Deck 14: Random Variables

Question 1

A company bids on two contracts.It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 if it gets the smaller contract.It estimates that there's a 20% chance of winning the larger contract and a 60% chance of winning the smaller contract. Create a probability model for the company's profit.Assume that the contracts will be awarded independently.

A) $$\begin{array} { l | l c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 & \$ 70,000 \\\hline \text { P(Profit) } & 0.32 & 0.48 & 0.08 & 0.12\end{array}$$
B) $$\begin{array} { l | l c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 & \$ 70,000 \\\hline \text { P(Profit) } & 0.08 & 0.6 & 0.2 & 0.12\end{array}$$
C) $$\begin{array} { l | c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 \\\hline \text { P(Profit) } & 0.2 & 0.6 & 0.2\end{array}$$
D) $$\begin{array} { l | l c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 & \$ 70,000 \\\hline \text { P(Profit) } & 0.32 & 0.48 & 0.08 & 0.8\end{array}$$
E) $$\begin{array} { l | c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 \\\hline \text { P(Profit) } & 0.32 & 0.48 & 0.08\end{array}$$

Answer 1

$$\begin{array} { l | l c c c } \text { Profit } & \$ 0 & \$ 20,000 & \$ 50,000 & \$ 70,000 \\\hline \text { P(Profit) } & 0.32 & 0.48 & 0.08 & 0.12\end{array}$$

Question 2

$$\begin{array} { c | c c c } x & 20 & 40 & 60 \\\hline P ( X = x ) & 0.25 & 0.30 & 0.45\end{array}$$

A)44
B)40
C)50
D)60
E)55

Answer 2

44

Question 3

$$\begin{array} { r | l c r r } \mathrm { x } & 4 & 8 & 12 & 16 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.1 & 0.4\end{array}$$

A)20.00
B)10.00
C)11.20
D)0.25
E)9.80

Answer 3

11.20

Question 4

A carnival game offers a(n)$80 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play and you're willing to spend up to $20 trying to win.You estimate that you have a(n)10% chance of hitting the balloon on any throw. Create a probability model for the number of darts you will throw.Assume that throws are independent of each other.Round to four decimal places if necessary.

A) $$\begin{array} { l | c r c c } \text { Number of Darts } & 1 & 2 & 3 & 4 \\\hline \text { P(Number of Darts) } & 0.1 & 0.09 & 0.0810 & 0.0656\end{array}$$
B) $$\begin{array} { l | c c c } \text { Number of Darts } & 1 & 2 & 3 \\\hline \text { P(Number of Darts) } & 0.1 & 0.09 & 0.0810\end{array}$$
C) $$\begin{array} { l | c c c c c } \text { Number of Darts } & 1 & 2 & 3 & 4 & 5 \\\hline \mathrm { P } \text { (Number of Darts) } & 0.1 & 0.09 & 0.0810 & 0.7290 & 0.0656\end{array}$$
D) $$\begin{array} { l | c r c c } \text { Number of Darts } & 1 & 2 & 3 & 4 \\\hline \text { P(Number of Darts) } & 0.1 & 0.09 & 0.0810 & 0.7290\end{array}$$
E) $$\begin{array} { l | c r c c } \text { Number of Darts } & 1 & 2 & 3 & 4 \\\hline \mathrm { P } ( \text { Number of Darts } ) & 0.1 & 0.1 & 0.1 & 0.7\end{array}$$

Question 5

A company is interviewing applicants for managerial positions.They plan to hire two people.They have already rejected most candidates and are left with a group of 9 applicants of whom 4 are women.Unable to differentiate further between the applicants,they choose two people at random from this group of 9.Let the random variable X be the number of men that are chosen.Find the probability model for X.

A) $$\begin{array} { l | c c c } \text { Number men } & 0 & 1 & 2 \\\hline \text { P(Number men) } & 0.167 & 0.410 & 0.278\end{array}$$
B) $$\begin{array} { l | c c c } \text { Number men } & 0 & 1 & 2 \\\hline \text { P(Number men) } & 0.198 & 0.494 & 0.309\end{array}$$
C) $$\begin{array} { l | c c c } \text { Number men } & 0 & 1 & 2 \\\hline \text { P(Number men) } & 0.278 & 0.556 & 0.167\end{array}$$
D) $$\begin{array} { l | c c c } \text { Number men } & 0 & 1 & 2 \\\hline \text { P(Number men) } & 0.167 & 0.278 & 0.278\end{array}$$
E) $$\begin{array} { c | c c c } \text { Number men } & 0 & 1 & 2 \\\hline \text { P(Number men) } & 11.077 & 66.462 & 66.462\end{array}$$

Question 6

Hugh buys $8000 worth of stock in an electronics company which he hopes to sell afterward at a profit.The company is developing a new laptop computer and a new desktop computer.If it releases both computers before its competitor,the value of Hugh's stock will jump to $21,000.If it releases one of the computers before its competitor,the value of Hugh's stock will jump to $17,000.If it fails to release either computer before its competitor,Hugh's stock will be worth only $5000.Hugh believes that there is a 40% chance that the company will release the laptop before its competitor and a 50% chance that the company will release the desktop before its competitor. Create a probability model for Hugh's profit.Assume that the development of the laptop and the development of the desktop are independent events.

