# Calculating probability of Y<100

Consider an airline selling tickets for a flight that can hold $$100$$ passengers. The probability that a passenger shows up is $$0.95$$, and the passengers behave independently. Define with the random variable $$Y$$ the number of passengers that show up to board on the flight.

Suppose the airline sold exactly 100 tickets. What is the probability that the flight departs with empty seats: $$\Pr(Y < 100)$$?

• What’s the probability that everyone shows up? – Joe Apr 2 at 18:59
• The probability that a passenger shows up is 0.95. – user909773 Apr 2 at 19:00
• Yes, the probability that any one specific person shows up is 0.95. So what’s the probability that all 100 passengers show up? – Joe Apr 2 at 19:01
• If you’re stuck on next steps, here’s a thought. You can rely on an independence assumption to compute that. Each passenger is independent. – Arya McCarthy Apr 2 at 19:03

Let $$E$$ denote the event that the plane departs with one or more empty seats, i.e. that $$Y<100$$. Let $$F$$ denote the event that the plane departs full, i.e. $$Y=100$$. Note that $$F$$ is the complement of $$E$$, i.e. $$P(E)+P(F)=1$$. Now, $$F$$ occurs if and only if every passenger shows up. Therefore, by independence
$$P(F) = 0.95^{100}\approx 0.006$$
$$P(E) = 1 - 0.95^{100} \approx 0.994.$$
The fact that $$E$$ occurs with fairly high probability provides some justification for the practice of overbooking.