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Most states and Canadian provinces have government-sponsored lotteries. Here is a simple lottery wager, from the Tri-State Pick $3$game that New Hampshire shares with Maine and Vermont. You choose a number with $3$digits from $0$to $9$; the state chooses a three-digit winning number at random and pays you $\backslash (500$if your number is chosen. Because there are $1000$numbers with three digits, you have a probability of $1/1000$of winning. Taking $X$to be the amount your ticket pays you, the probability distribution of $X$is

(a) Show that the mean and standard deviation of $X$are ${渭}_{\mathrm{X}}=\backslash )0.50$and ${蟽}_{\mathrm{X}}=\backslash (15.80$.

(b) If you buy a Pick 3 ticket, your winnings are $W=X-1$, because it costs $\backslash )1$to play. Find the mean and standard deviation of $W$. Interpret each of these values in context.

Step by step solution

01

Part (a) Step 1: Given Information 

Given in the question that a following table

02

Part (a) Step 2: Explanation 

The expected value is calculated by multiplying each possibility by its probability:

$$\begin{gathered} 渭=鈭�xP\left(x\right) \\ =\$0脳0.999+\$500脳0.001 \\ =\$0.50 \end{gathered}$$

The expected value of the squared variation from the mean is the variance:

$$\begin{gathered} {蟽}^2=鈭�(x-渭{)}^{2}P(x) \\ =(\$0-\$0.50{)}^2脳0.999+(\$500-\$0.50{)}^2脳0.001 \\ =249.75 \end{gathered}$$

The square root of the variance is the standard deviation:

$$\begin{gathered} 蟽=\sqrt{{蟽}^2} \\ =\sqrt{249.75} \\ 鈮�\$15.80 \end{gathered}$$

03

Part (b) Step 1: Given Information 

Consider the Result from exercise 42 a

${渭}_X=\$0.50$

${蟽}_X鈮�\$15.80$

04

Part (b) Step 2: Explanation 

Properties mean and standard deviation:

$$\begin{gathered} {渭}_{aX+b}=a{渭}_X+b \\ {蟽}_{aX+b}=a{蟽}_X \end{gathered}$$

Then we obtain for $W=X-1$:

localid="1649908788228" $$\begin{gathered} {渭}_W={渭}_{X-1} \\ ={渭}_X-1 \\ =\$0.50-\$1 \\ =-\$0.50 \end{gathered}$$

localid="1649908805592" $$\begin{gathered} {蟽}_W={蟽}_{X-1} \\ ={蟽}_X \\ =\$15.80 \end{gathered}$$

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