Probability your opponent has a pocket pair when you are paired versus not

Question

If I am holding a pocket pair does that effect the probability my opponent has a pocket pair.

Answer 1

I think if you have a pocket pair the probability your opponent has a pocket pair goes up slightly. You basically have less blockers.

you have pair the opponent = combin(4;2) * 12 + 1 = 73 = 5.959184%

you don't have a pair the opponent = combin(4:2) * 11 + 2 * combin(3;2) = 72 = 5.877551%

So the probability goes up 1/72 if you hold a pocket pair

Answer 2

When not holding a pocket pair:

((4 choose 2) * 11 + (3 choose 2) * 2) / (50 choose 2) = 5.87755%

When holding a pocket pair

((4 choose 2) * 12 + (2 choose 2) * 1) / (50 choose 2) = 5.95918%

So your opponent has a slightly better chance of getting a pocket pair when you hold one yourself, but the difference is indeed slight!