MATH 461- C13, Test 2, Spring 2007
April 20, 2007
Calculators, books, notes and extra papers are not allowed on this test
Show all of work to qualify for full credit
1. (10 points) At a certain bank, the amount of time (in minutes) that a customer spends
being served by a teller is an exponential random variable with parameter λ= 1/5.
(a) If there are no customers when you enter the bank, find the probability that your
serving time will not exceed 4 minutes.
(b) If there is a customer in service when you enter the bank, what is the probability
that he or she will still be with the teller after an additional 7 minutes?
Solution: Let Xbe the amount of time that a customer spends being served by a teller.
Then Xis exponential with parameter 1/5.
(a) P(X≤4) = 1 −e−4/5
(b) Let tbe amount of time the customer has been served before you entered the
bank. Then by the memoryless property of the exponential distribution P(X >
7 + t|X > t) = P(X > 7) = e−7/5.
2. (10 points) If 80 percent of the population of a large community is in favor of a proposed
rise in school taxes, use normal approximation to find the probability that a random
sample of 100 people will contain at least 90 who are in favor of the proposition.
Solution: Let X= the number of people in the random sample who are in favor of the
proposition. Then Xis hypergeometric. Since the population is large, hypergeometric
distribution can be approximated by a binomial distribution with parameters (100,0.8).
We approximate Xwith a normal variable with parameters (100(0.8),100(0.8)(0.2)) =
(80,16).
P(X≥90) = (continuity correction) = P(X≥89.5) = PX−80
√16 ≥89.5−80
√16
≈P(Z≥9.5
4) = P(Z≥2.375) = 1 −Φ(2.375) = (normal table)
= 1 −0.9912 = 0.0088 .
3. (10 points) Teams Aand Bplay a series of games. The series will end when one of
the teams wins 5 games. Suppose that team Awins each game with probability 1/4,
independently of the outcomes of other games. Find the probability that a total of 7
games are played.
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