A) $$\begin{array} { l | c c c } \text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \\\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3\end{array}$$
B) $$\begin{array} { | l | c c c | } \hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \\\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3\end{array}$$
C) $$\begin{array} { | l | c c c | } \hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \\\hline \text { P(Profit } ) & 0.2 & 0.2 & 0.3\end{array}$$
D) $$\begin{array} { l | c c c } \text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \\\hline \text { P(Profit) } & 0.2 & 0.9 & 0.3\end{array}$$
E) $$\begin{array} { l | c c c } \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \\\hline \text { P(Profit } ) & 0.9 & 0.2 & 0.3\end{array}$$

Question 7

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $20 plus an extra $60 for the ace of hearts.For any other card you win nothing. Create a probability model for the amount you win at this game.

A) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 60 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 39 } { 52 } & \frac { 4 } { 52 } & \frac { 4 } { 52 } & \frac { 1 } { 52 }\end{array}$$
B) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 80 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 36 } { 52 } & \frac { 12 } { 52 } & \frac { 3 } { 52 } & \frac { 1 } { 52 }\end{array}$$
C) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 80 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 32 } { 52 } & \frac { 16 } { 52 } & \frac { 3 } { 52 } & \frac { 1 } { 52 }\end{array}$$
D) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 60 \\\hline \text { P(Amount won) } & \frac { 36 } { 52 } & \frac { 12 } { 52 } & \frac { 3 } { 52 } & \frac { 1 } { 52 }\end{array}$$
E) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 80 \\\hline \text { P(Amount won) } & \frac { 36 } { 52 } & \frac { 12 } { 52 } & \frac { 4 } { 52 } & \frac { 1 } { 52 }\end{array}$$

Question 8

In a box of 7 batteries,3 are dead.You choose two batteries at random from the box.Let the random variable X be the number of good batteries you get.Find the probability model for X.

A) $$\begin{array} { l | c c c } \text { Number good } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Number good) } & 0.184 & 0.490 & 0.327\end{array}$$
B) $$\begin{array} { l | c c c } \text { Number good } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Number good) } & 0.143 & 0.571 & 0.286\end{array}$$
C) $$\begin{array} { l | c c c } \text { Number good } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Number good) } & 0.143 & 0.286 & 0.286\end{array}$$
D) $$\begin{array} { l | c c c } \text { Number good } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Number good) } & 0.286 & 0.571 & 0.286\end{array}$$
E) $$\begin{array} { l | c c c } \text { Number good } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Number good) } & 0.067 & 0.467 & 0.467\end{array}$$

Question 9

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls. Create a probability model for the number of children they will have.Assume that boys and girls are equally likely.

A) $$\begin{array} { l | c c c c } \text { Children } & 1 & 2 & 3 & 4 \\\hline \text { P(Children) } & 0.5 & 0.25 & 0.125 & 0.125\end{array}$$
B) $$\begin{array} { l | c c c c c } \text { Children } & 1 & 2 & 3 & 4 & 5 \\\hline \text { P(Children) } & 0.5 & 0.25 & 0.125 & 0.0625 & 0.0625\end{array}$$
C) $$\begin{array} { l | c c c } \text { Children } & 1 & 2 & 3 \\\hline \mathrm { P } \text { (Children) } & 0.5 & 0.25 & 0.25\end{array}$$
D) $$\begin{array} { l | c c c c } \text { Children } & 1 & 2 & 3 & 4 \\\hline \mathrm { P } \text { (Children) } & 0.5 & 0.25 & 0.125 & 0.0625\end{array}$$
E) $$\begin{array} { l | c c c c } \text { Children } & 1 & 2 & 3 & 4 \\\hline \text { P(Children) } & 0.25 & 0.25 & 0.25 & 0.25\end{array}$$

Question 10

You roll a fair die.If you get a number greater than 4,you win $70.If not,you get to roll again.If you get a number greater than 4 the second time,you win $30.Otherwise you win nothing. Create a probability model for the amount you win at this game.

A) $$\begin{array} { l | c c c } \text { Amount won } & \$ 70 & \$ 30 & \$ 0 \\\hline \text { P(Amount won) } & \frac { 2 } { 6 } & \frac { 8 } { 36 } & \frac { 16 } { 36 }\end{array}$$
B) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 100 & \$ 70 & \$ 30 & \$ 0 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 4 } { 36 } & \frac { 8 } { 36 } & \frac { 8 } { 36 } & \frac { 16 } { 36 }\end{array}$$
C) $$\begin{array} { l | c c c } \text { Amount won } & \$ 70 & \$ 30 & \$ 0 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 2 } { 6 } & \frac { 2 } { 6 } & \frac { 2 } { 6 }\end{array}$$
D) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 100 & \$ 70 & \$ 30 & \$ 0 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 4 } { 36 } & \frac { 2 } { 36 } & \frac { 2 } { 36 } & \frac { 16 } { 36 }\end{array}$$
E) $$\begin{array} { l | c c } \text { Amount won } & \$ 70 & \$ 30 \\\hline \text { P(Amount won) } & \frac { 2 } { 6 } & \frac { 4 } { 6 }\end{array}$$

Question 11

An insurance policy costs $400,and will pay policyholders $10,000 if they suffer a major injury (resulting in hospitalization),or $2,000 if they suffer a minor injury (resulting in lost time from work).The company estimates that each year 1 in every 3,000 policyholders may have a major injury,and 1 in 1,000 a minor injury. Create a probability model for the company's profit on this policy.

