Damn you. My latent OCD with probabilities and no sight of a 6O online calculator means my afternoon is now toast - sigh....
With 37 unknown cards, there are 1369 possible ways this hand ends up.
To win, Hand 2 needs:
i) Running hearts
ii) A jack (without a king or an eight)
iii) Running fives
iv) A three (as long as there is no A, 8, running clubs or case 3 to go with it)
v) A four (as long as there is no A, 8, running spades or case 4 to go with it)
vi) A five and a six (as long as they're both not spades or diamonds)
i) 10 * 9 = 90 ways for running hearts.
ii) (3 [jacks] * 32 [other cards exlcuding kings and eights] * 2 [turn and river]) = 192 ways to get a winning jack.
iii) 3 * 2 = 6 ways to get running fives.
iv) (2 [threes] * 36 [othercards] * 2 [turn or river]) - (2 [turn or river] * 1 [case Ace]) - (2 [turn or river] * 2 [eights]) - (1 [3 clubs] * 7 [another club]) - 1 [both threes] = 144 - 2 - 4 - 7 - 1 = 130 ways to get a winning three.
v) (2 [fours] * 36 [othercards] * 2 [turn or river]) - (2 [turn or river] * 1 [case Ace]) - (2 [turn or river] * 2 [eights]) - (1 [4 spades] * 10 [another spade]) - 1 [both fours] = 144 - 2 - 4 - 10 - 1 = 127 ways to get a winning four.
vi) (3 [fives] * 4 [sixes] * 2 [turn or river]) - 2 [both spades] - 2 [both diamonds] - 2 [both hearts counted in i)] = 18 ways to get a winning 56.
90 + 192 + 6 + 130 + 127 + 18 = 563 times Hand 2 will win.
563 / 1369 = 41.1%
To chop the hand we need:
i) the case two (without an eight)
ii) both threes
iii) both fours
iv) a five (without the board coming another of the same suit as the five)
i) 1 [case two] * 34 [all cards less the two eights] * 2 [turn or river] = 68 ways to get the two.
ii) 2 ways to get the case threes.
iii) 2 ways to get the case fours.
iv) (3 [fives] * 34 [other cards] * 2 [turn or river]) - (1 [5 spades] * 10 [other spades] * 2 [turn or river]) - (1 [5 diamonds] * 7 [other diamonds] * 2 [turn or river]) - (1 [5 hearts] * 9 [other hearts] * 2 [turn or river]) = 204 - 20 - 14 - 18 = 152 ways to get a tying five.
68 + 2 + 2 + 152 = 224 ways to chop.
224 / 1369 = 16.4%.
Therefore Hand 1 must win 42.5% of the time.
Please feel free to throughly destroy my wasted afternoon by pointing out where I've gone wrong
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