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Chapter 2 Discrete Random Variables and Probability Distributions
10. A multiple-choice examination contains 4 choices for each of 100 questions. a. Find the exact probability that a student who guesses misses, at most, 4
questions.
b. Approximate the probability in part a using the Poisson distribution. 11. The number of earthquakes of destructive magnitude in California follows a Pois-
son distribution with one such earthquake expected each year. What is the proba- bility of at least 3 such earthquakes in a six-month period?
12. A quality control inspector follows the following plan in inspecting soccer balls
that are produced according to a Poisson process with 4 soccer balls expected each minute. The produced balls fall into a bin that automatically empties at the end of
each minute. If the bin collects exactly 3 balls, the inspector takes them out for possible inspection of 10 seconds each. He flips a fair coin for each and inspects
them only if a head appears. If the bin should contain 5 balls, he spends 5 seconds inspecting each ball. Otherwise, the inspector does not inspect the output. What is
the average amount of time per minute spent in inspecting the soccer balls?
13. Major crimes are reported at an average rate of 5 per night in a given police
precinct. The number of these crimes is assumed to follow a Poisson distribution.
a. What is the probability that on a given night no more than 3 major crimes will
be reported?
b. What is the chance that a full week will pass with no more than 3 major crimes
reported on any of the 7 nights?
14. An airline knows that 10 of the people holding reservations for a certain flight
will not appear. The plane holds 90 people. Use the Poisson approximation in answering the following questions:
a. If 95 reservations have been sold, what is the probability that everyone who
appears for the flight can be accommodated?
b. How many reservations should be sold so the probability that the airline can
accommodate everyone who appears is at least 0.99?
15. Molecules of a rare gas occur at an average rate of 3 per cubic foot of air and
follow a Poisson distribution.
a. What is the probability that a cubic foot of air contains none of the molecules? b. What is the probability that 3 cubic feet of air contain exactly 4 of the
molecules?
c. How much air must be taken as a sample to make the probability at least 0.99
that at least one molecule will be found?
16. A librarian shelves 1000 books per day. If the probability that any particular book
is misshelved is 0.001 and if the books are shelved independently of one another,
2.14 The Poisson Process
169 a. What is the probability that at most 2 books are misshelved?
b. Approximate the probability in part a using the Poisson distribution. 17. A popular chocolate chip cookie “guarantees” at least 16 chocolate chips per
cookie. The actual number of chocolate chips per cookie, however, is a Pois- son random variable. What must be the average number of chips per cookie if
approximately 95 or more of the cookies are to meet the guarantee?
18. A bakery makes a batch of 1000 chocolate chip cookies and adds n chocolate chips
to the batter for each batch and mixes the batter well. Under these assumptions, the number of chocolate chips per cookie should follow a Poisson distribution.
a. If n = 4900, what is the probability that at least 2 chips are in a randomly
