b$)$ Question $4(30$ points$)$

A flight operator sells tickets for a flight with a capacity of $150$ seats. The operator overbooks the flight to account for no$-$shows. Due to historical data, they know that the number of no$-$shows follows a uniform distribution between $2$ and $7$ passengers.

a$)$ The operator must compensate a passenger who is bumped with $$500.$ The regular price for each seat is $$200,$ and you can assume there is enough demand to fill all $150$ seats. If a passenger is a no$-$show and the flight isn't overbooked, the operator can call a passenger from the waitlist to fill the seat, but since this is last$-$minute, the operator only charges $$50$ for the seat. What is the optimal number of overbooked seats?

$C_u=200-50=$$$150,C_o=500$

$S{L}^{**}=\frac{c_u}{C_u+C_o}=\frac{150}{150+500}=0\mathrm{.}23$

$Q^{**}=a+(b-a)S{L}^{**}=2+5**\mathrm{.}23=3\mathrm{.}15$