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

a) N (A∩B) = 0.9

b) N (A∩B) = 4.9

c) N (A or B) = 4.2

Step-by-step explanation:

Given that A and B each randomly, and independently, choose 3 of 10 objects;

P (A) = P (B) = 3/10 = 0.3

P (A') = P (B') = 1 - 0.3 = 0.7

a) chosen by both;

Probability of being chosen by both;

P (A∩B) = 0.3 * 0.3 = 0.09

Expected Number of objects being chosen by both;

N (A∩B) = P (A∩B) * N (total) = 0.09*10

N (A∩B) = 0.9

b) not chosen by either A or B;

Probability of not being chosen by either A or B;

P (A'∩B') = 0.7 * 0.7 = 0.49

Expected Number of objects being chosen by both;

N (A'∩B') = P (A'∩B') * N (total) = 0.49*10

N (A∩B) = 4.9

c) chosen by exactly one of A and B.

Probability of being chosen by exactly one of A and B

P (A∩B') + P (A'∩B) = 0.3*0.7 + 0.7 * 0.3 = 0.42

Expected Number of objects being chosen by both;

N (A or B) = 0.42 * 10

N (A or B) = 4.2