Probability of 2 different people choosing some of the same marbles from a jar

Question

There are 20,412 unique marbles in a jar. One person picks 11 marbles, and puts them back and the jar is thoroughly mixed. Another person then picks 5 marbles. What is the chance that 2 of the 5 marbles were the same as 2 of the 11 that the first person picked? (This is actually a genetics question, there are about 20,214 genes).

Answer 1

Without Loss of Generality, consider the first person to have chosen the first 11 genes (you get the same result for any 11)

Then, if the second person chooses 5, the probability that he chose 2 of the first 11 in 5 samples is

C(11,2)C(20412-11,5-2)/C(20412,5) = C(11,2)C(20401,3)/C(20412,5) = 55*1414943996600/2.9514*10¹⁹=

2.63675 * 10^-6

Alternatively, using the hypergeometric distribution, in Excel, = hypgeom.dist(2,5,11,20412,0) =

2.63675 * 10^-6