poker odds question: high card ace winning on river versus two players

Verdue167

Verdue167

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What are the odds that, in a 3 handed hand, and all three players hold three different non-paired hands, like AJ-QT-93 (so 6 different cards) and all 3 players check all streets, and all of them miss the board? (so that hero's ace high wins?)
 
vinnie

vinnie

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This is a very tricky question, as the odds will change subtly if players hold similar cards. For example, AK vs KQ vs KQ is going to have more "wins" for AK than AJ vs QT vs 93.

For the original hands you posted the odds will be slightly less than 7.17% that Ace high will win. {1 - (28*27*26*25*24) / (46*45*44*42*41)} I say slightly less because this ignores the possibility of a flush, straight, or split happening. This is just no player holding a pair at the end.

If you are asking about your odds when you hold AK (unimproved) and there are two unknown hands that are with you at showdown, it's also depends on the board cards and if there are straight or flush possibilities. But, the odds should be better than the 7% listed above. We already know that you missed, so it's just the odds that they don't hold any of those cards.
 
Verdue167

Verdue167

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This is a very tricky question, as the odds will change subtly if players hold similar cards. For example, AK vs KQ vs KQ is going to have more "wins" for AK than AJ vs QT vs 93.
Thanks for your reply. I meant 6 different cards, so no player holds a card (K,Q,J , etc) that any other player holds. And my question was also based on ignoring any straights and flush possibilities. And actually, which i forgot to state, with 5 different board cards ( so an unpaired board, or no trips or quads for that matter)
If you are asking about your odds when you hold AK (unimproved) and there are two unknown hands that are with you at showdown, it's also depends on the board cards and if there are straight or flush possibilities. But, the odds should be better than the 7% listed above. We already know that you missed, so it's just the odds that they don't hold any of those cards.
I assume it will be very close to 7%
 
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