You and a friend are playing rock-paper-scissors.4 each of you throws rock with probability r, paper with probability p, and scissors with probability s. (you both use the
same probabilities because you are equally smart.)
a) use lagrange multipliers to show that your probability of winning is maximized if
you use r = p = s =1/3
.
b) for rock awareness month, it is decided that games won by throwing rock will earn
you $1.50, while games won by throwing paper or scissors will only earn you $1. find
the probabilities that maximize your expected winnings.
c) what happens if the payout for rock is raised to $2?