A) $$\begin{array} { l | c | c | c } \text { Profit } & \$ 400 & - \$ 10,400 & - \$ 2,400 \\\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001\end{array}$$
B) $$\begin{array} { l | c | c | c } \text { Profit } & \$ 400 & \$ 10,000 & \$ 2,000 \\\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001\end{array}$$
C) $$\begin{array} { l | c | c | c } \text { Profit } & \$ 400 & \$ 10,400 & \$ 2,400 \\\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001\end{array}$$
D) $$\begin{array} { l | c | c | c } \text { Profit } & \$ 400 & \$ 9,600 & \$ 1,600 \\\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001\end{array}$$
E) $$\begin{array} { l | c | c | c } \text { Profit } & \$ 400 & - \$ 9,600 & - \$ 1,600 \\\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001\end{array}$$

Question 12

Your school's soccer team plays two games against another soccer team.The probability that your team wins the first game is 0.6.If your team wins the first game,the probability that they also win the second game is 0.7.If your team loses the first game,the probability that they win the second game is 0.5.Let the random variable X be the number of games won by your team.Find the probability model for X.

A) $$\begin{array} { l | c c c } \text { Games won } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Games won) } & 0.12 & 0.46 & 0.42\end{array}$$
B) $$\begin{array} { l | l c c } \text { Games won } & 0 & 1 & 2 \\\hline \mathrm { P } \text { (Games won) } & 0.2 & 0.2 & 0.42\end{array}$$
C) $$\begin{array} { l | l c c } \text { Games won } & 0 & 1 & 2 \\\hline \text { P(Games won) } & 0.2 & 0.5 & 0.3\end{array}$$
D) $$\begin{array} { l | l c c } \text { Games won } & 0 & 1 & 2 \\\hline \text { P(Games won) } & 0.2 & 0.18 & 0.42\end{array}$$
E) $$\begin{array} { l | l c c } \text { Games won } & 0 & 1 & 2 \\\hline \text { P(Games won) } & 0.2 & 0.38 & 0.42\end{array}$$

Question 13

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).Let X be the sum of the two cards.Find the probability model for X.

A) $$\begin{array} { l | l c r r } \mathrm { X } & 1 & 9 & 10 & 11 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4\end{array}$$
B) $$\begin{array} { l | l r r r } \mathrm { X } & 12 & 19 & 20 & 21 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 6 & 1 / 6 & 1 / 3 & 1 / 3\end{array}$$
C) $$\begin{array} { l | l r r r r } \mathrm { X } & 12 & 19 & 20 & 21 & 22 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5 & 1 / 5\end{array}$$
D) $$\begin{array} { l | l r r r r } \mathrm { X } & 2 & 12 & 19 & 20 & 21 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 12 & 1 / 12 & 1 / 6 & 1 / 3 & 1 / 3\end{array}$$
E) $$\begin{array} { l | l r r r } \mathrm { X } & 12 & 19 & 20 & 21 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4\end{array}$$

Question 14

You have arranged to go camping for two days in March.You believe that the probability that it will rain on the first day is 0.5.If it rains on the first day,the probability that it also rains on the second day is 0.5.If it doesn't rain on the first day,the probability that it rains on the second day is 0.3.Let the random variable X be the number of rainy days during your camping trip.Find the probability model for X.

A) $$\begin{array} { l | c c c } \text { Rainy days } & 0 & 1 & 2 \\\hline \text { P(Rainy days) } & 0.35 & 0.5 & 0.15\end{array}$$
B) $$\begin{array} { l | c c c } \text { Rainy days } & 0 & 1 & 2 \\\hline \text { P(Rainy days) } & 0.35 & 0.25 & 0.25\end{array}$$
C) $$\begin{array} { l | l l c } \text { Rainy days } & 0 & 1 & 2 \\\hline \text { P(Rainy days) } & 0.25 & 0.5 & 0.25\end{array}$$
D) $$\begin{array} { l | c c c } \text { Rainy days } & 0 & 1 & 2 \\\hline \text { P(Rainy days) } & 0.35 & 0.15 & 0.25\end{array}$$
E) $$\begin{array} { l | l l l } \text { Rainy days } & 0 & 1 & 2 \\\hline \text { P(Rainy days) } & 0.35 & 0.4 & 0.25\end{array}$$

Question 15

You roll a pair of fair dice.If you get a sum greater than 10 you win $50.If you get a double you win $40.If you get a double and a sum greater than 10 you win $90.Otherwise you win nothing. Create a probability model for the amount you win at this game.

A) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 40 & \$ 50 & \$ 90 \\\hline \text { P(Amount won) } & \frac { 28 } { 36 } & \frac { 5 } { 36 } & \frac { 2 } { 36 } & \frac { 1 } { 36 }\end{array}$$
B) $$\begin{array} { l | l l l } \text { Amount won } & \$ 0 & \$ 40 & \$ 50 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 27 } { 36 } & \frac { 6 } { 36 } & \frac { 3 } { 36 }\end{array}$$
C) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 40 & \$ 50 & \$ 90 \\\hline \text { P(Amount won) } & \frac { 27 } { 36 } & \frac { 5 } { 36 } & \frac { 3 } { 36 } & \frac { 1 } { 36 }\end{array}$$
D) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 40 & \$ 50 & \$ 90 \\\hline \text { P(Amount won) } & \frac { 26 } { 36 } & \frac { 6 } { 36 } & \frac { 3 } { 36 } & \frac { 1 } { 36 }\end{array}$$
E) $$\begin{array} { l | c c c c } \text { Amount won } & \$ 0 & \$ 40 & \$ 50 & \$ 90 \\\hline \text { P(Amount won) } & \frac { 27 } { 36 } & \frac { 6 } { 36 } & \frac { 2 } { 36 } & \frac { 1 } { 36 }\end{array}$$

Question 16

A carnival game offers a(n)$80 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play and you're willing to spend up to $20 trying to win.You estimate that you have a(n)12% chance of hitting the balloon on any throw. Create a probability model for the amount you will win.Assume that throws are independent of each other.Round to four decimal places if necessary.

A) $$\begin{array} { l | l c c c c } \text { Amount won } & \$ 75 & \$ 70 & \$ 65 & \$ 60 & - \$ 20 \\\hline \mathrm { P } \text { (Amount won } ) & 0.12 & 0.1056 & 0.0929 & 0.0818 & 0.0720\end{array}$$
B) $$\begin{array} { l | l c c c } \text { Amount won } & \$ 75 & \$ 70 & \$ 65 & \$ 60 \\\hline \mathrm { P } \text { (Amount won) } & 0.12 & 0.1056 & 0.0929 & 0.6815\end{array}$$
C) $$\begin{array} { l | l c c c c } \text { Amount won } & \$ 80 & \$ 75 & \$ 70 & \$ 65 & - \$ 20 \\\hline \mathrm { P } \text { (Amount won } ) & 0.12 & 0.1056 & 0.0929 & 0.0818 & 0.0720\end{array}$$
D) $$\begin{array} { l | l c c c c } \text { Amount won } & \$ 75 & \$ 70 & \$ 65 & \$ 60 & - \$ 20 \\\hline \mathrm { P } \text { (Amount won } ) & 0.12 & 0.1056 & 0.0929 & 0.0818 & 0.5997\end{array}$$
E) $$\begin{array} { l | l c c c c } \text { Amount won } & \$ 80 & \$ 75 & \$ 70 & \$ 65 & - \$ 20 \\\hline \mathrm { P } \text { (Amount won } ) & 0.12 & 0.1056 & 0.0929 & 0.0818 & 0.5997\end{array}$$

Question 17

$$\begin{array} { l | r r r r } \mathrm { x } & 100 & 200 & 300 & 400 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.3 & 0.4 & 0.2 & 0.1\end{array}$$

A)220
B)250
C)210
D)200
E)150

Question 18

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card).If you get a club the second time,you win $30.Otherwise you win nothing. Create a probability model for the amount you win at this game.

A) $$\begin{array} { l | r r r } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \text { P(Amount won) } & \frac { 1 } { 4 } & \frac { 21 } { 64 } & \frac { 27 } { 64 }\end{array}$$
B) $$\begin{array} { l | c c c c c } \text { Amount won } & \$ 80 & \$ 60 & \$ 30 & \$ 0 \\\hline P ( \text { Amount won) } & \frac { 1 } { 4 } & \frac { 9 } { 64 } & \frac { 3 } { 16 } & \frac { 27 } { 64 }\end{array}$$
C) $$\begin{array} { l | c c c } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \mathrm { P } \text { (Amount won) } & \frac { 1 } { 4 } & \frac { 3 } { 16 } & \frac { 9 } { 16 }\end{array}$$
D) $$\begin{array} { l | c c c c c } \text { Amount won } & \$ 110 & \$ 80 & \$ 30 & \$ 0 \\\hline P ( \text { Amount won) } & \frac { 1 } { 16 } & \frac { 3 } { 16 } & \frac { 3 } { 16 } & \frac { 9 } { 16 }\end{array}$$
E) $$\begin{array} { l | r c c } \text { Amount won } & \$ 80 & \$ 30 & \$ 0 \\\hline \text { P(Amount won) } & \frac { 1 } { 4 } & \frac { 1 } { 4 } & \frac { 1 } { 2 }\end{array}$$

Question 19

$$\begin{array} { c | l l l } \mathrm { x } & 0 & 1 & 2 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.5 & 0.2 & 0.3\end{array}$$

A)1.00
B)0.60
C)1.20
D)0.80
E)0.33

Question 20

$$\begin{array} { r | r } \mathrm { x } & \mathrm { P } ( \mathrm { X } = \mathrm { x } ) \\\hline 0 & 0.50 \\1 & 0.12 \\2 & 0.16 \\3 & 0.20 \\4 & 0.02\end{array}$$

