Some Probability Paradoxes in Choice from among Random Alternatives

Abstract The probability P(X>Y) can be arbitrarily close to 1 even though the random variable X is stochastically smaller than Yi the probabilities P(X <Y), P(Y <Z), P(Z <X) can all exceed 1/2; it is possible that P(X = min X, Y, Z) <P(Y = min X, Y, Z) <P(Z = min X, Y, Z) even though P(X <Y), P(X <Z), P(Y <Z) all exceed 1/2. This article examines these paradoxes and extensions of them, and discusses the difficulties they cause in the problem of choosing from among possible random losses or payoffs; as in choosing from among possible statistical decision procedures or from among possible wagers.

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