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Assignment 13.
Paradoxes of Indifference

1. Keynes was indifferent to the color of an unseen book. Is is red? Black? Blue? Applying the principle of indifference to these three colors led to a simple paradox of indifference. Using Keynes' example as a model, find or invent another example with a similar structure.

2. A square has been selected randomly on the diamond-shaped checkerboard above. What is the probability that the center square has been selected, according to the principle of indifference...
a. ...if random means indifferent over which square is chosen, ignoring whether they are light or dark?
b. ... if "random" means indifferent over whether the chosen square is light or dark; and if dark, indifferent over each dark square?

Hint: you will need to use the product rule of probability for 2b:
P(center square chosen)
    = P(center square is chosen | chosen square is dark) x P(chosen square is dark)

3. It turns out to be impossible to be indifferent over both rows and columns at the same time; and then assign probabilities in accord with these dual indifferences. To see this:
a. If we are indifferent over whether the chosen square is in any of the five horizontal rows, what is the probability that the chosen square is the square marked a?
b. If we are indifferent over whether the chosen square is in any of the five horizontal rows, what is the probability that the chosen square is the square marked b.
c. If we are indifferent over whether the chosen square is in any of the five vertical columns, what is the probability that the chosen square is in the center column that contains squares a and b?
d. Are your answers to questions 3a, 3b and 3c consistent with one another? If not, what do you conclude?