Here is a quick way to get a rough approximation: The average number of other big pairs you are up against is 9*19/(50 choose 2) = 171/1225 = .13959. That is greater than the probability you are up against a big pair because there may be more than one. To estimate the probability, the probability that a Poisson distribution with mean x is greater than 0 is 1-exp(-x). 1-exp(-171/1225) = .130287.
That isn't exact because the Poisson distribution is a rough approximation here. I'm not sure it is even better than 1-(1-19/1225)^9 = .131238, which assumes your opponents' hands are independent. If I get bored, I may do the exact calculation later, using
this method. However, I think these estimates are off by less than 0.2%.