Here goes, I tried explaining this the other day and the person just got lost but I'll try.
I'll Q-Q as an example. If you have Q-Q before the flop then there are 50 cards that you cannot see. Therefore we are getting 3 cards from this 50 in order to make the flop. There C(50,3)=50!/3!47! = 19,600 possible flops.
What we are interested in is how many of these flops will contain an Ace or a King (as they are overcards to the pocket Queens). We figure this out by working how many flops do not contain an Ace or King and subtracting it from the total number of flops (19,600).
So they are 50 remaining cards after taking out the QQ in our hand, once we take out the Aces and Kings there 42 cards left giving us
C(42,3) = 42!/3!39! = 11,480 flops that do not contain an Ace or a King.
So we take the 11,480 flops that have no Ace or King and subtract it from the total number of flops (19,600)
19,600 - 11,480 = 8,120
So there are 8,120 flops that will contain an Ace or King, and the probabilty of and an overcard to the QQ appearing is 8,120/19,600 = 0.414 or 41% of the time.
To calculate the probabilty of an over card on the whole board similar calculations are performed. There are C(50,5) = 2,118,760 possible boards and there are C(42,5) = 850,668 boards without an Ace or a King. So the number of boards with an overcard is 1,268,092. Therefore the probabilty of an overcard appearing on the board is 0.599 or 60%.
N.B - I am willing to bet that most of you that read this far down before thinking to hell with it will be thinking what the hell does C(50,3) = 50!/3!47! mean. C(50,3) is a short way of writing the possible combinations of 3 cards drawn from a deck of 50. And 50!/3!47! is basically the calculation that is used written in a shorter way. The '!' means Factorial, a factorial of any whole number is found by multiplying all the positive number equal to or below that number, e.g. the factorial of 3 is 1 x 2 x 3 = 6, while the factorial of 5 is 1 x 2 x 3 x 4 x 5 = 120. The sign is used to save time and in a lot case paper, because in this calculation we need the factorial of 50, which would take up a hell of lot of paper seeing as the answer to it is 65 numbers long.
So in order to calculate the possible number of flops we are choosing 3 cards from 50, when the numbers are put into the calculation we 50!/3!47! = 19,600. All the calculations for set of probablilites are done in the same way obviously as the pocket pair gets the lower the possible flops with overcards gets higher.
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