Am I Getting My Fair Share? (Dumb Question)

[Onaflag]

I did try to search for the answer, found nothing, and decided it was because it is so ridiculously simple. The "simple" things in life often throw me. Just ask my wife. [img]/images/graemlins/blush.gif[/img]

Let's say I have 15,000 hands in my database. If there are 169 possible starting card combinations, should I have (on average) been dealt a certain pair of cards 88.75 times?

So, if it shows 40 AA or 38 KK or something similar, can I safely assume that I haven't seen my fair share of AA (or whatever) and expect it to average out in a certain number of hands.

Reminds me of Sklansky's coin flip for $1. Sure, I may lose the first 10 tries, but it will all average out in the end. So, how far is the end and when do I get my fair share? [img]/images/graemlins/grin.gif[/img]

Onaflag...........

[DougOzzzz]

The possible starting hands are not all equal in frequency. Any pair occurs 1 in 221 times, not 169. By my calculations any offsuited combination occurs about 1 in 108 hands, while any suited combination occurs 1 in 352 hands. So after 15,000 hands, AA should come up about 68 times.

[uuDevil]

[ QUOTE ]
The possible starting hands are not all equal in frequency. Any pair occurs 1 in 221 times, not 169. By my calculations any offsuited combination occurs about 1 in 108 hands, while any suited combination occurs 1 in 352 hands. So after 15,000 hands, AA should come up about 68 times.

[/ QUOTE ]
I think a couple of these numbers are slightly off.

There 4 ways to be dealt suited cards:
P(suited hand)=4/C(52,2)=1/109.5 or 108.5:1

There are 12 ways to be dealt an unsuited hand:
P(unsuited hand)=12/C(52,2)=1/331.5 or 330.5:1

For the OP, the NUMBER of AA hands you are dealt will not necessarily "average out" after some number of hands. It is the PROPORTION of AA hands that will approach it's expected value. So after 15,000 hands if you have gotten AA 53 times (15 less AA hands than you "should" have), then after 150,000 hands you may end with 600 AA hands (80 fewer than expected). You have now been shorted more AA hands, but the proportion of AA you received has actually gotten closer to 1/221 (.004525). The proportion has gone from 53/15000 (.003533) to 600/150000 (.004000).

[DougOzzzz]

Oops. I see where I went wrong. I calculated the probability of getting any non-pair first... then calculated the probability that ANY two cards are suited or off suited. Thus, offsuited hands got a little more likely and suited hands got a little more likely, since all pairs of course are off suit.