Looking back at my post on Broadway vs Small Pairs, I realize I made a mistake in one of the calculations. In the post, I said this:

There are 2 more Js which you don’t want, so there are 47 possibilities for the second card, and 46 possibilities for the 3rd card of the flop. So in total, there are 3 * 47 * 46 ways of hitting just one J. It doesn’t matter if the J happens on the first, second or third card on the flop, though, so you can multiply that by 3. There are 50 * 49 * 48 possible flops. So in total you have a (3 * 47 * 46 * 3) / (50 * 49 * 48) = 16.55% chance of hitting just one J.

For you to have top kicker as well, there has to be no Q, K or A on the flop (an A gives you two pair, and we’ll look at that later). There are 4Q + 4K + 4A = 12 total cards that you want to avoid. The chances of one of these cards showing (by the same reasoning as above) is therefore (12 * 37 * 36 * 3) / (50 * 49 * 48) = 40.78%. The chances of there being no overs on the flop are thus 100 – 40.78 = 59.22%.

The chance of you getting TPTK with AJ requires both to happen, and therefore is 16.55% * 59.22% = 9.80%.

I was wrong. I should not be taking the odds of a flop with a J, and the odds of a flop with no overs and multiplying them. If we hit a J on one of the flop cards, we’re only interested in the remaining two, so that initial calculation was wrong. The right calculation should be (3 * 35 * 34 * 3) / (50 * 49 * 48) = 9.11%. I have updated the original post to reflect this.

The difference in results is very small, but the mathematical process behind it is very different, so I felt it needed correcting. I do believe that it’s not just the numbers that matter, it’s the process with which you arrive at the numbers.

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