Consider the following version of the continuity axiom. If A is preferred to B is preferred to C, then there exist some probabilities p and q such that gamble 1 is preferred to B is preferred to gamble 2, where in gamble 1 there is probability p of getting A and probability 1-p of getting C, and in gamble 2 there is probability q of getting A and probability 1-q of getting C. Suppose A = $10,000,001, B = $10,000,000, C = 50 years in prison.

a. Do you think it is really true that there are any values of p and q such that preferring gamble 1 to B and preferring B to gamble 2 really describes your preferences?

b. Psychological studies suggest that most people cannot distinguish between very small probabilities, i.e., that their preferences over gambles in which there is a small probability of a very bad outcome are unaffected by exactly how small the probability is. Does this show that there is something wrong with von Neumann and Morgenstern's theory?