A)1.12
B)0.97
C)1.52
D)1.02
E)1.62

Question 21

Hugh buys $8000 worth of stock in an electronics company which he hopes to sell afterward at a profit.The company is developing a new laptop computer and a new desktop computer.If it releases both computers before its competitor,the value of Hugh's stock will jump to $22,000.If it releases one of the computers before its competitor,the value of Hugh's stock will jump to $16,000.If it fails to release either computer before its competitor,Hugh's stock will be worth only $5000.Hugh believes that there is a 70% chance that the company will release the laptop before its competitor and a 60% chance that the company will release the desktop before its competitor.Find Hugh's expected profit.Assume that the development of the laptop and the development of the desktop are independent events.

A)$9200
B)$17,200
C)$12,200
D)$15,920
E)$11,300.00

Question 22

The number of golf balls ordered by customers of a pro shop has the following probability distribution. $$\begin{array} { r | r | r | r | r | r } \mathrm { x } & 3 & 6 & 9 & 12 & 15 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.14 & 0.12 & 0.36 & 0.28 & 0.10\end{array}$$

A)6.57
B)8.28
C)9
D)9.24
E)8.82

Question 23

In a box of 7 batteries,6 are dead.You choose two batteries at random from the box.Let the random variable X be the number of good batteries you get.Find the expected value of X.

A)μ = 0.36
B)μ = 0.92
C)μ = 0.14
D)μ = 1.71
E)μ = 0.29

Question 24

Your school's soccer team plays two games against another soccer team.The probability that your team wins the first game is 0.4.If your team wins the first game,the probability that they also win the second game is 0.4.If your team loses the first game,the probability that they win the second game is 0.3.Let the random variable X be the number of games won by your team.Find the expected value of X.

A)μ = 0.70
B)μ = 0.56
C)μ = 0.74
D)μ = 0.50
E)μ = 0.80

Question 25

A contractor is considering a sale that promises a profit of $24,000 with a probability of 0.7 or a loss (due to bad weather,strikes,and such)of $16,000 with a probability of 0.3.What is the expected profit?

A)$8000
B)$16,800
C)$28,000
D)$21,600
E)$12,000

Question 26

A company is interviewing applicants for managerial positions.They plan to hire two people.They have already rejected most candidates and are left with a group of 10 applicants of whom 4 are women.Unable to differentiate further between the applicants,they choose two people at random from this group of 10.Let the random variable X be the number of men that are chosen.Find the expected value of X.

A)μ = 1.50
B)μ = 1.20
C)μ = 0.80
D)μ = 0.57
E)μ = 0.93

Question 27

Jo is a hairstylist.The probability model below describes the number of clients that she may see in a day. $$\begin{array} { l | c l l c c c } \text { Number of clients } & 0 & 1 & 2 & 3 & 4 & 5 \\\hline \text { Probability } & 0.1 & 0.1 & 0.2 & 0.25 & 0.2 & 0.15\end{array}$$ What is the expected value of the number of clients that Jo sees per day?

A)2.7
B)3.00
C)2.9
D)2.80
E)2.50

Question 28

You pick a card from a deck.If you get a club,you win $90.If not,you get to draw again (after replacing the first card).If you get a club the second time,you win $30.If not,you lose. Find the expected amount you will win.

A)$30.00
B)$45.00
C)$32.34
D)$36.56
E)$28.13

Question 29

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. $$\begin{array} { l | l l l l } \text { Number of storms } & 0 & 1 & 2 & 3 \\\hline \text { Probability } & 0.1 & 0.3 & 0.5 & 0.1\end{array}$$ How many storms can the town expect each August?

A)1.5
B)1.9
C)2.0
D)1.7
E)1.6

Question 30

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $30 plus an extra $60 for the ace of hearts.For any other card you win nothing.Find the expected amount you will win.

A)$8.08
B)$4.62
C)$6.92
D)$6.35
E)$7.50

Question 31

The probabilities that a batch of 4 computers will contain 0,1,2,3,and 4 defective computers are 0.4096,0.4096,0.1536,0.0256,and 0.0016,respectively.Find the expected number of defective computers in a batch of 4.

A)0.80
B)0.89
C)1.21
D)0.70
E)2.00

Question 32

An insurance policy costs $140 per year,and will pay policyholders $15,000 if they suffer a major injury (resulting in hospitalization)or $7000 if they suffer a minor injury (resulting in lost time from work).The company estimates that each year 1 in every 2200 policyholders will have a major injury and 1 in every 400 a minor injury.What is the company's expected profit on this policy?

A)$115.68
B)$99.32
C)$130.68
D)-$24.32
E)-$297.95

Question 33

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the expected number of children they will have.Assume that boys and girls are equally likely.Round your answer to three decimal places.

A)1.625
B)1.750
C)2.500
D)1.938
E)1.875

Question 34

You have arranged to go camping for two days in March.You believe that the probability that it will rain on the first day is 0.5.If it rains on the first day,the probability that it also rains on the second day is 0.7.If it doesn't rain on the first day,the probability that it rains on the second day is 0.2.Let the random variable X be the number of rainy days during your camping trip.Find the expected value of X.

