Probability theory, which is a basis for decision theory has always
been plagued by paradoxes, ever since the St. Petersburg paradox of the
18th century. There have been many interesting paradoxes since then.
Some are paradoxes only because people seem to act in an "irrational"
way. Some are paradoxes because the problem makes unrealistic
mathematical assumptions. And some actually do appear to be paradoxes.
Thus, can I know something about the color of my hat merely by seeing
the hats that others are wearing, even when all colors are chosen
randomly and independently?
Can two people both get richer by exchanging envelopes that they are
holding?
Can a drug be good for men and good for women but bad for adults?
We will discuss these questions, as well as prove a surprising new
result about probabilistic conditionals.
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