An airline has collected an i.i.d. sample of $10000$ flight reservations and figured out that in this sample $5$ percent of passengers who made a reservation did not show up for the flight. They introduce a policy to sell $100$ tickets for a flight that can hold only $99$ passengers. Consider the following process of generating the two samples:
- We sample $10100$ passenger show up events independently at random according to an unknown distribution $p$.
- And then we split them into $10000$ passengers in the collected sample and 100 passengers booked for the $99$-seats flight.
Bound the probability of observing a sample of $10000$ with $95$% show ups and a $99$-seats flight with all $100$ passengers showing up by following the above sampling protocol. If you do things right, you can get a bound of about $0.0062$.
My idea was to apply Hoeffding's inequality but i do not quite understand how to apply it in this case. I also thought about using that $P(A)\geq P(A \text{and} B)$ but i think i might have misunderstood the assignment.
Each flight is going to be an "experiment" on its own. All you know is the $p$ you (the problem) calculated (i.e., the 5%), and that $p$ is a good approximation because of how big the survey was.
My first inclination would be to model this problem as a binomial random variable on $100$ seats.
That could theoretically be a pain in the butt, because of how the CDF is structured, but this time you only need to deal with the probability of a single event occurring...
If the problem involved complicated CDF calculations, I'd be looking at using a Central Limit theorem.
Good luck on your exam.