A)μ = 1.05
B)μ = 0.85
C)μ = 1.2
D)μ = 0.95
E)μ = 0.8

Question 35

You roll a pair of dice.If you get a sum greater than 10 you win $60.If you get a double you win $20.If you get a double and a sum greater than 10 you win $80.Otherwise you win nothing.You pay $5 to play.Find the expected amount you win at this game.

A)$8.33
B)$3.33
C)$5.00
D)$5.56
E)$3.89

Question 36

A company bids on two contracts.It anticipates a profit of $70,000 if it gets the larger contract and a profit of $40,000 if it gets the smaller contract.It estimates that there's a 30% chance of winning the larger contract and a 50% chance of winning the smaller contract.Find the company's expected profit.Assume that the contracts will be awarded independently.

A)$41,000
B)$57,500
C)$112,500
D)$47,000
E)$24,500

Question 37

A carnival game offers a $120 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $9 to play and you're willing to spend up to $36 trying to win.You estimate that you have a 10% chance of hitting the balloon on any throw.Find the expected amount you will win.Assume that throws are independent of each other.

A)$13.01
B)$10.32
C)-$14.76
D)-$11.16
E)$19.32

Question 38

The accompanying table describes the probability distribution for the number of adults in a certain town (among 4 randomly selected adults)who have a college degree. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.4096 \\1 & 0.4096 \\2 & 0.1536 \\3 & 0.0256 \\4 & 0.0016\end{array}$$

A)1.21
B)0.70
C)2.00
D)0.95
E)0.80

Question 39

Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $$\$ 4.00$$ for rolling a 6 or a 3,nothing otherwise.What is the expected amount you win?

A)$2.00
B)-$0.67
C)$4.00
D)-$2.00
E)$0.00

Question 40

Sue Anne owns a medium-sized business.The probability model below describes the number of employees that may call in sick on any given day. $$\begin{array} { c | c c c c c } \text { Number of Employees Sick } & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.3 & 0.15 & 0.05\end{array}$$ What is the expected value of the number of employees calling in sick each day?

A)2.00
B)1.00
C)1.65
D)1.70
E)1.75

Question 41

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).Let X be the sum of the two cards.Find the expected value of X.

A)19.0
B)18.0
C)20.5
D)18.5
E)20.0

Question 42

A couple plans to have children until they get a boy,but they agree that they will not have more than four children even if all are girls.Find the standard deviation of the number of children the couple have.Assume that boys and girls are equally likely.Round your answer to three decimal places.

A)1.109
B)0.992
C)1.053
D)0.984
E)1.173

Question 43

A company bids on two contracts.It anticipates a profit of $70,000 if it gets the larger contract and a profit of $30,000 if it gets the smaller contract.It estimates that there's a 10% chance of winning the larger contract and a 60% chance of winning the smaller contract.Find the standard deviation of the company's profit.Assume that the contracts will be awarded independently.

A)$30,758
B)$24,863
C)$25,632
D)$29,477
E)$28,195

Question 44

The random variable x is the number of houses sold by a realtor in a single month at the Sendsom's Real Estate Office.Its probability distribution is as follows.Find the standard deviation of the number of houses sold. $$\begin{array} { c | c } \text { Houses Sold } ( \mathrm { x } ) & \text { Probability P(x) } \\\hline 0 & 0.24 \\\hline 1 & 0.01 \\\hline 2 & 0.12 \\\hline 3 & 0.16 \\\hline 4 & 0.01 \\\hline 5 & 0.14 \\\hline 6 & 0.11 \\\hline 7 & 0.21\end{array}$$

A)6.86
B)2.62
C)1.62
D)2.25
E)4.45

Question 45

You pick a card from a deck.If you get a club,you win $80.If not,you get to draw again (after replacing the first card).If you get a club the second time,you win $30.If not,you lose.Find the standard deviation of the amount you will win.

A)$1112.11
B)$38.07
C)$30.68
D)$28.35
E)$33.35

Question 46

$$\begin{array} { c | r c r } \mathrm { x } & 0 & 1 & 2 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.7 & 0.2 & 0.1\end{array}$$

A)0.66
B)0.68
C)0.72
D)0.44
E)0.69

Question 47

A carnival game offers a $80 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play and you're willing to spend up to $20 trying to win.You estimate that you have a 8% chance of hitting the balloon on any throw.Find the standard deviation of the number of darts you throw.Assume that throws are independent of each other.

A)0.94
B)0.88
C)0.79
D)1.01
E)0.81

Question 48

A teacher grading statistics homework finds that none of the students has made more than three errors.14% have made three errors,25% have made two errors,and 39% have made one error.Find the standard deviation of the number of errors in students' statistics homework.

A)0.97
B)0.93
C)0.89
D)1.07
E)0.82

Question 49

The probabilities that a batch of 4 computers will contain 0,1,2,3,and 4 defective computers are 0.4746,0.3888,0.1195,0.0163,and 0.0008,respectively.Find the standard deviation of the number of defective computers.

