Hi--This might be a dumb question, but I wonder if anyone could tell me just what is the probability of hitting your hand, and by that I mean making at least one pair by the river. I had always heard that it was about one-third or roughly 33%. I never actually did out any of the math behind those numbers on my own, but I just accepted that info, and it seemed about right to me.

In the most recent issue of Card Player magazine (Vol. 31, No. 11; May 23, 2018) there is an article on page 26 talking about certain considerations when playing Ace-King, and the author states that "if you get to see all five cards, you'll make at least a pair about 60% of the time." This just seems wrong, and it seems to go against everything that I have previously heard. So, can anyone help me out here, and tell me if what I've always heard in the past is right, or is this guy right about his 60% figure? I tend to doubt the latter, but thought that I would ask anyway. Also, I am not sure if there would be any difference if one were playing at, say, a six-handed table versus a nine-handed table, but I most typically play at a nine-handed table, just in case that might make a difference.

33% is rough estimate for the flop, not all 5 cards. For all five cards, it should be a bit more than 50%, not sure how much exactly (simply extrapolating from classic coinflip scenario + cases when you hit and they hit a set). Im guessing that 60% number includes cases where there is a pair or better on board - even if you miss, you still technically have a pair or better

Excluding straights, flushes and board pairs, etc. the probability of getting a pair or better by the river with 2 unpaired cards is

Pr =1- C(44,5)/C(50,5) = 48.7%

The number of no-hit cards is 52-8 = 44. The equation then represents the 1 - Probability board has only no-hit cards.

The number of opponents has no effect if no other conditions are given.

I have a program that runs through all possible boards when holding a specific starting hand in NLHE and classifies the poker hands that result.

The table below shows the results for AKs and AKo. The rows at the bottom of the table show the percentages of boards that result in One Pair or Better and High Card Only respectively in the two cases.

Category	AKs starting hand	AKo starting hand
Royal Flush	1,084	94
Straight Flush	78	122
Four of a Kind	2,668	2,668
Full House	47,124	47,124
Flush	138,296	41,562
Straight	65,508	69,954
Three of a Kind	92,004	93,808
Two Pair	469,092	480,080
One Pair	916,776	965,568
High Card	386,130	417,780
.
Total Boards	2,118,760	2,118,760
.
One Pair or Better Count	1,732,630	1,700,980
High Card Only Count	386,130	417,780
.
One Pair or Better Pct	81.78%	80.28%
High Card Only Pct	18.22%	19.72%

Of course, the above methodology may not be ideal for OP's inquiry since, for example, it classifies boards such as AQ973 (resulting in a pair of Aces) together with Q9722 (resulting in a pair of Deuces) into the "One Pair" bucket which is probably not what OP had in mind or what Card Player calculated. I will leave it to others to pursue if they want further clarity into this question.

Bottom line: many of these types of questions are amenable to combinatoric or brute force calculations which are often fairly straightforward to perform.