Formula for making a single pair on the flop

Question

I'm trying to get the probability of pairing only one hole card on the flop. According to this calculator,

http://www.texasholdem-poker.com/calculator?card1=8&card2=13&chance_format=probability&decimals=9

the probability is 0.269387755

How do I get that number ^^

The formulas I have found online give me slightly different results.

-- this result is for a pair or better I believe and I only want a pair, not better
(6/50) + (6/49) + (6/48) = 0.36744897959184

-

-- chance of one not pairing  *  chance of one pairing --
((44/50) * (43/49) * (42/48)) * ((6/50) + (6/49) + (6/48)) = 0.24829052478134
-- close but still off.

-

-- first card pairs, second two cards miss
-- I believe the *3 is to ignore the order of the flop
((6/50) * (44/49) * (43/48)) * 3 = 0.28959183673469
-- again, close but off

These are all I could really find online, any ideas what's wrong here? By "what's wrong", I mean why do none of these three calculations equal the probability listed on the website?

Answer 1

The probability would be calculated by taking the odds of having 1 card pair and then the other two cards NOT pairing or forming a set. Calculate this for the three combinations (1st card pairs, 2nd card pairs, and third card pairs) and add them together:

(6/50) * (44/49) * (43/48) = 0.096530612
(44/50) * (6/49) * (43/48) = 0.096530612
(44/50) * (43/49) * (6/48) = 0.096530612
                             -----------
                     (SUM) = 0.289591837

NOTE 1:
The number of cards that are appropriate for each step are:

Line 1:

• (6/50) - Any of the 6 cards that would pair your hand
• (44/49) - Any of the cards except for the 3 that would pair your other card and the 2 that would give you a three of a kind
• (43/48) - The same as the previous line (40 if we exclude the previous board card pairing as in example 2 below)

Line 2 and 3 are similar, except that we delay the pairing of the hole card until the 2nd and 3rd card.

NOTE 2:
This also assumes that you are ignoring the possibility of the board pairing (in addition to pairing one of your hole cards). If we want to exclude that possibility (which is the same as the answer given on the calculator that you used), then we would use:

(6/50) * (44/49) * (40/48) = 0.089795918
(44/50) * (6/49) * (40/48) = 0.089795918
(44/50) * (40/49) * (6/48) = 0.089795918
                             -----------
                     (SUM) = 0.269387755

NOTE 3:
If you want to get even more accurate, you would also need to include the probability of not pairing either of your hole cards, but the board pairing (or making a three of a kind) independant of your hole cards. This is slightly more complicated, but we have already determined how they came up with their answer.

NOTE 4:
This also assumes that you weren't dealt a pair in the first place.

Answer 2

.367 is correct. The rule of thumb pros use is "a third of the time," and your math is right.

Answer 3

With combinations

match a card combin(6,1)
other combin(44,2)
flops combin(50,3)

combin(6,1) x combin(44,2) / combin(50,3) = 0.2895918367 = 1 / 3.4531360113

Answer 4

The flop = card 3, card 4 and card 5...card 3 will be paired to one of your two private cards.

this is the card to be paired: card 3 = combin (2 c 1) x combin (3 c 1) = 6

these two cards do not pair to any of your private cards nor do they pair to each other: card 4 and 5 = combin (11 c 2 ) x combin (4 x 1) x combin (4 x 1) = 880

the denominator is all possible three card combinations of the remaining 50 cards: combin (50 c 3) = 19600

So its 6 x 880 / 19600 = 5280 / 19600 = 26.94%