Holding 2♥ -2♦, how often do you flop a set?

There are a number of ways to figure this out.  Figure out how many possible combinations of cards there are on a flop and, of those, figure out how many of them contain the cards you are looking for.

You could make a list and start counting, but that would take a long time.  Better to use a little math to figure it out.  Of course, somebody has already figured this out, like some ancient mathematician trying to figure out how many color combinations you could weave into a rug with a given set of colored yarns.    If you let n be the number of cards left in the deck to choose from and let k be the number of cards you are choosing, then the number of possible combinations you can get (without regard to the order of the cards) can be represented by a formula that looks like n! / k!(n-k)!.   

Fortunately, we have Google.  We don't need to multiply out all of those factorials.  We don't even need to know the formula or call it the binomial coefficient.  We just need to know how many cards we are choosing from the deck and how many cards are in the deck.

Google built a tool right into their search engine to make it easy for us. Just type in "50 choose 3" where 50 is the number of cards in the deck and 3 is the number of cards.  It will return the number of possible combinations.

We can use this handy tool to help us figure out our solution

How many possible flop combinations are there?  There are 52 cards in the deck and we are holding two of them.  The flop contains 3 cards,chosen from the remaining 50 so:

50 Choose 3 = 19,600  The number of possible flop combinations

Seeking 2♣ plus any combination of any of the other 49 cards

49 choose 2 = 1176

Seeking 2♠ plus any combination of any of the other 49 cards

49 choose 2 = 1176

Note that 1176 contains flops that have both 2's and another card and 2♣-2♠-X is the same as 2♠-2♣-X so we need to pull out the duplicates.

Subtract out duplicates of 2♣-2♠ plus any of remaining 48 = 48

1176 + 1176 – 48 = 2304

So now we know that of a possible 19,600 flops, 2,304 of them contain one or two deuces and give us a very strong hand.

 2304 /19,600 = 11.755%

Of course, there is more than one way to skin a cat.  My old poker buddy who was a college professor and did a lot of research using statistics and probability didn't like my alternative method because it was subject to a tiny little bit of rounding error.  But it works.  It is a method of figuring out how many times you miss catching a deuce on the flop and simply subtracting that from 100% to see how often you hit it.

Alternate method:

2 of 50 is 4%, so 96% of the time, the first card misses.

Of the 96% of the time that the first card misses, 47 of 49 or 95.92% of the time second card also misses and 95.92% of 96% is 92.08% of the time.

Of the 92.08% of the time that the first two miss, 46 of 48 or 95.83% of the time the third card misses also. This means that 95.83% of the 92.08% or 88.24% of the time all three miss which means that 11.76% of the time, you flop three or four of a kind.

By either method, you will flop a set or quads about once every 8.5 times you see a flop or, saying it another way, the odds against flopping a set are 7 ½ to 1.