Question 1

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

Accepted Answer

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

Question 2

A slot machine at a casino pays out an average of $0.90,with a standard deviation of $120.It costs a dollar to play.If gamblers play this machine 6,969,600 times in a month,what is the probability that the casino will come out ahead? Assume that the casino's total profit follows a Normal model.

Accepted Answer

A) 0.023 
B) 0.014 
C) 0.977 
D) 0.955 
E) 0.986

Question 3

Suppose that 2.4% of people are left handed.If 30 people are selected at random,what is the standard deviation of the number of right-handers in the group?

Accepted Answer

A) 0.70272 
B) 0.72 
C) 0.85 
D) 0.84 
E) 5.41

Question 4

On one tropical island,hurricanes occur with a mean of 2.26 per year.Assuming that the number of hurricanes can be modeled by a Poisson distribution,find the probability that there will be no hurricanes next year.

Accepted Answer

A) 0.2358 
B) 0.7642 
C) 0.0462 
D) 0.0939 
E) 0.1044

Question 5

A company purchases shipments of machine components and uses this acceptance sampling plan: Randomly select and test 20 components and accept the whole batch if there are fewer than 3 defectives.If a particular shipment of thousands of components actually has a 4% rate of defects,what is the probability that this whole shipment will be accepted?

Accepted Answer

A) 0.9561 
B) 0.0439 
C) 0.0364 
D) 0.5141 
E) 0.1458

Question 6

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?

Accepted Answer

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

Question 7

Suppose the probability of a major earthquake on a given day is 1 out of 10,000.Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 2000 days.

Accepted Answer

A) 0.1813 
B) 0.1637 
C) 0.8187 
D) 0.8363 
E) 0.0175

Question 8

A tennis player makes a successful first serve 63% of the time.If she serves 8 times,what is the probability that she gets at least 3 first serves in? Assume that each serve is independent of the others.

Accepted Answer

A) 0.8693 
B) 0.0336 
C) 0.1307 
D) 0.0971 
E) 0.9664

Question 9

In a study,38% of adults questioned reported that their health was excellent.A researcher wishes to study the health of people living close to a nuclear power plant.Among 10 adults randomly selected from this area,only 3 reported that their health was excellent.Find the probability that when 10 adults are randomly selected,3 or fewer are in excellent health.

Accepted Answer

A) 0.4336 
B) 0.3093 
C) 0.2017 
D) 0.5664 
E) 0.2319

Question 10

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.

Accepted Answer

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

Question 11

An airline has found that on average,8% of its passengers request vegetarian meals.On a flight with 360 passengers the airline has 35 vegetarian dinners available.What is the probability that it will be short of vegetarian dinners?

Accepted Answer

A) 0.081 
B) 0.919 
C) 0.067 
D) 0.394 
E) 0.606

Question 12

A basketball player has made 70% of his foul shots during the season.If he shoots 4 foul shots in tonight's game,what is the probability that he makes all of the shots?

Accepted Answer

A) 0.0081 
B) 0.12 
C) 0.28 
D) 0.2401 
E) 0.7

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

We draw a card from a deck 40 times to find the distribution of the suits.After each draw the card is replaced.

Accepted Answer

A) Yes. 
B) No.More than two outcomes are possible. 
C) No.The draws are not independent of each other. 
D) No.40 is more than 10% of 52. 
E) No.The chance of getting each suit changes from one draw to the next.

Question 16

Police estimate that in one city 83% of drivers wear their seat belts.They set up a safety roadblock,stopping cars to check for seat belt use.If they stop 30 cars during the first hour,what is the standard deviation of the number of drivers expected to be wearing their seat belts?

Accepted Answer

A) 5.1 
B) 24.9 
C) 4.23 
D) 2.06 
E) 4.99

Question 17

An archer is usually able to hit the bull's-eye 82% of the time.He buys a new bow hoping that it will improve his success rate.During the first month of practice with his new bow he hits the bull's-eye 363 times out of 430 shots.Is this evidence that with the new bow his success rate has improved? In other words,is this an unusual result for him? Explain.

Accepted Answer

A) No; we would normally expect him to make 352.6 bull's-eyes with a standard deviation of 63.47.363 is 0.2 standard deviations above the expected value.That's not an unusual result. 
B) Yes; we would normally expect him to make 352.6 bull's-eyes with a standard deviation of 7.97.363 is 1.3 standard deviations above the expected value.That's an unusual result. 
C) Yes; we would normally expect him to make 352.6 bull's-eyes with a standard deviation of 63.47.363 is 0.2 standard deviations above the expected value.That's an unusual result. 
D) No; we would normally expect him to make 352.6 bull's-eyes with a standard deviation of 7.97.363 is 1.3 standard deviations above the expected value.That's not an unusual result. 
E) No; we would normally expect him to make 352.6 bull's-eyes with a standard deviation of 18.78.363 is 0.6 standard deviations above the expected value.That's not an unusual result.

