Probabilistic Insurance II $($Kahnemann and Tversky, Econometrica, $1979).{?}^1$ Back at the airline ticket

counter, the clerk is disappointed that you turned down the probabilistic coverage from Assignment

III. The facts are the same: you are indifferent between the "regular" policy that offers full insurance

with premium $R,$ and no coverage at all. Now she offers you another kind of policy. You now only pay

$(\frac{1}{2})R$ up front. If your luggage is not lost, then you owe nothing more. If your luggage is lost, however,

then a fair coin is flipped: if it comes up heads, then you pay the remaining half of the premium and

are reimbursed for the value $L$ of your luggage; if it comes up tails, then your premium is refunded

and you bear the entire loss. Show that if you have risk$-$averse EU preferences, then you prefer the

new policy to the regular policy. $[$Hint: Which is bigger $u(W-\frac{R}{2})$ or $u\frac{W}{2}+u\frac{W-R}{2}?]$