Q:

Statistics help, probability and expected value

A bus has 15 designated stops other than its starting/ending point. During its last trip, the bus only stops at a designated stop if any passenger wishes to get down there.
Suppose on the last trip yesterday the bus started with 20 passengers. From the driver’s point of view they were equally likely to get down at any of the 15 stops.
Calculate the expected number of stops made by the bus during that trip.
I'm really unsure what to do here, at first I thought it was a uniform distribution but then I didn't think you could do that with n people. If anyone can help, can you please tell me what method you used so I know for next time! Thanks!

A:

For $i=1$ to $15$, define the indicator random variable $X_i$ by $X_i=1$ if the bus stops at stop $i$ and by $X_i=0$ otherwise.  The number $Y$ of stops is then given by
$$Y=X_1+X_2+\cdots+X_{15}.$$
By the linearity of expectation, we have
$$E(Y)=E(X_1)+E(X_2)+\cdots +E(X_{15}).$$
To find $E(X_i)$, we find $\Pr(X_i=1)$.
The probability that $X_i=1$ is $1-p$ where $p$ is the probability that no one wants to step off at stop $i$. We have $p=(14/15)^{20}$.

