[ Geometric progression ]

At an Antarctic station, there are $n$ polar explorers, all of different ages. With a probability $p$, any two explorers will develop a friendly relationship, independently of other likes or dislikes. 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 $E I_{k}=(1-$ $p)^{k-1}$. 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 $p=1 \mathrm{E}=1$.

## Answer

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