Expected Value Problems
Brown Pre-calculus, p. 634
PT: Expected Value, Sample Space
RW: Test Taking
On a multiple-choice test, a student is given five possible answers for each
question. The student receives 1 point for a correct answer and loses ¼
point for an incorrect answer. If the student has no idea of the correct
answer for a particular question and merely guesses, what is the student’s
expected gain or loss on the question?
Suppose also that on one of the questions you can eliminate two of the
five answers as being wrong. If you guess at one of the remaining three
answers, what is your expected gain or loss on the question?
Brown Pre-calculus, p. 634
PT: Expected Value
RW: Farming/Reading a Table
A dairy farmer estimates for the next year the farm’s cows will produce
about 25,000 gallons of milk. Because of variation in the market price of
milk and cost of feeding the cows, the profit per gallon may vary with the
probabilities given in the table below. Estimate the profit on the 25,000
gallons.
Gain per
gallon
Probability
$1.10
0.30
$0.90
0.38
$0.70
0.20
$0.40
0.06
$0.00
0.04
-$0.10
0.02
Brown Pre-calculus, p. 634
PT: Expected Value
RW: Life Insurance
At many airports, a person can pay only $1.00 for a $100,000 life
insurance policy covering the duration of the flight. In other words, the
insurance company pays $100,000 if the insured person dies from a
possible flight crash; otherwise the company gains $1.00 (before
expenses). Suppose that past records indicate 0.45 deaths per million
passengers. How much can the company expect to gain on one policy?
On 100,000 policies?
Brown Pre-calculus, p. 635
PT: Expected Value
RW: Bidding on jobs
A construction company wants to submit a bid for remodeling a school.
The research and planning needed to make the bid cost $4000. If the bid
were accepted, the company would make $26,000. Would you advise the
company to spend the $4000 if the bid has only 20% probability of being
accepted? Explain your reasoning.
Brown Pre-calculus, p. 635
PT: Expected Value
RW: Consumer Economics
Suppose the warranty period on your family’s new television is about to
expire and you are debating about whether to buy a one-year maintenance
contract for $35. If you buy the contract, all repairs for one year are free.
Consumer information shows that 12% of the televisions like yours
require an annual repair that costs $140 on the average. Would you advise
buying the maintenance contract? Explain your reasoning.
Shaquille O'Neal is one of the best players on the professional basketball
team the Los Angeles Lakers. Shaq, as he is nicknamed, stands 7' 1" tall
and weighs 330 pounds. Most of the shots he takes are close to the basket,
and because he is so big other players have a hard time stopping him from
making baskets. In fact, he makes 57.2% of his shots, which is
impressive given that most players make about 45%.
In basketball, when a player trying to make a shot is hit on the body by
someone on the opposing team, thereby causing the player to miss the
shot, the player gets to take two free shots from 15 feet away from the
basket. These shots are called foul shots. Shaq does not shoot foul shots
very well. In fact, he makes only 51.3% of his foul shots.
Regular shots are worth two points. Foul shots are worth one point each.
Because Shaq is less likely to make foul shots, one strategy is to foul him
whenever he touches the ball. This strategy has been nicknamed the
"hack-a-Shaq." Let's see if the hack-a-Shaq pays off.
a. Calculate the expected value of the number of points Shaq scores on
one regular shot (not foul shots) at the basket (i.e., he makes or misses the
shot). Regular shots are the ones he has a 57.2% chance of making.
b. Assume that all foul shots are independent events (examinations of foul
shooting records suggest this is approximately true). Calculate the
expected value of the number of points Shaq gets when he shoots two foul
shots.
c. Compare the EV for foul shots to the EV for regular shots. Based on
these expected values, should opposing teams adopt the hack-a-Shaq?
You can choose to play one of two games. Each game costs one dollar to
play.
Game 1: A wheel with three numbers on it--zero, one, and two--is spun so
that there is a 40% chance that the wheel lands on zero, a 10% chance the
wheel lands on one, and a 50% chance the wheel lands on two. You get
back the amount in dollars of the number that the wheel lands on.
Game 2: A different wheel with three numbers on it--zero, one, and two-is spun so that there is a 5% chance that the wheel lands on zero, a 80%
chance the wheel lands on one, and a 15%chance the wheel lands on two.
You get back the amount in dollars of the number that the wheel lands on.
a. What is the expected amount of money that you will get back from
Game 1? What is the expected amount of money that you will get back
from Game 2? (Don't forget to answer both of these questions.)
b. If you played these games every second of every day for the rest of
your life, which statement is most likely to be true: a) you will make more
money playing Game 1; (b) you will make more money playing Game 2;
or, (c) you will make the same amount of money playing either game.
Justify your answer in two or less sentences.
c. In Game 1, what is the probability that you will not lose money? In
Game 2, what is the probability that you will not lose money?