A small regional carrier accepted 17 reservations for a...

A small regional carrier accepted 17 reservations for a particular flight with 14 seats. 9 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 51% chance, independently of each other. (Report answers accurate to 4 decimal places.) Find the probability that overbooking occurs. Find the probability that the flight has empty seats.

Transcript

00:01 The scenario for this question is a small regional aircraft carrier accepted 17 reservations for a flight that has 14 seats.

00:11 We were told that nine of the reservations went to regular customers who we know will arrive for the flight.

00:16 So that means there is an extra eight reservations that went to non -regular customers.

00:23 And we're told that the probability of each of these eight non -regular customers showing up is 51 percent, or 0 .51.

00:32 And we're asked to find the probability that an overbooking occurs.

00:38 So let's first define a random variable x as the number of the non -regular customers out of eight you show up for the flight.

00:44 So here each of these eight customers can be thought of as bernoulli trials with two outcomes of interest, either show up for the flight or not.

00:52 And it's given in the question that they arrive independently.

00:56 And so the number of successes in a fixed number of independent bernoulli trials is a binomial a random variable.

01:04 So here we can say x is a binomial based on eight trials and probability of success on each trial .5 .1.

01:11 Now in order to have an overbooking, we need more than 14 customers in total to show up.

01:18 We have the nine regular customers who are showing up.

01:21 So we need more than five of the non -regular customers showing up.

01:26 So we want to find the probability that x is greater than five.

01:30 This would result in an overbooking.

01:33 This can be expressed as one minus, the probability that x is at most 5.

01:42 And we can use software such as excel to solve this.

01:45 So we have 1 minus the probability that x is at most 5.

01:48 In excel, we started computation with an equal sign.