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) the probability that Petr Ivanovich is satisfied;

b) the expected number of satisfied men.
a) Pyotr Ivanovich is dissatisfied only if men are sitting on both sides of him. The probability of this is $49 / 99 \cdot 48 / 98=8 / 33$.

Therefore, the probability of the opposite event "Pyotr Ivanovich is satisfied" is 25/33.

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

## Answer

a) $25 / 33 ;$ b) $1250 / 33$