Nice, so this does help me to start to understand the math.
What I get from that article on stackexchange makes sense as far as it goes: Pick any card at random, 1 in 52. For the second card to be of the same rank, there are 3 of those cards in the remaining 51, which works out to 1-in-17, which is indeed 5.9%.
But that’s not quite right, is it? That 5.9% would be the odds of the first card in the flop pairing with the next card in the flop. But we get THREE cards, so that means we get TWO chances to pair the first card, since the flop gives us 3 cards, don’t we?
This is something like the “birthday problem” in probability, which I’ve read about numerous times, but still struggle to understand/explain.
So in a similar, but simpler, probability problem, if I were to flip a coin two times, what are the odds of it being heads both times or tails both times? 50%. But what are the odds of getting two heads or two tails after 3 flips? 100%. (Unless we’re not counting trials where it’s heads or tails all three times, I suppose, in which case it’d be 4/6.)
So this makes me think that the odds of the first card in the flop pairing with either the second or third card in the flop should be something more like:
3/51 + 3/50, which would be about 11.8%, or about double the 5.9% you’re telling me, isn’t it?
But wait, that’s just the odds of the FIRST card in the flop paring with the second or third card: XXY or XYX.
We can also get a pair from the second card and the third card matching each other in rank, if the second card from the flop doesn’t match the first: YXX.
So that would give us an additional ~5.9% (ignoring the slight change in the odds from removing the first couple cards from the deck as they’ve already been drawn). I guess to be precise, the odds of cards 2 and 3 in the flop pairing with each other are 3/49, or about 6.1%.
We don’t have to consider further combinations involving the third card, as they’re already accounted for in the considerations of the first two cards.
We could add in the odds of all three flop cards being of the same rank: (3/51) * (2/50) = 0.24% but it adds a nearly negligible increase to the overall odds of the flop containing a Pair, and anyway 3 of a kind is a drastically different hand than a pair is (since it makes quads and full houses much more likely) so I don’t really want to add that quarter-percent chance into this.
So that makes it more like 18% of the time that any one of the three cards in the flop will pair with either of the other two. Which is still not that much of the time, but I can buy something that is 18% likely happening in streaks a lot more frequently than something that’s only 6% likely happening in streaks. (Of course any probabilistic outcome will happen in streaks given true randomness, but it’s more likely to observe streaks when the odds of the thing happening are higher.)
And then, another thing I’m still not sure about is how drawing the first 19 cards at random for the 9 starting seats hole cards + the burn card will affect the above odds. Does the randomness over quadrillions of trials just cancel itself out, leaving the odds still at 5.9%? I have no idea to work out that probability.