Part I. True or False (2 pts. Each)
1. Inductive arguments always infer a universal statement from a set of particular statements.
2. Inductive arguments are evaluated as either valid or weak.
3. The conclusion of an inductive argument follows probabilistically from the premises.
4. An inductive argument is weak if the truth of the premises does not increase the likelihood of the conclusion.
5. An inductive argument is strong if the conclusion is likely to be true given that the premises are true.
6. Statistical generalizations can be strong regardless of whether the sample is randomized.
7. It is important for an inference to the best explanation to be falsifiable.
8. The probability of a disjunction is calculated using multiplication.
9. The probability of two events occurring together is calculated using addition.
10. The expected monetary value of an outcome, unlike the overall value of an outcome, does not depend on an individual's values and preferences.
Part II. Matching. Problems 1-5 display the right side of equations that you should be familiar with. Match them with what they represent from the word bank below. (3 pts. Each)
A. Expected monetary value.
B. Probability of two events happening together (with independence).
C. Probability of two events happening together (without independence).
D. Probability of an event not happening.
E. Expected overall value.
F. Probability of at least one of two events happening (with exclusivity).
G. Probability of at least one of two events happening (without exclusivity).
H. Bayes' Theorem.
1. ?=Pr(h1) + Pr(h2)
2. ?=Pr(h1) Pr(h2)
3. ?=1 - Pr(h).
4. ?= (Pr(h) net gain) - (Pr(not h) net loss).
5. ?=Pr (h1) + Pr(h2) - Pr( h1 & h2)
Part II. (1) Identify the argument as inductive or deductive. (2) If the argument is inductive, then identify the kind of inductive argument it is (statistical generalization, argument from analogy, etc.). (3) If the argument is inductive, identify whether the argument is strong or weak, and explain why. (4) If the argument is deductive, identify whether the argument is valid or invalid, and explain why. (3 pts. each)
1. I know that I am conscious. You have the same body-type, and react to stimuli in roughly the same way as I do. Therefore, you are probably conscious (not a zombie).
2. I have lots of friends. Most of them think that I would make a great president. So most Americans would probably agree.
3. The Matterhorn is higher than Mount Whitney, and Mount Whitney is higher than Mount Rainier. The obvious conclusion is that the Matterhorn is higher than Mount Rainer.
4. Because we both live in Utah, and I have seen a rattlesnake, it follows that you must have seen a rattlesnake.
5. James fished all day in the Allegheny River, but did not catch any fish. James infers from this that there are no fish in the Allegheny River.
Part IV. Probability. Compute the probability. Show your work. (5 pts. Each)
1. What is the probability of picking a black jack from a standard 52-card deck?
2. What is the probability of drawing at least one ace from a standard 52-card deck on two draws if the first card is replaced before the second is drawn?
3. Given an urn containing three red balls, four green balls, and five yellow balls, what is the probability of drawing a red ball on a single draw?
4. What is the probability of getting heads on three successive tosses of a coin?
5. What is the probability of drawing either an ace or a king on a single draw from a standard 52-card deck?
Decisions under Ignorance (25 pts.)
You are offered two jobs, but you can only take one. Job Y has offered you a salary of $40,000/year. Job Z has offered you a salary of $60,000/year. Assume that Job Y and Job Z are equally desirable in all respects other than their expected monetary value. Job Z is a new company that may go out of business, while Job Y is an established company that (let's assume) has a 0 percent probability of going out of business. Determine which choice each of the following rules favors: (1) rule of dominance; (2) principle of insufficient reason; (3) maximax rule; (4) minimax rule. Explain your reasoning.