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Assignment 13.
Paradoxes of Indifference

For submission

A square has been selected among those of the diamond-shaped checkerboard above. Use Keynes' Principle of Indifference to determine the probability that the selected square is the square marked "a."

1. If our judgments about the selected square are indifferent over all the squares of the figure, not distinguishing the light from the dark, what is the probability that the selected square is "a"?

2a. If our judgments about the selected square are indifferent over which of the horizontal rows of the checkerboard contains the selected square, what is the probability that the second row contains the selected square?

2b. Under the same indifference as 2a, what is the probability that the selected square is "a"?

3a. If our judgments about the selected square are indifferent over whether it is dark or light, what is the probability that the selected square is dark?

3b. If we could know that a dark square has been chosen but we would otherwise be indifferent in our judgments over which that dark square is, what is the probability that the selected square is "a"?

3c. If the answers to 3a and 3b are combined, what is the probability that selected square is "a"?

4. Your answers to questions 1-3 should be different. What is the significance of that difference for Keynes' Principle of Indifference?

For Discussion

Not for submission

A. The common presumption is that, whenever our intuitions and the probability calculus disagree about something chancy, then our intuitions are wrong. Is this presumption correct? What justifies it?

B. The solution to the paradoxes of indifference in the chapter is to allow that some cases of indefiniteness cannot be represented by the additive measures of probability theory. Is this too severe a response? Should we be so ready to abandon probabilities?

C. The solution considers two options: a non-probabilistic representation of indefiniteness or abandoning any comparison of belief or strengths of support for mutually exclusive outcomes. Which is preferable?

D. In decision making, should we always choose the option with the highest expected value? By this criterion, we should never buy a lottery ticket or take out insurance. How should we decide these cases?

E. This chapter used expected values to guide strategies in various gambling games. If the expectation is negative, the games should be avoided. if you must play them, the harm is minimized by making the fewest, largest bets possible, as opposed to many smaller bets. If the expectation is positive, the recommendations were reversed. How does this theory match your practical experience with games of chance (if you have any)?