Jonathan and Roberto Enjoy Playing Poker

Essay

Jonathan and Roberto enjoy playing poker. Jonathan's utility as a function of winning a poker hand is U_J = {
.
Roberto's utility as a function of winning a poker hand is U_R = {
.
Unfortunately for Jonathan, he has a habit of whistling only when he gets a full-house or better. Roberto, however, has not noticed this habit. Roberto currently has three-of-a-kind (which will lose to a full-house or better). Roberto believes that the probability Jonathan can beat his three-of-a-kind is ¹/₁₀. Roberto could choose to fold or play the hand. Calculate Roberto's expected utility according to his beliefs. Jonathan is currently whistling. How much could Roberto increase his utility by recognizing Jonathan's whistling habit?