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Brown Pre-calculus, p.627

M: Availability

PT: Conditional Probability, Bayes’ Theorem

RW: Insurance

An auto insurance company charges younger drivers a higher premium than it does older drivers because younger drivers as a group tend to have more accidents.  The company has 3 age groups: Group A includes those under 25 years old, 22% of all its policyholders.  Group B includes those 25-39 years old, 43% of all its policyholders, Group C includes those 40 years old and older.  Company records show that in any given one-year period, 11% of its Group A policyholders have an accident.  The percentages for groups B and C are 3% and 2%, respectively.

b)   Suppose Mr. X has just had a car accident.  If he is one of the company’s policyholders, what is the probability that he is under 25?

Say that this company not only classifies drivers by age, but in the case of drivers under 25 years old, it also notes whether they have had a driver’s education course.  One quarter of its policyholders under 25 have had a drivers education course and 5% of these have an accident in a one-year period. Of those under 25 who have not had a driver’s education course, 13% have an accident within a one-year period.  A 20-year-old woman takes out a policy with this company and within one year she as an accident.  What is the probability that she did not have a driver’s education course?

Brown Pre-calculus, p. 628

M: Availability

PT: Conditional Probability, Bayes’ Theorem

RW: Disease Testing

A medical research lab proposes a screening test for a disease.  To try out this test, it is given to 100 people, 60 of whom are known to have the disease and 40 of whom are known not to have the disease.  A positive test indicates the disease and a negative test indicates no disease.  Unfortunately, such medical tests can produce two kinds of errors:

2)   A false positive test:  For the 40 people who do not have the disease, this screening test indicates that 10 do have it.

a)   Which of the false tests do you think is more serious and why?

b)   Incorporate the facts given above into a tree diagram.  (Be sure to convert the given integers into probabilities.)

c)   Suppose the test is given to a person whose disease status is unknown.  If the test result is negative, what is the probability that the person does not have the disease?

1)   A false negative result?

2)   A false positive result?

b)      What is the probability that:

1)   The diagnostic test gives the correct result?

2)   A person with a positive diagnostic test has the disease?

3)   A person with a negative diagnostic test does not have the disease?

Key to Problem Bank: