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An Introduction to Decision Theory

Martin Peterson

Chapter 7

The philosophy of probability - all with Video Answers

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Chapter Questions

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Problem 1

In Chapter 6 we showed that $p(A)+p(\neg \mathrm{A})=1$. Verify that this theorem comes out as true in the classical interpretation.

Nick Johnson
Nick Johnson
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Problem 2

Summarise the major problems with the classical interpretation. In your view, which problem poses the greatest difficulty for the classical interpretation?

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Problem 3

Prove that $p(A)+p(\neg \mathrm{A})=1$ in the frequency interpretation (without using Kolmogorov's axioms).

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01:37

Problem 4

According to the frequency interpretation, probability is always defined relative to some reference class. What does this mean, and why might this spell trouble?

Christopher Stanley
Christopher Stanley
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02:19

Problem 5

By definition, unique events (such as marriages or stock market collapses) never occur more than once. What do advocates of the following interpretations have to say about unique events: (a) the classical interpretation, (b) the frequency interpretation, (c) the propensity interpretation and (d) the subjective interpretation?

Alexa Moschella
Alexa Moschella
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Problem 6

Consider the decision matrix below. You know that your marginal utility of money is decreasing, but you don't know by how much.
(a) Suppose that you are indifferent between Option 1 and Option 2 . What can you conclude about your subjective probability for the two states?
(b) Suppose that you strictly prefer Option 1 to Option 2. What can you conclude about your subjective probability for State 1, as compared to State 2 ?
(c) Suppose that the $$\$ 100$$ outcome is replaced with a black box with unknown contents, and that you feel indifferent between the two options. What can you conclude about your subjective probability for States 1 and 2?
$$
\begin{array}{lcr}
\hline & \text { State 1 } & \text { State 2 } \\
\hline \text { Option 1 } & \$ 100 & 0 \\
\text { Option 2 } & 0 & \$ 100 \\
\hline
\end{array}
$$

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Problem 7

You are indifferent between receiving $$\$ 10,000$$ for sure or five times that amount if and only if a Pakistani wins a gold medal in the next Olympic Games. Your utility for money is linear. What is your subjective probability that a Pakistani will win a gold medal in the next Olympic Games?

Victor Salazar
Victor Salazar
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01:00

Problem 8

How did Savage define acts?

Dominick Hankle
Dominick Hankle
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Problem 9

What is the sure-thing principle? Why is it controversial?

Susan Hallstrom
Susan Hallstrom
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01:08

Problem 10

Explain briefly, using a non-technical vocabulary, how Savage manages to derive both a utility function and a subjective probability function from a set of preferences over uncertain prospects.

Heather Zimmers
Heather Zimmers
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Problem 11

Explain the idea that the utility of an outcome may sometimes be statedependent.

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01:04

Problem 12

What is a Dutch Book?

Narayan Hari
Narayan Hari
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02:12

Problem 13

Construct a Dutch Book to show that if $A$ is impossible, then $p(A)=0$.

Manisha Sarker
Manisha Sarker
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01:31

Problem 14

Explain the difference between the betting approach to subjective probability and the qualitative approach taken by DeGroot. What are the merits and drawbacks of each approach?

Joshua Sieverding
Joshua Sieverding
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