11. Polar Friends (from 8th grade, 2 points). At an Antarctic station, there are $n$ polar explorers, all of different ages. With a probability $p$, any two polar explorers will develop a friendly relationship, independently of other affinities 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.
Solution. We will assign the polar explorers numbers from 1 to $n$ according to their age - the older they are, the smaller the number. The polar explorer with number $j$ can receive advice from $j-1$ older polar explorers. Therefore, he will not receive any advice only if he is not friends with any of them. The probability of this is $(1-p)^{j-1}$. Therefore, the event "polar explorer $j$ did not receive any advice at all" has an indicator $I_{j}$ with the expected value $\mathrm{E} I_{j}=(1-p)^{j-1}$. The total number of polar explorers who did not receive any advice is

$$
I_{1}+I_{2}+\ldots+I_{n}
$$

Let's move on to the expected value:

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

Answer: $\frac{1-(1-p)^{n}}{p}$.