Not sure if this is the correct spot to ask this but I’m doing some math incorrectly I think so I’m just asking for a little help... you can go to almost any website that talks about post flop odds for specific hands and find out something like the odds to hit a pair or better post flop with something like AK (~32%) and these websites will show you how to do this math... but this is extremely generic... what I can’t seem to find anywhere is how to do this same type of math while adding a factor of any kind, even a simple one... Let’s say I want to know how to do the math, same question, but I change one card (AQ and KQ), and I want to have not just a pair post flop but top pair or better (meaning the times with AQ where I flop a Q but a K also flops, or KQ where I flop a K or Q but an A also flops don’t count in the end value because I don’t have top pair)... I’m just not finding any websites that show how to do the math to minus events correctly from a specific range so even though ive learned how to calculate basic odds I can’t seem to always apply them to figure out the odds of a specific event... A link to somewhere they talk about this in greater detail would be amazing but just showing how the math would be done for the above scenarios should allow me to infer ways to answer deeper questions that have been bothering me on my own... Thanks in advance for any help you can provide!

Do you mean, for example, you have KQ and you are afraid to be beat by AK when the flop pairs your Q?

If yes, the simple thing to do is recognize the outs that a better hand has, which is 3 (A,A,A provided the board didn’t hit 10,J,Q, but let’s keep it simple by assuming you know the better starting hand only has 3 outs to beat you). If you know how many outs they have to beat you then you can use the formulas you know to find out their odds of drawing to win (roughly 14% for the turn, and 7% on the river).

In the above example if the flop did hit something stupid like 6,10,Q, then they have more outs (6) for a 26% on the turn and 13% chance on the river. Knowing this information and having watched your opponent you can bet appropriately to get the maximum value for your hand.


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No, I guess it is confusing... in the original example it’s a starting hand (example AQ/KQ) and I’m asking how the math is done to determine the chance to flop top pair or better (shown so I can see how to remove a range from a previous set of calculated math, example AQ flopping a Q but a K also appears on the flop as well)

For heavens sake use paragraphs...

It's a case of working out the chances of hitting top pair or better, which it sounds like you know how to do, and then subtract the cases you don't want, i.e. in your second post it's exactly the number of cases where the flop is precisely KQx, which is just a combinatorics question. It seems of limited value though, as it'll either not change the result much, and it doesn't really consider how good your situation is - it tell you the same answer with AQ on a Q72 rainbow board and a QJT monotone board

The general answer to your math question is simple in theory, sometimes hard in practice.

In general, for card deals or similar problems, the probability of an event occurring is

P(E) = Number of outcomes favorable to the event / Number of total outcomes

Let’s use your example as I understand it. You have AQ. You want to know the probability that the flop contains a Q giving you possible top pair but the flop also contains a K so top pair may not be true.

First, you have to be careful to define a successful event. We’ll assume here that a "successful" flop is Q K x where x is neither a Q, K, or A. To get the numerator, we seek the number of combinations (outcomes, ways) to get such a flop. We know two cards have been dealt – your A and Q, so, there are 50 remaining cards, 3 are queens, 4 are kings, 3 are aces and 40 are other ranks, or there are 40 x’s.

Thus, the number of way to flop Q, K, x using combinatorics is

Number of Q K x combos = C(3,1)*C(4.1)*C(40,1) = 3*4*40 = 480, the numerator.

The number of possible flops from 50 cards is C(50,3) = 19.600, the denominator

P(AQ flops a Q and K and rank below Q) = 480/19600 = 2.45%

There are other ways this can be solved.

This problem was relatively easy but for other similar type problems you have to be careful not to double count some outcomes or leave others out. Also defining success can be tricky – here for example I’m not totally sure I answered OP’s question but am very sure I correctly answered what I think OP’s question is.