Deck 7: Expectations and Decisions

Question 1

A fair game is defined to be a game in which

A) the probabilities of all events are equal.
B) the sum of all probabilities equals 1.
C) $E=1$
D) $E=0$

Answer 1

$E=0$

Question 2

For a payoff table containing company profit values, a pessimistic manufacturer would most likely use the _______ criterion.

A) minimax
B) $\operatorname{maximax}$
C) maximin
D) minimin

Answer 2

maximin

Question 3

The mathematical expectation is always

A) a probability.
B) the most likely value to occur.
C) a long run expected average.
D) a positive value.

Answer 3

a long run expected average.

Question 4

You are given two states of nature $S 1$ and $S 2$ and two courses of action $A 1$ and $A 2$ with a profit payoff table.


-Using the table above, answer the following: If the probabilities of $S 1$ and $S 2$ are, respectively, $\frac{4}{5}$ and $\frac{1}{5}$ , the expected profit if $A 1$ is selected is

A) 36 .
B) 34 .
C) 18 .
D) 26 .

Question 5

You are given two states of nature $S 1$ and $S 2$ and two courses of action $A 1$ and $A 2$ with a profit payoff table.


-Using the table above, answer the following: The action which maximizes expected profit is

A) $A 1$ since $34>26$ .
B) $A 1$ since $36>18$ .
C) $A 2$ since $34>26$ .
D) $A 2$ since $36>18$ .

Question 6

You are given two states of nature $S 1$ and $S 2$ and two courses of action $A 1$ and $A 2$ with a profit payoff table.


-Using the table above, answer the following: The decisions $A 1$ and $A 2$ have the same expected profit if the probabilities of $S 1$ and $S 2$ are, respectively,

A) $\frac{1}{2}$ and $\frac{1}{2}$ .
B) $\frac{1}{5}$ and $\frac{4}{5}$ .
C) $\frac{1}{3}$ and $\frac{2}{3}$ .
D) $\frac{2}{3}$ and $\frac{1}{3}$ .

Question 7

If the prediction of a consultant must be exact with no reward for being close, the best prediction is

A) the mean.
B) the median.
C) the mode.
D) none of these

Question 8

If the amount that the consultant is paid is reduced by a penalty which is 20 times the size of the error, the prediction which maximizes his payment is

A) the mean.
B) the median.
C) the mode.
D) none of these

Question 9

If the amount that the consultant is paid is reduced by a penalty which is 20 times the square of the size of the error, the prediction which maximizes his payment is

A) the mean.
B) the median.
C) the mode.
D) none of these

Question 10

For a payoff table containing company profit values, an optimistic manufacturer would most likely use the _______ criterion.

A) minimax
B) $\operatorname{maximax}$
C) maximin
D) minimin

Question 11

If 600 lottery tickets are sold for a cash price of $\$ 150$ , what is the mathematical expectation of a person who buys one of these tickets for $\$ 1$ ?

Question 12

A grab-bag contains 6 packages worth $\$ 2$ each, 11 packages worth $\$ 3$ each, and 8 packages worth $\$ 4$ each. Is it reasonable to pay $\$ 3.50$ for the option of selecting one of these packages at random?

Question 13

A real estate investor can buy a house for $\$ 70,000$ . He estimates that he can sell it in a year for $\$ 70,000, \$ 72,000$ , $\$ 75,000, \$ 80,000$ , and $\$ 82,000$ with probabilities, respectively, of $0.15,0.20,0.35,0.18$ , and 0.12 . If he buys the house, what is his expected gain in one year?

Question 14

Two finalists are in a tennis match with the winner to receive $\$ 40,000$ and the loser $\$ 25,000$ . What are the two players' mathematical expectations if they are equally matched?

Question 15

Two finalists are in a tennis match with the winner to receive $\$ 40,000$ and the loser $\$ 25,000$ . What are the two players' mathematical expectations if their probabilities of winning are $\frac{2}{3}$ and $\frac{1}{3}$ ?

Question 16

An entrepreneur is considering building a new restaurant. She knows that if the restaurant is built and the economy improves there will be a profit next year of $\$ 70,000$ ; if the restaurant is built and there is a recession, there will be a loss next year of $\$ 10,000$ . If she buys a particular established restaurant and the economy improves, there will be a profit of $\$ 40,000$ next year; if she buys the established restaurant and there is a recession, there will be a profit of $\$ 8,000$ next year.

-Construct a payoff table for the situation described above.

Question 17

An entrepreneur is considering building a new restaurant. She knows that if the restaurant is built and the economy improves there will be a profit next year of $\$ 70,000$ ; if the restaurant is built and there is a recession, there will be a loss next year of $\$ 10,000$ . If she buys a particular established restaurant and the economy improves, there will be a profit of $\$ 40,000$ next year; if she buys the established restaurant and there is a recession, there will be a profit of $\$ 8,000$ next year.
-For the situation above, find the expected profit of perfect information if the probability of a recession is $\frac{1}{4}$ .

Question 18

An entrepreneur is considering building a new restaurant. She knows that if the restaurant is built and the economy improves there will be a profit next year of $\$ 70,000$ ; if the restaurant is built and there is a recession, there will be a loss next year of $\$ 10,000$ . If she buys a particular established restaurant and the economy improves, there will be a profit of $\$ 40,000$ next year; if she buys the established restaurant and there is a recession, there will be a profit of $\$ 8,000$ next year.
-For the situation above, find the expected profit of perfect information if the probability of a recession is $\frac{1}{3}$ .

Question 19

The following payoff table shows two surveys of data for the monthly rental prices for equivalent apartments in four different locations in a certain city.

