An arriving plane carries $r$ families. A total of $n_j$ of these families have cheked in a total of $j$ pieces of luggages, so we have $\sum_j n_j = r$.Suppose that when the plane lands, the $N=\sum_j jn_j$ pieces of luggage come out of the plane in random order. As soon as a family collects all of its luggages, it immediately departs from the airport. If the Sanchez family chcked in $j$ pieces of luggage, find the expected number of families that depart after they do.
I am thinking that maybe a recurrence relation on $E(j,N)$ can give us some clues, but I cannot find a nice closed form for it. Could someone elighten me on how to get started?
Guide:
Apply linearity of expectation:
For every family except the Sanchez family find the probability that it will depart after the Sanchez family.
Then the expected number of families that depart after the Sanchez family will equalize the summation of these probabilities.
Formally let $F$ be an index set for these families and for $f\in F$ let $X_f$ denote the random variable that takes value $1$ if the corresponding family departs after the Sanchez family and let it take value $0$ otherwise.
Then you are after:$$\mathbb E\sum_{f\in F}X_f=\sum_{f\in F}\mathbb EX_f=\sum_{f\in F}P(X_f=1)$$
For any of these families that checked in $j$ pieces of luggage (just like the Sanchez family) this probability is $0.5$ and there are $n_j-1$ such families.
For a family that checked in $k\neq j$ pieces of luggage things are more complicated.
In that case you must find the number of strings with - let's say - $k$ letters $X$ and $j$ letters $S$ (corresponding with Sanchez) that end with an $X$ and divide it by the total number of such strings.