Question

Suppose that you are offered the following “deal.” You roll a die. If you roll a six, you win $\backslash (10.$ If you roll a four or five, you win $\backslash )5.$ If you roll a one, two, or three, you pay $\$6.$

a. What are you ultimately interested in here (the value of the roll or the money you win)?

b. In words, define the Random VariableX.

c. List the values that X may take on.

d. Construct a PDF.

e. Over the long run of playing this game, what are your expected average winnings per game?

f. Based on numerical values, should you take the deal? Explain your decision in complete sentences.

Short answer

(a) We are interested in the money we win.

(b) The random variable X defines the money we win or lost.

(c) $X=\left\{-6,5,10\right\}$.

(d) $$\begin{gathered} P\left(X=-6\right)=\frac{3}{6}=\frac{1}{2} \\ P\left(X=5\right)=\frac{2}{6}=\frac{1}{3} \\ P\left(X=10\right)=\frac{1}{6} \end{gathered}$$

(e) The expected earning per game is $\$0.33$.

(f) Yes, I think we should play the game.

Step-by-step solution

Given information (part a)

If you roll a die and you get $6$, you win $\$10$.

If you roll a die and you get$4$, you win $\$5.$

If you roll a die and you get $1,2,3,$ you lose $\$6.$

Explanation (part a)

Based on the die roll, we're looking for the winning sum. We win $\$10$ if we roll a six, $\$5$ if we roll a four or five, and we lose $\$6$if we roll $1,2,3.$

Given information (part b)

If you roll a die and you get $6,$ you win $\$10$.

If you roll a die and you get $4$, you win $\$5$.

If you roll a die and you get $1,2,3,$you lose $\$6.$

Explanation (part b)

The amount of money we gained or lost is denoted by the letter X. There are three possible outcomes: the first is to win$\$10$, the second is to win $\$5$, and the third is to lose $\$6.$

Given information (part c)

If you roll a die and you get $6,$ you win $\$10.$

If you roll a die and you get$4$, you win $\$5.$

If you roll a die and you get $1,2,3,$ you lose $\$6.$

Explanation (part c)

The amount of money we gained or lost is denoted by the letterX. There are three possible outcomes: first, win $\$10$, second, win $\$5$, and third, lose $\$6.$ As a result, theX values are $10,5,-6.$

Given information (part d)

If you roll a die and you get $6$, you win $\$10.$

If you roll a die and you get $4$, you win $\$5.$

If you roll a die and you get $1,2,3,$ you lose $\$6.$

Explanation (part d)

The amount of money we gained or lost is denoted by the letter X. There are three possible outcomes: the first is to win $\$10,$ the second is to win $\$5$, and the third is to lose $\$6.$ As a result, the X values are $10,5,-6.$The likelihood of winning $10$ is $\frac{1}{6}$ because we can only win $10$if we roll $6$, the probability of winning$5$ is $\frac{1}{3}$because we can only win $5$ if we roll $4$ or$5$, and the probability of losing $6$ is 3/6 because we can only lose$6$ if we roll $1,2,3.$

Given information (part e)

If you roll a die and you get$6$, you win $\$10$.

If you roll a die and you get $4,$ you win $\$5.$

If you roll a die and you get$1,2,3,$you lose $\$6.$

Explanation (part e)

$X$	$P\left(X\right)$	$X.P\left(X\right)$
$10$	$\frac{1}{6}$	$\frac{10}{6}$
$5$	localid="1649951074291" $\frac{2}{6}$	$\frac{10}{6}$
localid="1649951004032" $6$	localid="1649951088276" role="math" $\frac{3}{6}$	localid="1649951035940" $\frac{18}{6}$
Total	$1$	$\frac{2}{6}=0.33$

Given information (part f)

If you roll a die and you get $6$, you win $\$10.$

If you roll a die and you get $4,$you to win $\$5.$

If you roll a die and you get $1,2,3,$to lose $\$6.$

Explanation (part f)

Yes, I believe we should play the game because the estimated earning per game is $0.33,$ implying that you should win $33\%$of the time.