Quote:
The other night when playing limit hold ’em at The Bellagio, a young lady managed to get dealt aces two hands in a row. She couldn’t believe her good fortune and wanted to know what the odds of that happening were. Another player spoke up and told her that it was over 40,000-to-1. Of course this is wrong.
I think he was trying to say that two hands never exist in isolation, and the size of the hand series determines the answer. But the answer was not wrong. The only reasonable way to answer that question is with the frequency it will occur on average.
To illustrate this, let's look at some progressions.
If only two hands are played, the odds they will both be AA is ~48840:1
If 100 hands are played, the odds that two consecutive ones are AA is ~493:1
If 34000 hands are played, the odds that two consecutive ones are AA is ~1:1 (50% chance).
If 230,000 hands are played, the probability that two consecutive ones are AA is ~99%.
But the key question is, how often will this event occur
on average? And the answer is exactly the same as the odds of it occuring on the next two hands. It will happen on average once for every ~48,840 hands played.
So I say the answer was precisely right. It is in fact the average frequency that this event will occur.