Question

53. The Tri - State Pick $3$Refer to Exercise $42$. Suppose you buy a $\$1$ ticket on each of two different days.
(a) Find the mean and standard deviation of the total payoff. Show your work.
(b) Find the mean and standard deviation of your total winnings. Interpret each of these values in context.

Short answer

(a) The mean and standard deviation of the total payoff are $\$1.00$and $\$22.34$.

(b) The mean and standard deviation of your total winnings are $-\$1.00$and $\$22.34$.

Step-by-step solution

Part (a) Step 1: Given information

We have to refer the exercise 42 and find the mean and standard deviation of the total payoff.

Part (a) Step 2: Calculate the mean and standard deviation of total payoff

According to the information, the population mean $\left(\mu \right)=\$0.50$
Population standard deviation $\left(\sigma \right)=\$15.80$
Let's estimate the value of the mean and standard deviation of the total payoff:
$$\begin{gathered} {\mu }_{X+X}={\mu }_X+{\mu }_X \\ =0.50+0.50 \\ =\mathrm{\$}1 \end{gathered}$$

Then,

$$\begin{gathered} {\sigma }_{X+X}=\sqrt{{\sigma }_X^2+{\sigma }_X^2} \\ =\sqrt{{15.80}^2+{15.80}^2} \\ =22.34 \end{gathered}$$

Part (b) Step 1: Given information

Refer the exercise 42. We have to find the mean and standard deviation of the total winnings.

Part (b) Step 2: Calculate the mean and standard deviation of total winnings

Let's consider the total payoff is$X$.
Hence, $$\begin{gathered} P=X+X-2 \\ =2X-2 \end{gathered}$$

Now find the mean and standard deviation of the total payoff as:

$$\begin{gathered} {\mu }_{X+X-2}={\mu }_X+{\mu }_X-2 \\ =2\left(0.50\right)-2 \\ =-\mathrm{\$}1 \end{gathered}$$

Then,

$$\begin{gathered} {\sigma }_{X+X-2}=\sqrt{{\sigma }_X^2+{\sigma }_X^2} \\ =\sqrt{{15.80}^2+{15.80}^2} \\ =22.34 \end{gathered}$$