So, yesterday I was bored and decided to conduct a little experiment. I shuffled the deck and drew 2 cards. I did this 100 times.

It took me 50 hands to get QQ and 73 to get AA. If you play uber tight, that's a boring session.

All toll I got about 15 playable hands.

Here's an interesting thing I also discovered. If you play any Ace you get, it could degrade your game by as much as 10%.

Add in any suited cards with that, and you're looking at a 15% leak.

Try it out, it's a fun experiment.

You done good: The odds of being dealt AA are 220 to 1
Of being dealt QQ or higher 72 to 1.

What are you considering playable hands? Hands you can play from any position?

How boring. If you try again, those numbers can change and instead of 73 it could become 5th or 2782th.




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This is the kind of thing computers are perfect for. An old laptop could do this experiment with 10 million hands in a few seconds. And its answers would match those predicted by mathematics (barring a miracle).

100 hands gives you a 36.46% chance of getting AA at least once. You have a 50% chance with 153 hands, so 153 is the median. The average is 221 which gives you a 63.3% chance.

In general, performing the average number of trials gives you a probability of approximately (e-1)/e which is 63.212%. That's the Poisson approximation and its accuracy increases as N approaches infinity. (So for instance if winning the lottery is 1 in 200 million, then if you play it 200 million times you have a 63.212% chance of winning at least once in that span.)