Full House With Pocket Pair - Gambling and Probability

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Full House With Pocket Pair

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11-21-2016 , 06:06 PM

statmanhal

Pooh-Bah

Join Date: Jan 2009 Posts: 4,987

Will someone confirm or refute the following:

Given a pocket pair, the probability of exactly a full house by the river, including a full house board, is 8.57%.

Thanks.

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11-21-2016 , 08:16 PM

Buzz

gramps

Join Date: Sep 2002 Posts: 18,254

Quote:

Originally Posted by statmanhal

Will someone confirm or refute the following:

Given a pocket pair, the probability of exactly a full house by the river, including a full house board, is 8.57%.

Thanks.

I get 8.77%.
Using AA as a prototype, Hero makes a full house if the board has any trips or
any pair (but aces) plus an ace
trips -> 12*4*46*45/2-12*4*1
pair -> 12*6*2*44*43/2
12*4*46*45/2-12*4*1+12*6*2*44*43/2=185856

50*49*48*47*46/1/2/3/4/5=2118760

185856/2118760=0.08772

Buzz

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11-22-2016 , 01:28 AM

whosnext

Carpal \'Tunnel

Join Date: Mar 2009 Posts: 6,730

Here is my take assuming we are talking about Hold-Em. Suppose pocket pair = XX.

Case 1. Board = XYYWZ
Number = C(2,1)*C(12,3)*C(3,1)*C(4,2)*C(4,1)*C(4,1) = 126,720

Case 2. Board = XYYZZ
Number = C(2,1)*C(12,2)*C(4,2)*C(4,2) = 4,752

Case 3. Board = XYYYZ
Number = C(2,1)*C(12,2)*C(2,1)*C(4,3)*C(4,1) = 4,224

Case 4. Board = YYYWZ
Number = C(12,3)*C(3,1)*C(4,3)*C(4,1)*C(4,1) = 42,240

Case 5. Board = YYYZZ
Number = C(12,2)*C(2,1)*C(4,3)*C(4,2) = 3,168

Grand Total = 181,104

Total number of boards possible in this situation is C(50,5) = 2,118,760.

So percentage is 181,104 / 2,118,760 = 8.548%.

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11-22-2016 , 12:36 PM

statmanhal

Pooh-Bah

Join Date: Jan 2009 Posts: 4,987

Oops. I see what I did. I left out the XYYYZ combo and double counted the XYYZZ combo. Not sure where we differ from Buzz.

Thanks guys.

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11-22-2016 , 02:26 PM

whosnext

Carpal \'Tunnel

Join Date: Mar 2009 Posts: 6,730

Quote:

Originally Posted by statmanhal

Oops. I see what I did. I left out the XYYYZ combo and double counted the XYYZZ combo. Not sure where we differ from Buzz.

Thanks guys.

Buzz's method also double-counts the XYYZZ case (two pair on board with an ace).

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