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Alice and Bob are playing a card game with the following rules:

  1. Alice selects two piles of 3 cards from a standard 52-card deck and places the piles in front of her, face up. Bob can see these cards.
  2. Bob takes the deck and selects a 5-card poker hand without showing his cards to Alice. Bob is only allowed to pick a straight or lower (no flush, full house etc.)
  3. Alice puts away one of her piles, leaving just 3 cards in front of her.
  4. Bob turns over his cards. Alice can use any 2 cards from Bob's hand along with her own 3 cards to make a 5-card poker hand. (Alice's hand is not restricted to straight-or-lower like Bob's.) The player with the better hand wins €1 from the other player. If both have the same hand, nobody wins anything.

Can either player expect to win this game in the long run? What strategy should the players use?

Additional note:

Both A,K,Q,J,10 and 5,4,3,2,A are valid straights in this game (note added since there are some forms of poker where this is not the case).

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  • $\begingroup$ Is A2345 considered a valid straight? And if so, is it considered the lowest of all straights? $\endgroup$ – JS1 Jul 1 at 20:30
  • $\begingroup$ @JS1 Yeah, A2345 is the lowest (5-high) straight. $\endgroup$ – jafe Jul 1 at 20:47
  • $\begingroup$ @JS1 That said, it's not a completely universal rule (in lowball poker, for example). So I've added a note. $\endgroup$ – jafe Jul 1 at 20:53
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Alice's two piles should be 555 and TTT.

Each of these piles makes Alice have at least a three-of-a-kind, which beats any pair or two-pair hand. Additionally, if Bob has a higher three-of-a-kind hand such as AAA, Alice can make a full house to beat it. Therefore Bob needs to select a straight in order to not lose immediately.

Any straight from A2345 to TJQKA must contain either a 5 or a T. So Alice has a 50/50 chance of winning, because for any straight that Bob selects, she can pick the "correct" pile and make a 4-of-a-kind, or she can pick the "wrong" pile and lose with a 3-of-a-kind.

So in the end, if Alice picks 555 and TTT, and Bob selects any straight at random, neither player should be expected to win any money in the long run.

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  • $\begingroup$ Additionally, if Bob tries to get any three-of-a-kind, Alice can take 2 of them to win with a Full House (regardless of which hand she kept). $\endgroup$ – DqwertyC Jul 1 at 22:20
  • $\begingroup$ @DqwertyC Oops I forgot about 3 of a kind as Bob's hand. I'll edit to add that in there. $\endgroup$ – JS1 Jul 1 at 23:10
  • $\begingroup$ This is correct! $\endgroup$ – jafe Jul 2 at 5:27

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