Please Help me Figure This Out

So, I'm trying to find out, in a game of Texas Hold'Em...

if there is no pair on the board itself on the flop alone,
so not counting what I or anybody hit with their hole cards,
just what you see on the board/flop so far itself,

how would I calculate what percentage the board will pair by the time I get to the river?
(So counting it only happening only on the turn or river.)

These are the numbers I have so far, but don't know what to do to get the calculation I am looking for:

Scenario	Formula	Solution
the flop contains a pair	1 - combin(13,3) x 4^3 x combin(52,3)^-1	0.171765
the flop does not contain a pair	combin(13,3) x 4^3 x combin(52,3)^-1	0.828235
the board contains a pair (flop and turn combined)	1 - combin(13,4) x 4^4 x combin(52,4)^-1	0.323890
the board does not contain a pair (flop and turn combined)	combin(13,4) x 4^4 x combin(52,4)^-1	0.676110
the board contains a pair (flop, turn, and river combined)	1 - combin(13,5) x 4^5 x combin(52,5)^-1	0.492917
the board does not contain a pair (flop, turn, and river combined)	combin(13,5) x 4^5 x combin(52,5)^-1	0.507083

So, I'm guessing the formula to figure out the answer I'm looking for would be to multiply
0.828235 x 0.492917

(which comes out to 0.408251, by the way)

but I wasn't sure if I was thinking of it right.

The only reason I'm questioning my calculation is cause I read that
the board will pair 18% of the time on the turn itself and 24% of the time on the river,
so was trying to see what the percentage would be if it were to happen
either on the turn OR the river combined.

Hope this makes sense!

If anyone could help, it would be much appreciated!
TIA!

If I may reword the question, hopefully correctly:

The board contains three singletons. No other cards are known, including your own. What is the probability the turn and/or river do not pair the board? A two pair or three of a kind on the board, I assume, count as pairing.

The probability of not pairing the board is combin(10,2)*4^2/combin(49,2) = 61.22%.

So, the probability of pairing is 100%-61.22% = 38.78%.

Quote: Wizard

If I may reword the question, hopefully correctly:

The board contains three singletons. No other cards are known, including your own. What is the probability the turn and/or river do not pair the board? A two pair or three of a kind on the board, I assume, count as pairing.

The probability of not pairing the board is combin(10,2)*4^2/combin(49,2) = 61.22%.

So, the probability of pairing is 100%-61.22% = 38.78%.

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Yes, thank you Mike!

I guess to say the board "pairing or better" would count, glad you caught that!
And thank you for figuring that out for me!
What I read was that it was about a 2:1 chance the board would NOT pair on the turn or river had it NOT already paired on the flop, and your calculation comes pretty close, but I'm glad you also included it possibly pairing twice or even tripping by the river.

You're the best!