Consider the following lotteries defined over the three outcomes: M$250,$ M$50,$ M$0.$ L$=(0,1,0)$; L$=(0\mathrm{.}1,0\mathrm{.}89,0\mathrm{.}01)$; L$=(0,0\mathrm{.}11,0\mathrm{.}89)$; L$=(0\mathrm{.}1,0,0\mathrm{.}9).$ Note that lottery L$$ pays M$250$ with a probability of $0\mathrm{.}1,$ M$50$ with a probability of $0\mathrm{.}89$ and MO with a probability of $0\mathrm{.}01$; the other lotteries are interpreted similarly. a$)$ Show that if a decision maker satisfies EU theory and chooses L$$ over L$,$ the decision maker must also choose L$$ over L$2.[12][13]$ S b$)$ Under the assumptions that a decision maker satisfies EU theory and employs the Bernoulli utility function u$($x$)=$ x$1/2,$ calculate the certainty equivalents for lotteries L$1,$ L$1,$ L$2$ and L$2\text{'}$ to determine which pattern of choices $($i$.$e$.,$ either L$$ and L$$ are chosen, or L$$ and L$\text{'}$ are chosen$)$ will be made.