There are 8 students stepping into the elevator on the groun

There are 8 students stepping into the elevator on the ground floor in the a building (which has floors numbered {G, 1, 2, . . . , 11}. Each one of the students needs to get to a floor which is chosen uniformly at random from {1, ..., 11}. Let N be a random variable that counts the number of floors that the elevator stopped in (not counting G, where they got on), until the last student stepped out. Find the expected value of N.

Solution

Consider adding one student at a time. If the value is N after x students, then student x+1 has a (11-N)/11 = 1 - N/11 probability of choosing a new floor, so N increases by 1 - N/11

i.e. 1 student: N(1) = 1
2 students: N(2) = N(1) + 1 - N(1)/11
3 students: N(3) = N(2) + 1 - N(2)/11
etc

or counting u = 11-N, the number of floors the lift skips:
u(1) = 10
u(2) = u(1) - u(1)/11 = u(1) * 10/11 = 11 * (10/11)^2
u(3) = u(2) - u(2)/11 = u(2) * 10/11 = 11 * (10/11)^3