Suppose that A and B each randomly and independently choose

Suppose that A and B each randomly and independently choose 3 of 10 objects. Find the expected number of objects chosen by both A and B; not chosen by either A or B; chosen by exactly one of A and B.

Solution

Probability that item is chosen=0.3(3/10, assuming all items are chosen together)

The given distribution is a bernoulli distribution with p=0.3, q=1-0.3=0.7 for a single person

P(Selected by both) = 0.3*0.3 = 0.09

P(Selected by none) = 0.7*0.7 = 0.49

P(Selected by one) = 0.3*0.7+0.7*0.3 = 0.42(As when it is selected by one, it is not by other and vice versa, so added twice)

(a) Expected number of objects chosen by both= 10*p = 10*0.09 = 0.9~1

(b)Expected number of objects chosen by none = 10*p = 10*0.49 = 4.9~5

(c)Expected number of objects chosen by exactly one = 10*p = 10*0.42=4.2~4

This is assuming 10 objects are evetually chosen. If you want to choose 3, multiply with 3 to get the answer (= 0.27,1.47 and 1.26 respectively)