability that your pocket Aces will be the best hand at the showdown? The probability that the A-A will win the pot against one opponent with a smaller pair is about 80%. The same holds for most other starting hands with which opponents might pay to see the flop. If two opponents remain in the hand, then the probability of pocket Aces winning the pot, is: 80% x 80% = 64%.
Against three opponents, the probability drops to about 50% (80% x 80% x 80% = 51%) or almost even money. You will lose as often as you win. Would it be best to play against a single opponent, winning 80% of the time? (The odds favor you 4-to-1.)
That makes sense in a no-limit game, where you might extract lots of an opponent’s chips in one hand. However, in limit hold’em, playing against one opponent is far from optimum. The pot will be small – especially after the casino’s rake. In a limit game, considering both pot size and the probability of winning with pocket Aces, three or four opponents are optimum. A smart limit player bets/raises with that goal in mind, hoping to force out several (but not all) opponents against his A-A; but do you always want to raise?
• You’re on the Button with pocket Aces, several opponents stay to see the flop – no raises to you. Just call. Raising would alert your opponents to the strength of your hand. Then, on the flop, everyone likely would check to you and fold when you bet – even more so if you flop a set of Aces… That’s wasting your A-A.
• In a middle position, with two opponents calling to see the flop, raise to force out players behind you – helping toward your goal of 3-4 opponents seeing the flop.
• In an early position, assessment of your opponents should guide your decision as to whether to raise. At a tight table, raising may cause all your opponents to fold – wasting your A-A. At a loose table, raising is preferred to force out a few players so you can better achieve your goal to play against 3-4 players.
Although rare, pocket Aces is the best starting hand. Playing them for best results takes real skill.
Comments? George “The Engineer” Epstein can be contacted at [email protected]