# Bounding a probability of airplane

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:

1. We sample $$10100$$ passenger show up events independently at random according to an unknown distribution $$p$$.
2. 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.

• Not following. What "two samples" are you talking about? What is a "passenger show up event"? What is it you want to compute or estimate? – lulu Dec 16 '19 at 13:38
• Are you aware that (good) answers can be accepted $\left(\color{limegreen}{\checkmark} \right)$? – callculus Dec 16 '19 at 15:22

## 1 Answer

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.