Suppose there is an urn containing 300 balls, of those 100 balls are red, while the remaining 200 balls are either black

[answerhappygod]

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?