What is the probability that at least 1 passenger will get off the bus at each stop?

Question

There are 20 passengers in a bus. They will randomly get off the bus in the next 3 stations $ S_1, S_2, S_3 $. What is the probability that at least one passenger will get off the bus at each of the stations? (this implies that 0 passengers could get off at a station)

Answer 1

Major hints and problem setup:

Consider the three events $A_1,A_2,A_3$ where $A_i$ represents the event that noone gets off at stop $i$.

We have then the probability we are interested in originally is $Pr(A_1^c\cap A_2^c\cap A_3^c)=1-Pr(A_1\cup A_2\cup A_3)$

Using inclusion-exclusion this simplifies further:

$=1-Pr(A_1)-Pr(A_2)-Pr(A_3)+Pr(A_1\cap A_2)+Pr(A_1\cap A_3)+Pr(A_2\cap A_3)-Pr(A_1\cap A_2\cap A_3)$

Each term in the above should be straightforward to calculate.

For example $Pr(A_1\cap A_2)$ represents the probability that noone gets off at stop one and noone gets off at stop two, i.e. everyone gets off at stop three. Assuming that each person gets off uniformly and independently at random from one another at one of the three stops, each person individually has a $1$ in $3$ chance to get off at the third stop and this needs to occur for each of the twenty people so this occurs with probability $\frac{1}{3}\cdot\frac{1}{3}\cdot\frac{1}{3}\cdots \cdot \frac{1}{3}=\frac{1}{3^{20}}$. The other terms are calculated similarly.