Texas Holdem Hands Probability

[DoctorWard]

I count at least 20 double ups, however some might be triple ups?

For example, Ah Kh is a suited connector, all paint AND A-x suited.

I still get 210

[gaming_mouse]

[ QUOTE ]
I count at least 20 double ups, however some might be triple ups?

For example, Ah Kh is a suited connector, all paint AND A-x suited.


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I took this into account.

Why don't you show me your work, so I can double check that?

[DoctorWard]

Gaming Mouse

I have set up a spreadsheet for this so potentially it's easier for me to email it to you?

However, in the attempt of getting this resolved here are my calculations...
a) Pairs = 78 hands
b) Suited Connectors = 55 hands
c) All Paint (A, K, Q & J) = 120 hands
d) A-x suited = 48 hands

Firstly this differs from your workings in both (b) and (c)

Now going in order you look at pairs, so you get 78 hands. Then you add in the suited connectors of 55 hands and nothing here overlaps, therefore we are at 133 hands.

Next add in the all paint of 120 hands however, there are 36 overlaps here, so now we are at 217 hands.

Next add in the A-x suited of 48 hands however, there are a further 16 overlaps, so we get to 249 hands.

This is my result!

78 + 55 + 120 + 48 - 36 - 16 = 249 hands

[gaming_mouse]

[ QUOTE ]

a) Pairs = 78 hands
b) Suited Connectors = 55 hands
c) All Paint (A, K, Q & J) = 120 hands
d) A-x suited = 48 hands

Firstly this differs from your workings in both (b) and (c)

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How did you get 55 for suited connectors? There are 13 possible: A2-TJ is 10, JQ,QK,KA is 3 more. For each, there are 4 hands, one for each suit. 13*4 = 52.

There are 6 ways to combine A,K,Q & J. For each, there are 16 possible hands. 16*6 = 96.

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Next add in the all paint of 120 hands however, there are 36 overlaps here, so now we are at 217 hands.

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There are 3 connected hands among the 6 combinations of paint cards: JQ, QK, KA. For each one, 4 hands are also suited. Thus 4*3 = 12 overlaps.


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Next add in the A-x suited of 48 hands however, there are a further 16 overlaps, so we get to 249 hands.

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A2 and AK represent 4 overlaps each with suited connectors, for a total of 8 overlaps with suited connecters. In my original response, I forgot to take into account the Ax suited overlaps with paint: AKs, AQs, AJs -- gives us 12 more overlaps.

(78 + 52 + 86 + 48) - 12 - 8 -12 = 232

So I'm still getting a different answer. As you're still leaving out your work, I can't see where you (or I) am going wrong....

[DoctorWard]

OK here are my workings...
a) Suits = 13 different cards with 4 choose 2 suits = 13 x 4c2 = 78 hands
b) Suited connector = 13 combinations with 4 suits = 13 x 4 = 52 hands
c) All paint = 16 paint cards choose 2 = 16c2 = 120 hands
d) A-x suited = A with 12 other cards by 4 suits = 12 x 4 = 48 hands

Total = 298 hands

Agree now with all except (c), am confident of these workings!

Then the overlaps of A-x suited agree it is 12 overlaps
Then the overlaps of Suited connectors with All paint is 12 overlaps (A-K, K-Q & Q-J by the 4 suits)
Then the overlaps of Suits with All paint is 24 overlaps (A-A, K-K, Q-Q & J-J by 4 suits choose 2, i.e. 4 x 6 = 24)

Therefore we have
= 298 - 12 - 12 -24
= 250 hands!

[gaming_mouse]

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c) All paint = 16 paint cards choose 2 = 16c2 = 120 hands


Agree now with all except (c), am confident of these workings!


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Your calculation includes things like AA, KK, etc, which are already counted in the pairs.

[gaming_mouse]

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(78 + 52 + 86 + 48) - 12 - 8 -12 = 232



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In my orig post I made an arithmetic error. 86 above should be 96. The correct final answer is 242.

[jason1990]

Did you subtract AKs one too many times?

[gaming_mouse]

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Did you subtract AKs one too many times?

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Yep. I forgot the third term in inclusion-exclusion. Nice catch.