Question

A casino offers the following game: you draw one card from a standard 52-card deck. If you draw a jack, you win $1.25. If you draw a queen, you win $3.25. If you draw a king, you win $5.25 dollars. If you draw any ace except the ace of spades, you win $6.75. If you draw the ace of spades, you win $8.5.

The entry fee to play this game is $2. Compute the expected value of this gamble (include the entry fee in your expected value).

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} x& p(x) \\ \hline -2 & {9 \over 13} \\ -0.75 & {1 \over 13} \\ 1.25 & {1 \over 13} \\ 3.25 & {1 \over 13} \\ 4.75 & {3 \over 52} \\ 6.5 & {1 \over 52} \\ \end{array}$$

$$E(X)=(-2)(\dfrac{9}{13})+(-0.75)(\dfrac{1}{13})+(1.25)(\dfrac{1}{13})$$

$$+(3.25)(\dfrac{1}{13})+(4.75)(\dfrac{3}{52})+(6.5)(\dfrac{1}{52})$$

$$=-\dfrac{145}{208}$$