Suppose there is an urn containing 300 balls, of those 100 balls are red, while the remaining 200 balls are either black or white with unknown proportions. The balls are well-mixed that any individual ball is as likely to be drawn as any other. Consider the first pair of gambles: Gamble 1A: You receive $10 if you draw a red ball, $0 otherwise Gamble 1B: You receive $10 if you draw a black ball, $0 otherwise Then consider the second pair of gambles: Gamble 2A: You receive $10 if you draw a red or white ball, $0 otherwise Gamble 2B: You receive $10 if you draw a black or white ball, $0 otherwise Before starting on this exercise, think about which gamble you would prefer from each pair. (There is no right or wrong answer. Just choose what you like.)

Then consider the second pair of gambles: 

Gamble 2A: You receive $10 if you draw a red or white ball, $0 otherwise Gamble 2B: You receive $10 if you draw a black or white ball, $0 otherwise Before starting on this exercise, think about which gamble you would prefer from each pair. (There is no right or wrong answer. Just choose what you like.) 

(a) Normalise the utility from getting $0 to 0, and let u10 be the utility from getting $10. Let p be the proportion of black balls among the 200 balls of unknown colours (i.e., there are 200p black balls in the urn). Suppose an expected utility maximiser strictly prefers Gamble 1A among the first pair of gambles, what must his/her belief about p be?

(b) (Level B) Now suppose an expected utility maximiser strictly prefers Gamble 2B among the second pair of gambles, what must his/her belief about p be? 

(c) (Level B) Given your answer to (a) and (b), can an expected utility maximiser strictly prefers 1A to 1B, while strictly prefers 2B to 2A?