Problem $3($Required$,30$ marks$)$

An investor is facing two investment plans $($Plan X and Plan Y$)$ as follows:

Plan X Plan Y

Payoff Probability Payoff Probability

$10\mathrm{.}510\mathrm{.}45$

$20\mathrm{.}120\mathrm{.}3$

$30\mathrm{.}330\mathrm{.}1$

$40\mathrm{.}140\mathrm{.}15$

You are given that

$$ The investor has strict preference on these outcomes: For any $$ and

$,$ we have either $$ or $.$

$1$ is the best outcome and $4$ is the worst outcome.

Question: Suppose that the investor prefers Plan Y to Plan X$,$ what can you say

about the investor$$s preference between the outcomes $2$ and $3?$ Choose the best

answer below and provide full justification to your answer.

$($A$)$ The investor prefers $2$ to $3$

$($B$)$ The investor prefers $3$ to $2$

$($C$)$ The given information cannot indicate the investor$$s preference between

$2$ and $3.$

$(*$Note: If your answer is $($C$),$ you need to give an example which the investor prefers

$2$ to $3$ and another example which the investor prefers $3$ to $2.)$