A)0.75
B)0.56
C)1.01
D)0.70
E)0.87

Question 50

A police department reports that the probabilities that 0,1,2,and 3 burglaries will be reported in a given day are 0.52,0.42,0.05,and 0.01,respectively.Find the standard deviation of the number of burglaries in a day.

A)0.41
B)0.96
C)0.64
D)0.80
E)0.84

Question 51

You roll a pair of dice.If you get a sum greater than 10 you win $50.If you get a double you win $20.If you get a double and a sum greater than 10 you win a $70.Otherwise you win nothing.Find the standard deviation of the amount you win at this game.

A)$14.08
B)$274.31
C)$25.21
D)$14.91
E)$16.56

Question 52

$$\begin{array} { l | c c c c } \mathrm { x } & 100 & 200 & 300 & 400 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.4 & 0.3 & 0.1\end{array}$$

A)117.00
B)90.00
C)99.00
D)108.00
E)82.80

Question 53

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. $$\begin{array} { l | c c c c } \text { Number of storms } & 0 & 1 & 2 & 3 \\\hline \text { Probability } & 0.1 & 0.2 & 0.4 & 0.3\end{array}$$ What is the standard deviation of the number of storms in August?

A)0.88
B)0.94
C)0.66
D)0.75
E)0.77

Question 54

Find the standard deviation for the given probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.27 \\\hline 1 & 0.07 \\\hline 2 & 0.07 \\\hline 3 & 0.22 \\\hline 4 & 0.37\end{array}$$

A)2.87
B)1.28
C)1.73
D)2.73
E)1.65

Question 55

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $20 plus an extra $40 for the ace of hearts.For any other card you win nothing.Find the standard deviation of the amount you will win.

A)$10.53
B)$9.69
C)$11.59
D)$110.94
E)$122.09

Question 56

Jo is a hairstylist.The probability model below describes the number of clients that she may see in a day. $$\begin{array} { l | c c c c c c } \text { Number of clients } & 0 & 1 & 2 & 3 & 4 & 5 \\\hline \text { Probability } & 0.05 & 0.15 & 0.15 & 0.3 & 0.2 & 0.15\end{array}$$ What is the standard deviation of the number of clients that Jo sees per day?

A)1.31
B)1.41
C)2.19
D)1.99
E)1.55

Question 57

Sue Anne owns a medium-sized business.The probability model below describes the number of employees that may call in sick on any given day. $$\begin{array} { c | c c c c c } \text { Number of Employees Sick } & 0 & 1 & 2 & 3 & 4 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.05 & 0.4 & 0.25 & 0.2 & 0.1\end{array}$$ What is the standard deviation of the number of employees calling in sick each day?

A)1.20
B)1.09
C)1.31
D)1.19
E)0.98

Question 58

$$\begin{array} { c | c c c r } \mathrm { x } & 3 & 6 & 9 & 12 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3\end{array}$$

A)8.62
B)2.71
C)2.41
D)3.01
E)3.32

Question 59

The accompanying table describes the probability distribution for the number of adults in a certain town (among 4 randomly selected adults)who have a degree from a post-secondary institute.Find the standard deviation for the probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.0256 \\\hline 1 & 0.1536 \\\hline 2 & 0.3456 \\\hline 3 & 0.3456 \\\hline 4 & 0.1296\end{array}$$

A)0.96
B)1.12
C)0.99
D)2.59
E)0.98

Question 60

$$\begin{array} { c | c c c } \mathrm { x } & 20 & 40 & 60 \\\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.2 & 0.3 & 0.5\end{array}$$

A)14.84
B)13.28
C)15.62
D)15.31
E)14.21

Question 61

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + 2Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 60 & 12 \\\mathrm { Y } & 50 & 7\end{array}$$

A)? = 110,? = 18.44
B)? = 100,? = 18.44
C)? = 110,? = 26
D)? = 160,? = 18.44
E)? = 160,? = 26

Question 62

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 4Y - 5.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 190 & 16 \\\mathrm { Y } & 280 & 14\end{array}$$

A)? = 1115,? = 55.78
B)? = 1120,? = 56.22
C)? = 1115,? = 51
D)? = 1120,? = 56
E)? = 1115,? = 56

Question 63

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 4X + 19.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 100 & 14 \\\mathrm { Y } & 120 & 7\end{array}$$

A)? = 400,? = 75
B)? = 419,? = 75
C)? = 400,? = 56
D)? = 419,? = 59.14
E)? = 419,? = 56

Question 64

A company is interviewing applicants for managerial positions.They plan to hire two people.They have already rejected most candidates and are left with a group of 7 applicants of whom 6 are women.Unable to differentiate further between the applicants,they choose two people at random from this group of 7.Let the random variable X be the number of men that are chosen.Find the standard deviation of X.

