Calculating odds of flopping a pair

[AaronBrown]

First, 50 choose 3 is 19,600, not 15,180 (that's 46 choose 3, which does not help for this problem). Second, 5,676/15,180 is 37%, not 30%.

As fiskebent said, you've computed the probability of getting a pair and two cards that are not A or K (although they might form a pair between them).

To complete the problem, you can compute all the possibilities:

A or K/x/x 5,676
A/K/x 396
A/A/x or K/K/x 264
A/A/K or K/K/A 18
A/A/A or K/K/K 2

The total is 6,356. Divide by 19,600 to get 0.3243 or about 32%.

AaronBrown,

Great post. For someone just learning this stuff, could you possibly show the reasoning and equattions to get your other possibilities?

For example:
A or K/x/x = 5676
What are the details on deriving that value?
and A/K/x, etc.

I think I can really get this if you'll explain maybe 3 of those answers to me.

Thanks AaronBrown.

Eric

[AaronBrown]

Thanks for the kind words. Here they all are:

A or K/x/x 5,676

There are six cards that are A or K and 44 cards that are neither. 6*44*43/2 = 5,676. You have to divide by two because the last two cards are indistinguishable. This is the trickiest part of these calculations.

A/K/x 396

There are three Aces, three Kings and 44 other cards. 3*3*44 = 396.

A/A/x or K/K/x 264

3*2/2 = 3 ways to select two Aces out of three available, 44 ways to select x. 3*44 = 132. Double it for the K/K/x combinations to get 264.

A/A/K or K/K/A 18

3 ways to select the Aces, three ways to select one King out of three. 3*3 is 9. Double it for K/K/A to get 18.

A/A/A or K/K/K 2

One way to select three Aces, one way to select three Kings. 1 + 1 = 2.