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Probability Distributions

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1. A volunteer ambulance service handles 0 to 5 service calls on any given day. The probability distribution for the number of service calls is as follows.
Number of Service Calls Probability
0 .10
1 .15
2 .30
3 .20
4 .15
5 .10

a. Is this a valid probability distribution? Why or why not.
b. What is the probability of exactly 3 service calls?
c. What is the probability of 2 or more service calls?
d. What is the probability of 4 or less service calls?
e. What is the "expected value" [ E(x), the mean]of the distribution?
f. What is the standard deviation of the distribution?

2. Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a .90 probability of detecting a missile attack. This is a binomial distribution.
a. What is the "expected value" [ E(x), the mean]of the distribution?
b. What is the standard deviation of the distribution?
c. What is the probability that a single system will detect an attack?
d. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
e. If three detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
f. If the goal is ensure that the country is warned of a missile attack, would you recommend that multiple detection systems be used, based on your above results?

3. In a survey of MBA students, the following data were obtained on "students' first reason for application to the school in which they matriculated."

Reason for Application
Enrollment Status &#8595; School Quality Cost or Convenience Other Total
Full Time 421 393 76 890
Part Time 400 593 46 1039
Total 821 986 122 1929

a. What is the probability that a student drawn at random will be full time?
b. What is the probability that a student drawn at random will have chosen his/her school for either Quality or Cost?
c. If a student goes Part Time, what is the probability that school quality is the first reason? (HINT: probability of quality given part time).
d. If a student has chosen "Cost or Convenience", what is the probability that the student is full time? (HINT: probability of full time given "cost or convenience").
e. What is the probability that a student drawn at random will be both full time and have chosen his/her school for "other" reasons? (HINT: what is the intersection of Full time and Other?)
f. Compare your answers in (a) probability of full time and (d) probability of full time given "Cost or Convenience." Are enrollment status and reason for application independent or dependent? Why or why not (based on the first part of this sub-question).