-Use the situation above to solve the following: A builder wants to construct apartments of the type referred to in the table and must select one of the four locations A, B, C, or D. Which location should he choose if he wants to maximize his minimum rental charge?

Question 20

The following payoff table shows two surveys of data for the monthly rental prices for equivalent apartments in four different locations in a certain city.

-Use the situation above to solve the following: A builder wants to construct apartments of the type referred to in the table and must select one of four locations A, B, C, or D. Which location should he choose if he wants to maximize his maximum rental charge?

Question 21

A speculator evaluating a particular investment concludes that, if the company's balance sheet claims are true, the speculator's profit will be $\$ 15,000$ , and if the company's balance sheet claims are false, he will lose $\$ 5,000$ . If he makes no investmentrbethas no or loss.

-Construct a payoff table for the decision problem above.

Question 22

A speculator evaluating a particular investment concludes that, if the company's balance sheet claims are true, the speculator's profit will be $\$ 15,000$ , and if the company's balance sheet claims are false, he will lose $\$ 5,000$ . If he makes no investmentrbethas no or loss.

-In the situation above, if the speculator feels that the probabilities are above 0.20 and 0.80 that the balance sheet claims are true or false, respectively, what should he decide to do to maximize his expected profit?

Question 23

A speculator evaluating a particular investment concludes that, if the company's balance sheet claims are true, the speculator's profit will be $\$ 15,000$ , and if the company's balance sheet claims are false, he will lose $\$ 5,000$ . If he makes no investmentrbethas no or loss.

-Suppose that the speculator's accountant feels that the spectator rates the accuracy of the balance sheet claims too low, that the probabilities of the claims being true or false are 0.30 and 0.70 . Would it affect the speculator's decision if he assigned these probabilities to the two events?

Question 24

The values in the following table are the prices charged for three menu items at three different restaurants of equal quality:

-Use the situation above to solve the following: A person who has narrowed his choice of entree to one of the three items is planning on dining at one of the three restaurants and will make his final choice of entree after having a cocktail. Which restaurant would he go to if he applied the minimax criterion to the amount of money he will have to pay?

Question 25

The values in the following table are the prices charged for three menu items at three different restaurants of equal quality:

-Use the situation above to solve the following: A person who has narrowed his choice of entree to one of the three items is planning on dining at one of the three restaurants and will make his final choice of entree after having a cocktail. Which restaurant would he go to if he applied the minimin criterion to the amount of money he will have to pay?

Question 26

A salesperson must choose between a straight salary of $\$ 25,000$ and a salary of $\$ 20,000$ plus a bonus of $\$ 8,000$ if his sales exceed a certain quota. How does he assess his probability of exceeding the quota if he chooses the lower salary with the possibility of a bonus?

Question 27

A student wants to purchase a ticket to a very popular rock concert from a friend. He agrees to pay the friend $\$ 30$ , but will only get the ticket if he rolls a 5 or a 6 on an ordinary die, otherwise the friend keeps the ticket and the $\$ 30$ . What is the ticket worth to the student if he thinks that this arrangement is fair?

Question 28

The annual income (in thousands of dollars) of all ten entries in a lottery are $30,25,50,48,30,30,60,42,38,15$ , and their chances of winning are all equal.
-If we want to predict the annual income of the winner, what prediction maximizes the expected reward if there is a reward for being right but none for being close?

Question 29

The annual income (in thousands of dollars) of all ten entries in a lottery are $30,25,50,48,30,30,60,42,38,15$ , and their chances of winning are all equal.
-If we want to predict the annual income of the winner, what prediction maximizes the expected reward if there is a penalty equal to 15 times the size of the error?

Question 30

The annual income (in thousands of dollars) of all ten entries in a lottery are $30,25,50,48,30,30,60,42,38,15$ , and their chances of winning are all equal.
-If we want to predict the annual income of the winner, what prediction maximizes the expected reward if the penalty is equal to 15 times the square of the size of the error?

Question 31

What is Bayesian analysis?

Question 32

The maximin criterion applies only when the payoffs are quantities, such as profits or benefits, which we want to make large.

Question 33

If the payoffs are quantities such as losses or costs, which we want to make small, we use the minimax criterion.

Question 34

If the payoffs are losses or costs, the optimistic decision maker would use the minimax criterion.

Question 35

The mathematical expectation is a kind of average.

Question 36

In Bayesian analysis, probabilities are assigned to the states of nature.

Question 37

The maximum expected profit in a payoff table is calculated using the row amounts.

Question 38

In a payoff table, the columns indicate the decisions to be made.

Question 39

Suppose the mathematical expectation is given with the corresponding probabilities and all amounts but one amount $a$ . Solving for $a$ requires $E$ to be multiplied by a number.

Question 40

When applying the _______ criterion, we look for the largest value of each column of the payoff table and select the largest of these column values.

Question 41

In order to make a decision using the minimax criterion, we select the action that gives the _______ of the _______ opportunity losses.

Question 42

When calculating the mathematical expectation, amounts which represent losses are assigned _______ values.

Question 43

In Bayesian analysis, we select the action which promises the _______ expected profit or the _______ expected loss.

Question 44

A game in which the mathematical expectation of each player is zero is called a _______.

Question 45

Calculating the mathematical expectation requires having the values for the amounts and their corresponding _______.

Question 46

The expected profit with perfect information for a profit payoff table is calculated by using the values in each _______ (row, column).

Question 47

The expected value with perfect information is calculated by subtracting the maximum expected profit under uncertainty from _______.

Question 48

Given the profit payoff table below, if we use the maximin criterion we will choose action _______.

Question 49

Given the profit payoff table below, if we use the maximax criterion we will choose action _______.