Calculating odds of flopping a pair

[PJM1206]

According to Super System the odds of flopping a pair is approx 32%. This is where you hold AK and an Ace or King flops. So..... I tried to calculate this and came up with differnt answer approx 30% which is not good. Here is what I did:
I have AK so there are 6 cards left in the deck of 50 that will make my pair. So with that I used the following formula: 50 Choose 3 ( to get the number of possible 3 card flops which totaled to 15,180. Then I used 6 Choose 1 * 44 Choose 2 for a total of 5676. Here I am choosing 1 out of my possible 6 winning cards and 2 of the 44 loosing cards. Then 5767/15180= 30%..... What am I missing or is this correct.

You're missing the combinations where you flop more than 1 A/K.

I think it's easier to solve it if you look at the odds of missing. There are C(44,3) combinations that miss you completely and C(50,3) combinations in all. The chance of missing is C(44,3)/C(50,3). So the chance of hitting in some way is 1-C(44,3)/C(50,3) or ~32.4%.

Hope that helps.

[PJM1206]

ah said the blind man ---- thanks

[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%.

I'm new to poker odds, and still learning. Let me take a stab at this without using combinations.

To flop any pair, the odds of the first flop card completeing our pair would be:
6/50 or .12 or 12 percent
Then if we still don't have the pair:
6/49 or .122 or 12.2 percent
Then if we need the 3rd flop card to get our pair:
6/48 or .125 or 12.5 percent

Now, if we add probabilities, if we only need 1 of x number of results to get what we need, we would have:
12% + 12.2% + 12.5% or, 36.5%

I obviously must be doing something wrong. Can someone correcte the error of my ways? /images/graemlins/smile.gif

Eric

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.