Question $6$:

Consider the bet $($$$400,\frac{1}{3}$;$-$$$Y,\frac{2}{3}).$

$($a$)$ Suppose that Heidi is an expected utility maximizer with $u(x)=lnx,$ and her initial wealth is $$12000.$

$($i$)$ For what values of $Y$ does Heidi reject a single play of the bet?

$($ii$)$ For what values of $Y$ does Heidi accept two independent plays of the bet?

$($iii$)$ Is it possible for Heidi to reject the single bet but accept the aggregate bet?

Note: For part $($a$),$ report your answers to $3$ decimal points.