Solutions - 5120-01

(1) Compute $E[X]$:

There are $100$ total students.

Let $P[B_k]$ be the probability that Bus $k$ is selected, where $k\in \{1,2,3,4\}$.

Note that $P[B_k]=\frac{k}{10}$.

$$\begin{array}{c}E[X]=\sum _{k=1}^{4}P[B_k](10k)\\ \\ \gg \gg \sum _{k=1}^4{k}^2\gg \gg 30\end{array}$$

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(2) Compute $E[Y]$:

Let $P[B_k]$ be the probability that Bus $k$ is selected.

Since it’s based off the drivers, $P[B_k]=\frac{1}{4}$ for all $k$.

$$\begin{array}{c}E[Y]=\sum _{k=1}^{4}P[B_k]10k\\ \\ \gg \gg \frac{5}{2}\sum _{k=1}^{4}k\gg \gg 25\end{array}$$

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(3) Interpret solution:

For a driver, each bus is equally likely to be experienced.

For a student, you are more likely to find yourself on the bus with more students, so your experience is weighted towards the more crowded bus.