9. Round Table. Petr Ivanovich, 49 other men, and 50 women sit around a round table in a random order. We will call a man satisfied if a woman sits next to him. Find:

a) (6-11 grades, 1 point) the probability that Petr Ivanovich is satisfied;

b) (8-11 grades, 1 point) the expected number of satisfied men.
Solution. a) Pyotr Ivanovich is dissatisfied only if men are sitting on both sides of him. The probability of this is $\frac{49}{99} \cdot \frac{48}{98}=\frac{8}{33}$. Therefore, the probability of the opposite event “Pyotr Ivanovich is satisfied” is $\frac{25}{33}$.

b) The probability that one man is satisfied is $\frac{25}{33}$. This is the expected number of satisfied men among one man. Then, among 50 men, the expected number of satisfied men is $50 \cdot \frac{25}{33}=\frac{1250}{33}$.