Bad beat cross/post

[Slacker13]

I posted my bad beat story in the Internet forum and would like to see if anyone can figure out the probability of this situation.
First I include the original text from the post...

[ QUOTE ]
Heads up play against my 8 yr old daughter in a game of go fish. She is first to act, one by one she ask's me for a card until she had all my cards. We actually had the exact same cards in our hands. She takes me out and i didn't even get to act.

[/ QUOTE ]

What are the odds of two people starting with 10 cards and both having identical hands? My math skills are not good enough to even attempt this problem.

[elitegimp]

The total number of ways to deal the cards to your daughter:
50 choose 10 = 1*10^10 (huge number... not rounded in future calculations)

Number of ways to deal one player 10 unique cards:
(13 choose 10)*4^10 = 3*10^8

So your daughter has 10 unpaired cards 3*10^8/(1*10^10) of the time, or about 3% of the time (2.919% using unrounded numbers).

When this happens, there are 42 cards left. So there are 42 choose 10 = 8.5*10^8 ways to deal you a hand. Of those, you will get 1 each of the 3 remaining cards that show up in your daughter's hand 3^10 times, or 59049 times. Hence this happens 59049/(8.5*10^8), or 0.00697% of the time.

Now we just multiply:
Probability your daughter gets 10 unique cards and you get one of each of them is 3*10^8/(1*10^10) * 59049/(8.5*10^8), or 0.0002034% (about 491710:1).

This excludes other ways for her to win before you go --
She could be dealt 5 pairs, she could be dealt 4 pairs and you could have at least one of each of her other two cards, etc.

Someone lemme know if I'm double counting anything, I don't think I am [img]/images/graemlins/smile.gif[/img]

PS: this is the best bad-beat story I've ever heard!

[Slacker13]

Wow, very nice. Thanks for working this out.

[MickeyHoldem]

I think your thought process is good... but when I play Go Fish, we use a 52 card deck, and some of you calculations are off.

(13c10 * 4^10 / 52c10) * (3^10 / 42c10) = .00000076072539474
or about 1 in 1314535