Binomial Distribution (Statistics)

[haribol]

Hi guys, if you can help me with this problem it would be of great help

1) Pythag-Air-US Airlines has determined that 5% of its customers do not show up for their flights. If a passenger is bumped off a flight because of overbooking, the airline pays the customer $200. What is the expected payout by the airline, if it overbooks a 240 seat airplane by 5%?

PS: The answer is 0. Can you please explain your reasoning because I am completely lost in this one

Thanks a lot

 

[AKG]

It seems like an oddly worded question. By "it overbooks a 240 seat airplane by 5%", does that mean that 5% of the people booked are "extras" (i.e. 95% of the people booked = 240 people --> 252.63 people are booked) or that it books an extra 5% on top of the 240 people (i.e. 240 + 5% of 240 = the number of people booked = 252). Now, if it's the first one, then the answer is clearly zero. 5% of the people booked are extra, and 5% of the people do not show up, so chances are the people that have to be bumped off a flight is zero, so they have to payout $0. However, the problem with the first one is that it doesn't make sense to say they book 252.63 people. However, perhaps they mean that in total, of all their, say, 1000000 customers, 5% are overbooked, i.e. not 5% of 252.63, but 5% in general. If we go with the second option, then the answer is still zero. If they overbook by 5% as per the second definition, they book 252 people. 5% of them don't show, that's 12.6 people, so in total only 239.4 people show, .6 less than the maximum, so again on average no one has to be bumped off, and the payout is zero. Of course, there are statistical problems with this, and like I said the question is kind of vague, but two approaches both lead to the given answer, so it all works out.

 

[haribol]

Thanks AKG, I think I got it