I was recently watching a video on Stoxpoker, and the commentary was talking about 3betting OOP. The logic went something this:"it is easier to play a hand like AJ OOP then a small pocket pair because you will flop TPTK much more often then you will flop a set." That got me to thinking how often will we flop flop TPTK? In order to get TPTK we will need a T high flop. I know, roughly 1/3 of the time we will flop a pair, or 33%, and half of the pairs we flop will be Ts or about 16.5%. How do we subtract the number of times we flop a T, but it is not TP?

The bolded statement is wrong.

Holding AJ you will flop a J-high flop (giving you TPTK or better) about 10% of the time. Holding AT flopping a Ten high flop is even less, about 8%.

You hold AJ. There are C(50,3) or 19600 flops possible. Take out all the A,K,Q and you have C(39,3) or 9139 flops. Take out the Jacks too and you get C(36,3) or 7140 flops. So flops that have a Jack but no A,K,Q (when you hold AJ) is (9139-7140)/19600 =
10.2%

That said, you will have an Ax hand more often than you will have a pocket pair. There are 78 pairs. There are 192 Ax hands excluding AA, so about 2.5 times as often. But that wasn't the point in the bolded statement. If the video person actually said that, they were wrong.

Thank you very much. It sounded wrong to me. What kind of calculations are you doing? I understand that C(50,3) stands for 50 unknown cards, 3 being flopped, but how are you getting 19600? Also thank you for giving me both the answer for AJ and AT. I started writing the post thinking the video said AJ, then went back and double checked and realized he said AT and forgot to change the title.

C(50, 3), as you say, means choosing 3 cards from 50.

There are 50 * 49 *48 ways to choose these 3 cards (50 possibilities for the 1st card, 49 for the 2nd, 48 for the 3rd). However, a flop of Td 8c 6c is the same as a flop of 6c Td 8c, for example. The order of the cards doesn't matter. For any 3 cards, there are 6 possible ways of arranging them. So therefore, the total possible flops are 50*49*48 / 6 = 19600.

Another way of looking at this is that there are 3 more Tens in the deck. Your chances of getting exactly one T on the flop are 3/50 * 47/49 * 46/48 * 3 = 16.5%.

There are 4J + 4Q + 4K + 3A = 15 overs left in the deck. The chances of at least one of these showing on the flop are therefore 15/50 * 34/49 * 33/48 * 3 = 42.9%. Therefore, the chances of there being no overs on the flop would be about 100-42.9% = 57.1%.

The chances of both happening (thereby giving you TPTK) would therefore be 16.5% * 57.1% = 9.44%.

This number is slightly lower than fdel's because it ignores the possibility of flopping trip or even quad Ts. But the rough idea is the same.

When you have a pocket pair, you will flop a set (exactly a set, not quads) 2/50 * 48/49 * 47/49 * 3 = 11.5% of the time.

So no, you're slightly more likely to flop a set when you have a pocket pair, from a pure math perspective. However, the advantage that a AT hand has is that it has more outs. You're not just playing for a TPTK. You could hit a two pair, or if they are suited, a flush. Or even the nut straight. Much more options, as compared to the pocket pair which only improves to trips or a full house.

If you include all those possibilities the numbers might favor the AT (or at least even it out), just a gut feeling. I'm heading for lunch now, will crunch the numbers later.

Ok so I'm back again. More numbers.

On the flop, AT would get TPTK 9.44% of the time, as mentioned earlier. How else can it improve?
- 3.47% (6/50 * 5/49 + 6/50 * 44/49 * 5/48 + 44/50 * 5/49 * 5/48) of the time, it will improve to either Two Pair, Trips, or a Full House.
- 1.2% of the time, if they are suited, it will improve to a straight or a flush (11/50 * 10/49 * 9/48 = 0.84% for the flush, 12/50 * 8/49 * 4/48 = 0.33% for the straight).


So therefore you have a total of 14.07% chance to make a winning hand (TPTK or higher) on the flop.

For a pocket pair, you are essentially playing for a set. As mentioned, you have 11.5% chance of flopping either a set or a full house by matching your card (forgot to mention the inclusion of a full house in the above post). What else?
- 0.25% of the time, you'll flop a 4 of a kind
- 0.27% of the time, a full house by the board tripling up

Therefore you have a total of 12.02% odds of improving.

So there can be an argument made that ATs would be easier to play than 22. You do stand to hit more flops, if you consider the other options. With a small pocket pair, anything below 77, there's at least a 95.8% chance (for 66) that an over card will flop, so you only have about a 4.16% chance of having an overpair to the board.

Thank you for all this derrickwa. I thought the (50,3) meant 50*49*48, but I was forgetting to divide by 6 and your reasoning makes perfect sense now. Also, by breaking down TPTK and better hands, im guessing this is what the author of the stox video meant. Whether he had done all the math or was just speaking from general experience I am not sure.

Just one thing to add:

The probability of flopping *exactly* TPTK is as follows:

Number of J high flops with only one Jack = 3 * C(36,2) = 1890

Total # of flops = 19600

Odds of hitting exactly TPTK = 1890/19600 =~ 9.64%

The only difference between spadebidder and my answers is that I took out the circumstances where you hit quads (1 possibility) and trips (C(3,2)*36 = 108 possibilities) on the flop.

That's correct. I specified that my answer was for any J-high flop, which could give you better than TPTK.

Ok I realized I made a mistake in this. You can't directly multiply the two odds. Basically, you want to hit a T on one of the three cards. After this, there are 32 non-T, non-over cards remaining, and you want to hit two of these. So your odds of getting exactly TPTK is 3*(3*32*31)/(50*49*48) = 7.59%. That's about 1.85% less than my original calculation.

Adjusting my final conclusion with this change, you have a 12.22% chance of improving to at least TPTK with AT suited. This is very close to the 12.02% that of a pocket pair, but it is a negligible difference, imo.

maybe this is a dumb question but:
with AJ on an AKQ board, do you have TPTK, because J is the highest kicker you can have without making twopair?

Yeah. The reason I didn't consider the cases when you get TPTK by pairing the Ace is because there are a lot of situations (if the flop is A-6-4), for example, where you don't know if you have top kicker - it depends on what the other players hold. And I don't really know how to calculate the probability that your opponent has a A-Q or A-K. So I left that out.

But yeah, AJ on AKQ board is considered TPTK, I think.