[BruceZ]

You made some instructive errors on these.

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The odds of getting dealt any pocket pair is equal to: (52*3)/C(52,2) or 1/8.5

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1/17. You must divide the numerator by 2 if you use combinations in the denominator, or else you will count every combination twice. For example, you are counting AsKc and KcAs as 2 separate hands in the numerator, but only once in the denominator. I see that you are thinking "52 ways to pick the first card, and 3 ways to pick the second card", but that double counts the combinations because it counts each possible ordering of the 2 cards. However, you could do 52/52 * 3/51 = 1/17. Notice the subtle difference to the first method. When you use combinations you don't care about order, but when you use fractions, then you do consider the probability for the first card times the probability for the second card.

Also, for all of these, the number represents the probability, not the odds.


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The odds of the board containing both of the cards to match that pocket pair is:
2*1*C(48,3)/C(50,5)=~.016

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Just 1*C(48,3)/C(50,5) =~ 0.0082. There is only 1 KK, and then C(48,3) combinations for the remaining 3 cards. We are using combinations, so we don't say "2 kings for the first card times 1 king for the second card". This counts each KK twice, but obviously there is only 1, not 2.


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So the odds of being dealt a pocket pair and making quads with it is equal to:
1/8.5*.016=~.002

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1/17 * 0.0082 =~ 0.00048.


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The odds of being dealt the exact same pocket pair is equal to:
C(4,20)/C(52,2)=1/221

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The 1/221 is correct, and you mean C(4,2) = 6, not C(4,20).


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Once again the odds of the board containing the remaining 2 of these cards is ~.016

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~0.0082.


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So the odds of being dealt a specific pocket pair and making quads with it is:
.016*1/221=~.00007

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0.0082 * 1/221 =~ 0.000037.


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Therefore the odds of, on any two given hands, being dealt a pocket pair, making quads with it and then have the same thing happen the next hand is equal to:
.00007*.002=.0000001 or .00001%

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0.00048 * 0.000037 = 0.000000018 or without rounding until the end 1 in 56,378,481. Odds are 56,378,480-to-1.