Exam 7: Expectations and Decisions

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

The mathematical expectation is a kind of average.

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?

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 $\frac{1}{4}$ .

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?

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?

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 $\frac{1}{3}$ .

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.

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 fair game is defined to be a game in which

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

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

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

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?

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

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

The mathematical expectation is always

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

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

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