#4 Need this done asap please. Only need the answers in the format that the ques

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#4

Need this done asap please. Only need the answers in the format that the question requests. Thanks alot in advance!

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table 45 46 4748 49 50 51 52 53 54 55 p(y) 0.05 0.10 0.13 0.14 0.24 0.17 0.06 0.05 0.03 0.02 0.01 (a) What is the probability that the flight will accommodate all ticketed passengers who show up? (b) What is the probability that not all ticketed passengers who show up can be accommodated? (c) If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

Explanation / Answer

Since the probability distribution table has been given, all the parts can be answered by looking at the data.

(a)

We know that the flight has a maximum of 50 seats.

So,

P(flight can accomodate all ticketed passengers who show up) = P(number of ticketed passengers showing up is less than or equal to 50) = P(Y <= 50)

Now,

P(Y <= 50) = P(Y=45) + P(Y=46) + P(Y=47) + P(Y=48) + P(Y=49) + P(Y=50) = 0.05 + 0.10 + 0.13 + 0.14 + 0.24 + 0.17 = 0.83

(b)

This case can only occur when the number of passengers that show up exceed 50, which means Y > 50.

So,

P(Y > 50) = 1 - P(Y <= 50) = 1-0.83 = 0.17

(c)

If you are the first person on the stand by list, then you can get the accomodation if the number of ticketed passengers showing up is less than 50.

So,

P(Y < 50) = P(Y <= 50) - P(Y = 50) = 0.83-0.17 = 0.66

If you are the third person on the stand by list, then you can get the accomodation if the number of ticketed passengers showing up is less than 48.

So,

P(Y < 48) = P(Y=45) + P(Y=46) + P(Y=47) = 0.05 + 0.10 + 0.13 = 0.28

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