Constantly playing pocket aces: the pocket rocket paradox
How many people must be in a room with you before there's a better-than-50% chance that at least one of them shares your birthday? (Same month and day, not necessarily same year.) Most people would answer "uh... I guess 183 because that's more than half the number of days in a year." It turns out that, according to the so-called "birthday paradox," the answer is actually only 23. The reason is that people forget to include the chance that all of you share the same birthday, or that all but one of you were born on the same day, and so on. Although all those chances seem small, they add up to a lot, and that's why you need so few people in the room before it's likely you'll find at least one.
A fun application of this principle tells us how many online poker tables you must play at the same time to have a greater-than-50% chance of always playing at least one pocket-aces starting hand at any given moment. The chance of being dealt pocket aces is 1:220, or one out of every 221 hands. So you can think of this as each hand being a birthday in a year of 221 days. The math works out to be this:
- 2 tables: 0.45%
- 3 tables: 1.35%
- 4 tables: 2.69%
- 5 tables: 4.45%
- 6 tables: 6.62%
- 7 tables: 9.15%
- 8 tables: 12.03%
- 9 tables: 15.21%
- 10 tables: 18.67%
- 11 tables: 22.35%
- 12 tables: 26.21%
- 13 tables: 30.22%
- 14 tables: 34.32%
- 15 tables: 38.48%
- 16 tables: 42.66%
- 17 tables: 46.81%
- 18 tables: 50.90%
- 19 tables: 54.90%
- 20 tables: 58.78%
- 21 tables: 62.51%
- 22 tables: 66.07%
- 23 tables: 69.45%
- 24 tables: 72.63%
- 25 tables: 75.60%
- 26 tables: 78.36%
- 27 tables: 80.91%
- 28 tables: 83.24%
- 29 tables: 85.36%
- 30 tables: 87.28%
So there you have it: play 18 tables simultaneously, and there will be a 50% chance that you'll be playing pocket aces at any given time. Play 30 tables if you want to be guaranteed* to be doing so.
*"Guaranteed" defined as "better than the chances of aces over the hammer," because we know AA always beats 72o.
