Homework 1
Due time: Oct. 7th
1. Suppose mortgage interest tax deductions average $8,268 for people with incomes
between $50,000 and $200,000 and $365 for those with incomes of $40,000 to
$50,000. Suppose also the standard deviations of the housing benefits in these two
categories were equal to $2,750 and $120, respectively.
(a) Examine the two standard deviations. What do these indicate about the range of benefits
enjoyed by the two groups? (10 points)
(b) Repeat part (a) using the coefficient of variation as the measure of relative variation. (10
points)
2. Online marketplaces such as Craigslist have become a popular way for individuals to buy
and sell miscellaneous items. The table below shows the numbers of days products stayed
active (not sold) on one of these sites and also the price range of the items.
(a) Using the relative frequency approach to probability assessment, what is the probability
that a product will be on the website more than 7 days?
(5 points)
(b) Is the event 1-7 days on the website independent of the price $200–$500? (10 points)
(c) Suppose an item has just sold and was on the website less than 8 days, what is the most
likely price range for that item? (5 points)
3. Beacon Hill Trees & Shrubs currently has an inventory of 10 fruit trees, 8 pine trees, and
14 maple trees. It plans to give 4 trees away at next Saturday’s lawn and garden show
in the city park. The 4 winners can select which type of tree they want. Assume they
select randomly.
(a) What is the probability that all 4 winners will select the same type of tree? (10 points)
(b) What is the probability that 3 winners will select pine trees and the other tree will be a
maple? (5 points)
(c) What is the probability that no fruit trees and 2 of each of the others will be selected? (5
points)
4. Three shooters shoot at the same target, each of them shoots just once. The first one
hits the target with a probability of 70%, the second one with a probability of 80% and
the third one with a probability of 90%. What is the probability that the shooters will hit
the target
(a) at least once? (10 points)
(b) at least twice? (10 points)
5. A doctor is called to see a sick child. The doctor has prior information that 90% of sick
children in that neighborhood have the flu, while the other 10% are sick with measles.
Let F stand for an event of a child being sick with flu and M stand for an event of a child
being sick with measles. Assume for simplicity that there no other maladies in that
neighborhood.
A well-known symptom of measles is a rash (the event of having which we denote R).
Assume that the probability of having a rash if one has measles is P (R|M ) = 0.95. However,
occasionally children with flu also develop rash, and the probability of having a rash if one
has flu is P (R|F ) = 0.08. Upon examining the child, the doctor finds a rash. What is the
probability that the child has measles? (20 points)