Doubling down. recall the dice game played between alice and bob: alice rolls two dice, and tells bob the sum; then bob guesses a number from 1 to 6. bob wins the game if his guess appears on either of the two dice. now suppose alice and bob play for money. if bob wins, he wins a dollar, and if he loses, alice wins a dollar. we can model this using a random variable w representing bob’s winnings. we saw that using an optimal strategy, bob wins with probability 5/9. this means eii|-(1)(5/9)+(-1)(4/9)-1/9to make the game more interesting, alice allows bob to "double down": this means that after alice tells bob the sum, bob is allowed to double his bet from one dollar to two dollars, if he so chooses. give an optimal strategy for bob (for both his guess and the double-down decision) and compute e[w] for this strategy. hint: if z is a random variable representing the sum of the two dice, compute e[w] using the law of total expectation: