probability for a non-pair card to hit top pair

[Norb]

A question to the math experts. I want to calculate my probability to hit top pair with arbitrary kicker on the flop with any given non-pair hand. Is the formula below correct?

The chances of pairing the highest card on the flop with my first pocket card is 6 out of 50 and the chance, that my second pocket card will not hit the middle card on the flop is 44 out of 49 and the chance, that my second pocket card will not hit the lowest card on the flop is 43 out of 49. Therefore the probability of hitting top pair with any kicker (and nothing else) should be

(6/50)*(44/50)*(43/49) = 9.27%

Is this calculation correct?

How to calculate the probability that some single opponent on the table will flop TPTK with any random pocket cards? Could somebody help me constructing the formula to evaluate this?

Or does somebody know some available document that describes something like this?

Thanks for your help!

Best regards, Norbert

[Orlando Salazar]

I think you need to factor in the rank of your two holdings as they will influence how many higher cards can come. Try the problem with AK, then QK, then JT and you'll see what i mean

[jay_shark]

You hold something like xy where x differs from y . Clearly the answer should depend on the rank of cards you have . If you have A-K , then you're more likely to hit top pair than if you had something like j-10 . Here is the probability you flop top pair with j-10 where either a jack or a ten is the highest ranking card on the flop .

First lets calculate all the flops where a jack is the highest

P(j-x-y) , P(j-j-x) , P(j-j-j)| x or y is less than a jack :

(3*35c2 + 3c2*35 + 3c3)/50c3 =0.0964

Now lets calculate all the flops where a ten is the highest .

P(10-x-y) ,P(10-10-x) ,P(10-10-10)|x or y is less than a 10:

(3*32c2 + 3c2*32 +3c3 )/50c3 =0.0808

Add both and you get that the percentage of flops in which a jack or a ten is top pair is 17.72% .

[jgca]

Hi, Norbert. Here's a general formula, and how we get it:

Given your two cards, the total number of flops is 50 choose 3 = 50*49*48/3*2*1 = 19,600.

For an unpaired hand, the number of flops that yield one split pair, with your kicker and both other board cards all of different ranks, is:

55 (11 choose 2: different possible ranks of the other board cards) *
3 (suits for the pairing card) *
4 * 4 (suits of the other board cards) *
2 (either of your cards can be the one paired) =
5280.

Let m = number of ranks below your top card, excluding your bottom card, and
n = number of ranks below your bottom card. Then the number of top pair flops is:

(m choose 2) * 3 * 4 * 4 + (n choose 2) * 3 * 4 * 4 =

48 {(m)(m-1) / 2 + (n)(n-1) / 2} =

24 {(m)(m-1) + (n)(n-1)}.

For AK, m=11 and n=11, and as you can see, this works out to 5280. (Obviously all one pair flops for AK are top pair.) This is about 27% of all flops.

Here are some other examples. First column is # of flops that yield one pair. Second column is % of one pair flops that are top pair (i.e., divided by 5280). Third column is % of total flops that yield top pair (i.e., divided by 19,600).

AQ 4800 91% 24%
KQ 4320 82% 22%
KT 3504 66% 18%
K7 2640 50% 13%
J9 2352 45% 12%
87 960 18% 5%

Note that these numbers are for exactly one pair; that is, they exclude two pair (either from hitting your hand twice or from a split pair plus a paired board) and anything better. Obviously, it is impossible to flop a straight or flush and a pair at the same time, although you can easily have a pair plus a powerful draw.