BrainDen.com - Brain Teasers
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## Question

Coop is our weekly friendly poker games resident card counter and odds calculator. Fortunately, his poker face reads like a book and he can't keep an ash on his stogie when bluffing. As of late prop bets have been in favor at our game and last week Coop offered this one: "For five bucks I'll cut the deck, if it's a club I win, if not it's your turn and if you cut a Jack, Queen, King, or Ace the fiver's yours. We'll continue mano a mano until there's a winner shuffling after each cut if you want. I'll even agree to go around the table five times, twenty games in all. Everyone in?" Coop then pulled a c-note from his wallet and Huck, after a long pull on his bourbon and oblivious to the workout the ashtray was now getting, piped up first. "So thirteen clubs vs sixteen honors, I'm in." Rob, Vern and I reluctantly agreed even though we suspected Coop had an edge. Where we right?

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You were right. He has a 25% chance of winning on the first cut.

If he misses you have a 30.769% chance of winning on the second cut.

But, you only get that chance 75% of the time, making your chances of winning on the second cut just over 23% when calculated before the prop bet starts.

It continues from there for the third, fourth, etc... cuts but the trend is the same, his cut is always a higher chance than your's.

Edited by smoth333
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You were right.

Now what is the probability of Coop winning any given game?

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Coop has an edge because he goes first. He has 13/52 (or 25%) chance of winning without ever giving you a chance. Your probability of winning is 16/52 multiplied by 3/4 (the probability that Coop doesn't win on his turn), so your total probability of winning is 12/52.

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...one should be able to relate this to a single turn of the game since a draw brings you right back to where the game starts. but k-man, what happens the other 24/52 of the time?

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Now what is the probability of Coop winning any given game?

Just plotting this in Excel I get the overall probability of Coop winning is 52%.

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...one should be able to relate this to a single turn of the game since a draw brings you right back to where the game starts. but k-man, what happens the other 24/52 of the time?

Yeah, I missed the fact that the game continues after the first 2 draws if they're both unsuccessful. I'm being lazy right now - it's Friday

The probabilities of next turns remain the same, but they have a common multiplier, so adding them all up the ratio of 13/25 vs. 12/25 will remain. 13/25 = 52%.

Edited by k-man
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...found by k-man - the straight Monte Carlo. kinda appropriate for a card game puzzle.

Just plotting this in Excel I get the overall probability of Coop winning is 52%.

Can you find other methods of solving?

Oh, and happy Friday k-man.

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...found by k-man - the straight Monte Carlo. kinda appropriate for a card game puzzle.

Can you find other methods of solving?

Oh, and happy Friday k-man.

Thanks, you too

Our posts are crossing in time. I added the spoiler to my last post with another way to solve it and here is another...

Each turn has 3 outcomes - Coop wins (13/52), you win (12/52), nobody wins (27/52). Since you will keep trying until someone wins it means that the third outcome can be discarded and the total probabililty divided proportionally between the first 2 outcomes - 13+12=25, so 13/25 and 12/25 or 52% and 48% respectively.

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Prob of Coop winning a cut = c = .25

Prob of Huck winning a cut = h = .3077

Coop wins if he wins immediately, or if he and Huck both miss the cut and he wins later.

Prob of Coop winning the game W = c + (1-c)(1-h)W

Solving for W: W = c/(h+c-hc), which works out to .52

Edited by CaptainEd
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...the algebraic shuffle. smooth moves k-man.

Each turn has 3 outcomes - Coop wins (13/52), you win (12/52), nobody wins (27/52). Since you will keep trying until someone wins it means that the third outcome can be discarded and the total probabililty divided proportionally between the first 2 outcomes - 13+12=25, so 13/25 and 12/25 or 52% and 48% respectively.

still two more independant methods (that i can think of anyway)

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Prob of Coop winning a cut = c = .25

Prob of Huck winning a cut = h = .3077

Coop wins if he wins immediately, or if he and Huck both miss the cut and he wins later.

Prob of Coop winning the game W = c + (1-c)(1-h)W

Solving for W: W = c/(h+c-hc), which works out to .52

Yes indeed, CaptainEd, the actual algebraic equation - nicely done. to me this was the last method I got but maybe the most elegant.

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