Question

Priority Mail is the U.S. Postal Service’s alternative to commercial express mail companies such as FEDEX. An articles in the Wall Street Journal presented some interesting conclusions comparing Priority Mail shipments with the much less expensive first-class shipments. When comparing shipments intended for delivery in three days, first class deliveries failed to deliver on time 19% of the time, whereas Priority Mail failed 33%of the time. Note that at the time of the articles, first-class deliveries started as low as $0.34 and Priority mail started at$3.50.
If 10 items are to be shipped first-class to 10 different destinations claimed to be in a three-day delivery location, what is the probability that
A) 0 items will take more than three days?
B) Exactly 1 item will take more then three days?
C) 2 or more will take more then three days?
D) What are the mean and the standard deviation of the probability distribution?

Answer 1

If 10 items are to be shipped first-class to 10 different destinations claimed to be in a three-day delivery location,
what is the probability that A) 0 items will take more than three days?
that means all the ten items took less that 3 days
the probability it takes more than 3 days is \(.19\) so the probability it takes 3 or less is \(1-.19=.81\)
the probability that all ten make in in 3 or less days is therefore
\[(.81)^{10}\]

Answer 2

Exactly 1 item will take more then three days?
binomial
\[\binom{10}{1}(.19)^1\times (.81)^{9}=10\times .19\times (.81)^9\] and then a calculator

Answer 3

C) 2 or more will take more then three days?
since we have just computed both no items take more than three days, and one item takes more than three days, two or more means "not 0, not 1"
take the above two answers, add them up, and subtract the result from 1, since they are complimentary events

Answer 4

D) What are the mean and the standard deviation of the probability distribution?
since this is a binomial distribution, the mean is \(np=10\times .81\) assuming we are looking at getting there on time, not late
and the standard deviation is \(\sqrt{np(1-p)}=\sqrt{10\times .81\times .19}\)