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Old 02-12-2017, 06:44 AM   #1
Thamel18
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Red / Black flop, with cards known, odds?

A friend of mine wanted to know what the odds are of a flop being majority red or black, when you have 4 cards of the opposite color (playing Omaha). But, he's disagreeing with my math and providing none of his own. Are my calculations correct

Holding 4 black cards, betting red : 22 black & 26 red cards unknown.

Probability of flop being 3 red : (26/48) * (25/47) * (24/46) = 0.1503
Probability of flop being 2 red : (26/48) * (25/47) * (22/46) * 3 = 0.4134
Multiplied the second number by 3 because a flop can come 2 red in 3 different orders, but the chance of each happening is equivalent

Total probability = 0.5637
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Old 02-12-2017, 10:34 AM   #2
heehaww
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Re: Red / Black flop, with cards known, odds?

You're right and your friend is wrong. If he doesn't believe you, see if he's willing to bet on the outcome of a repeated experiment!
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Old 02-12-2017, 03:38 PM   #3
whosnext
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Re: Red / Black flop, with cards known, odds?

This has already been settled, but I'll take this opportunity to give another (equivalent) approach using pure combinations. Since the order of the cards on the flop does not matter (only their color), combinations are applicable.

As you state, if you hold 4 black cards, the three-card flop comes from a "virtual deck" containing 22 black cards and 26 red cards.

It is easy to tally the number of ways the flop can appear:

3 black, 0 red: C(22,3)*C(26,0) = 1,540
2 black, 1 red: C(22,2)*C(26,1) = 6,006
1 black, 2 red: C(22,1)*C(26,2) = 7,150
0 black, 3 red: C(22,0)*C(26,3) = 2,600

TOTAL: C(48,3) = 17,296

Then it is easy to compute the probability of a majority of red cards or any other probability of interest.
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