Chapter 10.2, Problem 40E

You $\left(R\right)$ and a friend $\left(C\right)$ are playing the following matrix game, where the entries indicate your winnings from $C$ in dollars . To encourage your friend to play, you pay her $\text{\$}4$ before each game. The jack, queen, king, and ace from the hearts, spades, and diamonds are taken from a standard deck of cards. The game is based on a random draw of a single card from these $12$ cards. The play is indicated at the top and side of the matrix.

$\begin{array}{l}C\\ \begin{array}{ccc}H& S& D\end{array}\\ R\begin{array}{c}J\\ Q\\ K\\ A\end{array}\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 6\\ \hfill 2& \hfill 2& \hfill 1\\ \hfill 4& \hfill 4& \hfill 7\\ \hfill 1& \hfill 2& \hfill 6\end{array}\right]\end{array}$

(A) If you select a row by drawing a single card and your friend selects a column by drawing a single card (after replacement), what is your expected value of the game?

(B) If your opponent chooses an optimal strategy (ignoring the cards) and you make your row choice by drawing a card, what is your expected value?

(C) If you both disregard the cards and make your own choices, what is your expected value, assuming that you both choose optimal strategies?