Question 18

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. $$\begin{array} { c | c } 
\text { Houses Sold } ( \mathrm { x } ) & \text { Probability P(x) } \
\hline 0 & 0.24 \
\hline 1 & 0.01 \
\hline 2 & 0.12 \
\hline 3 & 0.16 \
\hline 4 & 0.01 \
\hline 5 & 0.14 \
\hline 6 & 0.11 \
\hline 7 & 0.21
\end{array}$$

Accepted Answer

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

Question 19

The rate of defects among CD players of a certain brand is 1.3%.Use the Poisson approximation to the binomial distribution to find the probability that among 100 such CD players received by a store,there are exactly three defectives.

Accepted Answer

A) 0.1497 
B) 0.0748 
C) 0.0998 
D) 0.9002 
E) 0.1996

Question 20

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.

Accepted Answer

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)  $$\begin{array} { l | r c c } 
\text { Amount won } & \$ 80 & \$ 30 & \$ 0 \
\hline \text { P(Amount won) } & \frac { 1 } { 4 } & \frac { 1 } { 4 } & \frac { 1 } { 2 }
\end{array}$$

Exam 14: Random Variables

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

A slot machine at a casino pays out an average of $0.90,with a standard deviation of $120.It costs a dollar to play.If gamblers play this machine 6,969,600 times in a month,what is the probability that the casino will come out ahead? Assume that the casino's total profit follows a Normal model.

Suppose that 2.4% of people are left handed.If 30 people are selected at random,what is the standard deviation of the number of right-handers in the group?

On one tropical island,hurricanes occur with a mean of 2.26 per year.Assuming that the number of hurricanes can be modeled by a Poisson distribution,find the probability that there will be no hurricanes next year.

Suppose the probability of a major earthquake on a given day is 1 out of 10,000.Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 2000 days.

A tennis player makes a successful first serve 63% of the time.If she serves 8 times,what is the probability that she gets at least 3 first serves in? Assume that each serve is independent of the others.

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.

An airline has found that on average,8% of its passengers request vegetarian meals.On a flight with 360 passengers the airline has 35 vegetarian dinners available.What is the probability that it will be short of vegetarian dinners?

A basketball player has made 70% of his foul shots during the season.If he shoots 4 foul shots in tonight's game,what is the probability that he makes all of the shots?

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

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

We draw a card from a deck 40 times to find the distribution of the suits.After each draw the card is replaced.

Police estimate that in one city 83% of drivers wear their seat belts.They set up a safety roadblock,stopping cars to check for seat belt use.If they stop 30 cars during the first hour,what is the standard deviation of the number of drivers expected to be wearing their seat belts?

The rate of defects among CD players of a certain brand is 1.3%.Use the Poisson approximation to the binomial distribution to find the probability that among 100 such CD players received by a store,there are exactly three defectives.

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.

Exam 14: Random Variables

x481216P(X=x) 0.10.40.10.4\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}xP(X=x) ​40.1​80.4​120.1​160.4​​

A slot machine at a casino pays out an average of $0.90,with a standard deviation of $120.It costs a dollar to play.If gamblers play this machine 6,969,600 times in a month,what is the probability that the casino will come out ahead? Assume that the casino's total profit follows a Normal model.

Suppose that 2.4% of people are left handed.If 30 people are selected at random,what is the standard deviation of the number of right-handers in the group?

On one tropical island,hurricanes occur with a mean of 2.26 per year.Assuming that the number of hurricanes can be modeled by a Poisson distribution,find the probability that there will be no hurricanes next year.

Suppose the probability of a major earthquake on a given day is 1 out of 10,000.Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 2000 days.

A tennis player makes a successful first serve 63% of the time.If she serves 8 times,what is the probability that she gets at least 3 first serves in? Assume that each serve is independent of the others.

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.

An airline has found that on average,8% of its passengers request vegetarian meals.On a flight with 360 passengers the airline has 35 vegetarian dinners available.What is the probability that it will be short of vegetarian dinners?

A basketball player has made 70% of his foul shots during the season.If he shoots 4 foul shots in tonight's game,what is the probability that he makes all of the shots?

x100200300400P(X=x) 0.30.40.20.1\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}xP(X=x) ​1000.3​2000.4​3000.2​4000.1​​

x100200300400P(X=x) 0.20.40.30.1\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}xP(X=x) ​1000.2​2000.4​3000.3​4000.1​​

We draw a card from a deck 40 times to find the distribution of the suits.After each draw the card is replaced.

Police estimate that in one city 83% of drivers wear their seat belts.They set up a safety roadblock,stopping cars to check for seat belt use.If they stop 30 cars during the first hour,what is the standard deviation of the number of drivers expected to be wearing their seat belts?

The rate of defects among CD players of a certain brand is 1.3%.Use the Poisson approximation to the binomial distribution to find the probability that among 100 such CD players received by a store,there are exactly three defectives.

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.

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

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

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