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?

An annual marathon covers a route that has a distance of approximately 26 miles. winning times for this marathon are all over 2 hours. the following data are the minutes over 2 hours for the winning male runners over two periods of 20 years each. earlier period 14 12 15 22 13 10 19 13 9 14 20 18 16 20 23 12 18 17 6 13 recent period 7 11 7 14 8 9 11 14 8 7 9 8 7 9 9 9 9 8 10 8 (a) make a stem-and-leaf display for the minutes over 2 hours of the winning times for the earlier period. use two lines per stem. (use the tens digit as the stem and the ones digit as the leaf. enter none in any unused answer blanks. for more details, view how to split a stem.) minutes beyond 2 hours earlier period 0 1 2 (b) make a stem-and-leaf display for the minutes over 2 hours of the winning times for the recent period. use two lines per stem. (use the tens digit as the stem and the ones digit as the leaf. enter none in any unused answer blanks.) minutes beyond 2 hours recent period (c) compare the two distributions. how many times under 15 minutes are in each distribution