For all the following questions, suppose that you are facing the following gamble: $\frac{1}{3}$ probability of winning $$25$ and $\frac{2}{3}$ probability of winning $$4.$ Suppose, first, that Jon's utility function is $u(x)=\sqrt[\phantom{2}]{x}.$

$($a$)$ Compute the expected utility of the gamble. $(4$ points$)$

$($b$)$ Compute the certainty equivalent of the gamble. $(4$ points$)$

Suppose, next, that Stephen's utility function is $u(x)=x^2.$

$($c$)$ Compute the expected utility of the gamble. $(4$ points$)$

$($d$)$ Compute the certainty equivalent of the gamble. $(4$ points$)$

Finally:

$($e$)$ Is Jon risk averse or risk prone? $(2$ points$)$

$($f$)$ Is Stephen risk averse or risk prone? $(2$ points$)$