There are two boxes each has 10 balls numbered 1 through 10

There are two boxes, each has 10 balls numbered 1 through 10. A random sample of 6 balls is drawn, without replacement, from each box. Find the probability that the samples contain exactly 3 balls having the same numbers in common.

Solution

We have 20 balls in each boxes and we will take a sample of 6 balls without replacement (combinations) for each box

So the posibles cases are 10C6 = 210 for one box so for both boxes are 210+210 = 420 posibles cases

now the favorable cases are that exactly 3 balls having the same numbers in common

so that is 12C3 = 220 for 2 boxes

the probability is = 220/420 = 0.5238