We find the expected value of one play of the game. We do so by calculating the expected value of each card and summing them. The probability of drawing each card is $\dfrac{1}{52}$.
The expected value for drawing an Ace is equal to the winnings, which is $1$ dollar.
The expected value for drawing a $2$ through $10$ is equal to the winnings, which can be calculated by taking the average of the numbers from $2$ to $10$, giving us $\dfrac{2+3+4+5+6+7+8+9+10}{9}=6$.
The expected value for drawing a face card is equal to the winnings, which is $10$ dollars.
The expected value for drawing a $\clubsuit$ is equal to the negative winnings, which is $-1$.
Similarly, the expected value for drawing a diamond is equal to the winnings, which is $25$ dollars.
Since drawing the $\spadesuit$ doesn't change the amount of money won, the expected value for drawing a $\spadesuit$ is equal to the winnings, which is $0$ dollars.
Summing up all these individual expected values, we get that the total expected value is
\begin{align*}
&\left(\dfrac{1}{52} \cdot 1 \right) \\
+ &\left(\dfrac{9}{52} \cdot 6 \right) \\
+ &\left(\dfrac{12}{52} \cdot 10 \right) \\
+ &\left(\dfrac{4}{52} \cdot -1 \right) \\
+ &\left(\dfrac{13}{52} \cdot 25 \right) \\
+ &\left(\dfrac{13}{52} \cdot 0 \right) \\
&= \dfrac{1}{52} \cdot 1 + \dfrac{9}{52} \cdot 6 + \dfrac{12}{52} \cdot 10 - \dfrac{4}{52} \cdot 1 + \dfrac{13}{52} \cdot 25 \\
&= \dfrac{1}{52} + \dfrac{9 \cdot 6}{52} + \dfrac{12 \cdot 10}{52} - \dfrac{4}{52} + \dfrac{13 \cdot 25}{52} \\
&= \dfrac{1 + 54 + 120 - 4 + 325}{52} \\
&= \dfrac{496}{52} \\
&= 9.54,
\end{align*}rounded to the nearest cent. Therefore, a fair price to pay to play this game is $\boxed{\text{\$}9.54}$.
I draw a card from a standard $52$-card deck. If I draw an Ace, I win $1$ dollar. If I draw a $2$ through $10$, I win a number of dollars equal to the value of the card. If I draw a face card (Jack, Queen, or King), I win $10$ dollars. If I draw a $\clubsuit$, my winnings are subtracted, and if I draw a $\spadesuit$, then nothing happens. If I draw a diamond, I get $25$ dollars. (For example, if I draw the $8\clubsuit$, then I lose $16$ dollars.) What would be a fair price to pay to play the game? Express your answer as a dollar value rounded to the nearest cent.
Your answer should be a number with two digits after the decimal point, like $21.43$.
1 answer
1 answer