At an Antarctic station, there are $n$ polar explorers, all of different ages. With probability $p$, any two polar explorers establish friendly relations, independently of other sympathies or antipathies. When the wintering period ends and it's time to return home, in each pair of friends, the older one gives a friendly advice to the younger one. Find the expected number of those who did not receive any friendly advice.
Let $0<p<1$. We will assign the polar explorers numbers from 1 to $n$ according to their age - the older they are, the smaller the number. The explorer with number $k$ can receive advice from $k-1$ explorers older than him. Therefore, he will not receive any advice only if he is not friends with any of them. The probability of this is $(1-p)^{k-1}$. Therefore, the event "explorer $k$ did not receive any advice at all" has an indicator $I_{k}$ with the expected value $\mathrm{E} I_{k}=(1-p)^{k-1}$. The total number of explorers who did not receive any advice is $I_{1}+I_{2}+\ldots+I_{n}$.

$\mathrm{E}\left(I_{1}+I_{2}+\ldots+I_{n}\right)=(1-p)^{0}+(1-p)^{1}+\ldots+(1-p)^{n-1}=1 / p\left(1-(1-p)^{n}\right)$.

For $p=0$, obviously, $\mathrm{E}=n$, and for $p=1 \quad \mathrm{E}=1$.

## Answer

$p$ when $p=0 ; 1 / p\left(1-(1-p)^{n}\right)$ when $0<p<1 ; 1$ when $p=1$.