problem 90

(a): Let $I_{j}$ be the indicator variable that $j$ person in Bob’s sample is also sampled by Alice. Then, $P (I_{j} = 1) = \frac{1}{10}$ . Then, the expected number of people in Bob’s and Alice’s samples is $2$ .
(b): $| A \cup B | = 100 + 20 - | A \cap B | \Rightarrow E (| A \cup B |) = 100 + 20 - E (| A \cap B |) = 100 + 20 - 2 = 118$ .
(c): Let $I_{j}$ be the indicator random variable for couple $j$ being in Bob’s sample. Then $P (I_{j} = 1) = \frac{(\frac{998}{18})}{(\frac{1000}{20})}$ . Thus, the expected number of couples in Bob’s sample is $E (X) = 500 \frac{(\frac{998}{18})}{(\frac{1000}{20})} \approx 0.2$ .