Q. 7.8 - A First Course in Probability

Question

An arriving plane carries r families. A total of $n_j$ of these families have checked in a total of $j$ pieces of luggage, $\sum _i{n}_j=r$. Suppose that when the plane lands, the $N=\sum _{i}j{n}_j$ pieces of luggage come out of the plane in a random order. As soon as a family collects all of its luggage, it immediately departs the airport. If the Sanchez family checked in j pieces of luggage, find the expected number of families that depart after they do.

Step-by-Step Solution

Answer

The expected value is equal to $\sum _i{n}_i·\frac{i}{i+j}$.

Step 1: Given Information

An arriving plane carries $r$ families as $\sum _i{n}_j=r$.

Step 2: Explanation

The number of families that depart after Sanchez family can be described as $X=\sum _{k=1}^{r-1}{I}_k$

Observe that $k$ th family leaves later if and only if all Sanchez's luggages have been placed before the last luggage of $k$ th family. So, we can only consider $i+j$ luggages and the probability that the last luggage comes from the set that has $i$ members is equal to

$P\left(I_k=1\right)=\frac{i}{i+j}$

Step 3: Explanation

Now, we group random variables $I_k$ according to the number of luggages they posses. We have

$X=\sum _i\sum _{k_i=1}^{n_i}{I}_{k_i}$

Therefore, $E\left(X\right)=\sum _i\sum _{k_i=1}^{n_i}E\left(I_{k_i}\right)=\sum _i\sum _{k_i=1}^{n_i}P\left(I_{k_i}=1\right)$

$=\sum _i{n}_i·\frac{i}{i+j}$.

Step 4: Final answer

The expected value is equal to $\sum _i{n}_i·\frac{i}{i+j}$.