I got a question concerning calculating probability in random samples.

Let's assume I have two sets of the size n(1) and n(2). Both sets are made up of the same items but can have different sizes and thus partially overlap, i.e.:

n(1) =7, set1 = [1,2,3,4,5,6,7]

n(2) = 9, set2 = [1,2,3,4,5,6,7,8,9].

From each set, I draw a random sample without replacement of the sizes k(1) and k(2).

How can I calculate the probability to obtain the intersect of the size X in those 2 random samples. The 2 samples are indepent of each other P i. e:

k(1) = 4, randomSample1 = [1,2,7,6]

k(2) = 5, randomSample2 = [3,2,9,6,4]

intersect = [2,6], intersect size X = 2,

Following, my intuition I would calculate the probability for an overlap in the random samples as the conditional probability of the overlap in the random samples given the probability for the overlap in the two base sets. But obviously, my intuition fails me...

If you could point me in the right direction how to calculate this accurately, I would be very grateful, for I am nothing but a sinner in the hands of the angry god of probability. If you show me the light so that I can start to atone for my ignorance by studying probability.

Cheers

UsrX