A game consists of tossing a fair die. A player wins if the number is even and loses if the number is odd. A game consists of tossing a fair die. A player wins if the number is even and loses if the number is odd. The winning or losing (dollar) payoff is equal to the number appearing. Find the player's mathematical expectation $E$. Winning chances or even #: `2,4,6` Lossing chances or odd #: `1,3,5` So $2\times \frac{1}{2} + 4\times \frac{1}{2}+ 6\times \frac{1}{2} = 6$ $1\times \frac{1}{2} + 3\times \frac{1}{2}+ 5\times \frac{1}{2} = 4.5$ so $6 - 4.5 = \$1.50$ But the answer in my book says it is $\$0.50$ What am I doing wrong?

Your small mistake is that the probability of each outcome is not $1/2$ but $1/6$. (Each side of the die comes up with probability $1/6$.) Multiply all your equations by $1/3$, and your answer will be the same as that in the book.