Problem #PRU-65352

Problem

In a terrible thunderstorm, along a rope ladder, $n$ dwarfs ascend in a chain. If suddenly there is a thunderbolt, then from fear, every gnome, regardless of others, can fall with probability $p$ ($0<p<1$). If the dwarf falls, then he also hits all of the dwarfs that are below him. Find:

a) The probability that exactly $k$ dwarfs will fall.

b) The mathematical expectation of the number of fallen dwarfs.