A)σ = 0.49
B)σ = 0.42
C)σ = 0.45
D)σ = 0.20
E)σ = 0.55

Question 65

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $$Y _ { 1 } + Y _ { 2 }$$ $$\begin{array} { c | c c } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 150 & 12 \\\mathrm { Y } & 200 & 18\end{array}$$

A)? = 350,? = 21.63
B)? = 350,? = 30
C)? = 400,? = 25.46
D)? = 400,? = 36
E)? = 400,? = 6.00

Question 66

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 3X - Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 190 & 15 \\\mathrm { Y } & 170 & 19\end{array}$$

A)? = 60,? = 26
B)? = 740,? = 48.85
C)? = 400,? = 40.79
D)? = 400,? = 48.85
E)? = 400,? = 26

Question 67

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable Y - 19.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 300 & 29 \\\mathrm { Y } & 220 & 30\end{array}$$

A)? = 201,? = 35.51
B)? = 201,? = 11
C)? = 201,? = 23.22
D)? = 220,? = 30
E)? = 201,? = 30

Question 68

An insurance company estimates that it should make an annual profit of $150 on each homeowner's policy written,with a standard deviation of $6000.If it writes 7 of these policies,what are the mean and standard deviation of the annual profit? Assume that policies are independent of each other.

A)μ = $396.86,σ = $42,000
B)μ = $1050,σ = $294,000
C)μ = $1050,σ = $42,000
D)μ = $1050,σ = $15,874.51
E)μ = $396.86,σ = $15,874.51

Question 69

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + 6.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 30 & 9 \\\mathrm { Y } & 50 & 6\end{array}$$

A)? = 36,? = 15
B)? = 30,? = 9
C)? = 36,? = 10.82
D)? = 36,? = 9
E)? = 30,? = 15

Question 70

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 40 & 4 \\\mathrm { Y } & 70 & 7\end{array}$$

A)? = 2800,? = 11
B)? = 110,? = 65
C)? = 110,? = 8.06
D)? = 2800,? = 8.06
E)? = 110,? = 11

Question 71

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $$x _ { 1 } + x _ { 2 } + x _ { 3 }$$ $$\begin{array} { c | c c } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 12 & 2 \\\mathrm { Y } & 50 & 11\end{array}$$

A)? = 6,? = 19.05
B)? = 6,? = 5.74
C)? = 6,? = 33
D)? = 86.60,? = 19.05
E)? = 36,? = 3.46

Question 72

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 2X. $$\begin{array} { c | c c } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 40 & 5 \\\mathrm { Y } & 70 & 8\end{array}$$

A)? = 80,? = 5
B)? = 42,? = 5
C)? = 42,? = 10
D)? = 80,? = 10
E)? = 80,? = 20

Question 73

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 0.3Y. $$\begin{array} { c | c c } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 70 & 7 \\\mathrm { Y } & 60 & 8\end{array}$$

A)? = 18,? = 0.72
B)? = 60.3,? = 8
C)? = 60.3,? = 2.4
D)? = 18,? = 8
E)? = 18,? = 2.4

Question 74

An insurance policy costs $150 per year,and will pay policyholders $16,000 if they suffer a major injury (resulting in hospitalization)or $6000 if they suffer a minor injury (resulting in lost time from work).The company estimates that each year 1 in every 2000 policyholders will have a major injury and 1 in every 400 a minor injury.What is the standard deviation of the company's profit on this policy?

A)$582.92
B)$466.34
C)$447.68
D)$498.98
E)$550.28

Question 75

In a box of 8 batteries,5 are dead.You choose two batteries at random from the box.Let the random variable X be the number of good batteries you get.Find the standard deviation of of X.

A)σ = 0.78
B)σ = 0.40
C)σ = 0.47
D)σ = 0.59
E)σ = 0.63

Question 76

You have arranged to go camping for two days in March.You believe that the probability that it will rain on the first day is 0.4.If it rains on the first day,the probability that it also rains on the second day is 0.5.If it doesn't rain on the first day,the probability that it rains on the second day is 0.3.Let the random variable X be the number of rainy days during your camping trip.Find the standard deviation of X.

A)σ = 0.67
B)σ = 0.76
C)σ = 0.57
D)σ = 0.77
E)σ = 0.70

Question 77

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X - Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 220 & 16 \\\mathrm { Y } & 260 & 12\end{array}$$

A)? = -40,? = 28
B)? = 480,? = 20.00
C)? = -40,? = 20.00
D)? = -40,? = 10.58
E)? = -40,? = 4

Question 78

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).Let X be the sum of the two cards.Find the standard deviation of X.

A)0.92
B)3.54
C)0.96
D)4.72
E)22.25

Question 79

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $$X _ { 1 } + X _ { 2 }$$ $$\begin{array} { c | c c } & \text { Mean } & \text { SD } \\\hline \mathrm { X } & 14 & 3 \\\mathrm { Y } & 60 & 8\end{array}$$

A)? = 28,? = 4.24
B)? = 74,? = 8.54
C)? = 28,? = 6
D)? = 28,? = 2.45
E)? = 74,? = 11

Question 80

Your school's soccer team plays two games against another soccer team.The probability that your team wins the first game is 0.4.If your team wins the first game,the probability that they also win the second game is 0.5.If your team loses the first game,the probability that they win the second game is 0.2.Let the random variable X be the number of games won by your team.Find the standard deviation of X.

A)σ = 0.63
B)σ = 0.90
C)σ = 0.70
D)σ = 0.60
E)σ = 0.78