Question 1

If a student guesses on each question,what is the probability that the student will answer at least 6 questions correctly.

Accepted Answer

A) 0.172 
B) 0.205 
C) 0.828 
D) 0.377 
E) 0.6 
A) 0.172 
B) 0.205 
C) 0.828 
D) 0.377 
E) 0.6

Question 2

An airline,believing that 6% of passengers fail to show up for flights,overbooks.Suppose a plane will hold 320 passengers and the airline sells 335 seats.What is the probability the airline will not have enough seats and will have to bump someone?

Accepted Answer

A) 0.075 
B) 0.121 
C) 0.375 
D) 0.919 
E) 0.081 
A) 0.075 
B) 0.121 
C) 0.375 
D) 0.919 
E) 0.081

Question 3

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.

Accepted Answer

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)  $$\begin{array} { l | c c c c } 
\text { Amount won } & \$ 0 & \$ 15 & \$ 20 & \$ 80 \
\hline \text { P(Amount won) } & \frac { 36 } { 52 } & \frac { 12 } { 52 } & \frac { 4 } { 52 } & \frac { 1 } { 52 }
\end{array}$$ 
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 4

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. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 30 & 9 \
\mathrm { Y } & 50 & 6
\end{array}$$

Accepted Answer

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

Question 5

Suppose a computer chip manufacturer rejects 3% of the chips produced because they fail presale testing.If you test 3 chips,what is the probability that none of the chips fail?

Accepted Answer

A) 0.97 
B) 0.03 
C) 2.7 × $10 ^ { - 5 }$ 
D) 0.09 
E) 0.9127 
A) 0.97 
B) 0.03 
C) 2.7 × $10 ^ { - 5 }$ 
D) 0.09 
E) 0.9127

Question 6

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.

Accepted Answer

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

Question 7

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.

Accepted Answer

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 
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 8

Suppose that 10% of people are left handed.If 5 people are selected at random,what is the probability that exactly 2 of them are left handed?

Accepted Answer

A) 0.1458 
B) 0.0729 
C) 0.0100 
D) 0.0073 
E) 0.0081 
A) 0.1458 
B) 0.0729 
C) 0.0100 
D) 0.0073 
E) 0.0081

Question 9

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

Accepted Answer

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

Question 10

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.

Accepted Answer

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

Question 11

Sunita's job is to provide technical support to computer users.Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 4.8 calls per hour.What is the probability that in the next 15 minutes there will be no calls?

Accepted Answer

A) 0.3614 
B) 0.6988 
C) 0.3012 
D) 0.0082 
E) 0.0395 
A) 0.3614 
B) 0.6988 
C) 0.3012 
D) 0.0082 
E) 0.0395

Question 12

The amount of time it takes to serve each customer in a bank is a random variable,X,with a mean of ? minutes and a standard deviation of ? minutes.When you arrive at the bank there are three customers in front of you.If the times for the three customers are independent of one another,which of the following shows the correct method for calculating the variance of your wait time?

Accepted Answer

A) Var( $X _ { 1 }$ + $x _ { 2 }$ + $X _ { 3 }$ )= $3 ^ { 2 }$ Var(X)= 9 $\sigma ^ { 2 }$ 
B) Var(3X)= 3Var(X)= 3 $\sigma ^ { 2 }$ 
C) Var( $X _ { 1 }$ + $X _ { 2 }$ + $X _ { 3 }$ )= $\sqrt { 3 }$ Var(X)= $\sqrt { 3 }$ $\sigma ^ { 2 }$ 
D) Var(3X)= $3 ^ { 2 }$ Var(X)= 9 $\sigma ^ { 2 }$ 
E) Var( $X _ { 1 }$ + $X_ { 2 }$ + $X _ { 3 }$ )= Var( $X _ { 1 }$ )+ Var( $X _ { 2 }$ )+ Var( $X _ { 3 }$ )= 3 $\sigma ^ { 2 }$ 
A) Var( $X _ { 1 }$ + $x _ { 2 }$ + $X _ { 3 }$ )= $3 ^ { 2 }$ Var(X)= 9 $\sigma ^ { 2 }$ 
B) Var(3X)= 3Var(X)= 3 $\sigma ^ { 2 }$ 
C) Var( $X _ { 1 }$ + $X _ { 2 }$ + $X _ { 3 }$ )= $\sqrt { 3 }$ Var(X)= $\sqrt { 3 }$ $\sigma ^ { 2 }$ 
D) Var(3X)= $3 ^ { 2 }$ Var(X)= 9 $\sigma ^ { 2 }$ 
E) Var( $X _ { 1 }$ + $X_ { 2 }$ + $X _ { 3 }$ )= Var( $X _ { 1 }$ )+ Var( $X _ { 2 }$ )+ Var( $X _ { 3 }$ )= 3 $\sigma ^ { 2 }$

Question 13

Janet is planning to rent a booth at a festival for a day to sell clothes that she has made.She sells jackets for $\$ 80$ and skirts for $\$ 50$ Her past experiences suggests that sales of jackets will have a mean of 6.5 with a standard deviation of $\sigma _ { j }$ ,and sales of skirts will have a mean of 12.6 with a standard deviation of $\sigma _ { s }$ .The cost of renting the booth for the day is $\$ 180$ Let the random variable J represent the number of jackets that Janet sells and the random variable S represent the number of skirts that Janet sells.Then Janet's net income,I,is given by the expression: I = 80J + 50S - 180
Find an expression for the standard deviation of Janet's net income.Assume that sales are independent of each other.

Accepted Answer

A) ? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$ 
B) ? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } - 180 }$ 
C) ? = $\sqrt { 80 \sigma _ { j } ^ { 2 } + 50 \sigma _ { s } ^ { 2 } }$ 
D) ? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } }$ 
E) ? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$ - 180 
A) ? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$ 
B) ? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } - 180 }$ 
C) ? = $\sqrt { 80 \sigma _ { j } ^ { 2 } + 50 \sigma _ { s } ^ { 2 } }$ 
D) ? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } }$ 
E) ? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$ - 180

Question 14

Sue buys a large packet of rice.The amount of rice that the manufacturer puts in the packet is a random variable with a mean of 1016 g and a standard deviation of 8 g.The amount of rice that Sue uses in a week has a mean of 200 g and a standard deviation of 29 g.If the weight of the rice remaining in the packet after a week can be described by a Normal model,what is the probability that the packet still contains more than 831.0 g after a week?

Accepted Answer

A) 0.655 
B) 0.274 
C) 0.691 
D) 0.309 
E) 0.345 
A) 0.655 
B) 0.274 
C) 0.691 
D) 0.309 
E) 0.345

Question 15

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.

Accepted Answer

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

Question 16

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.

Accepted Answer

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)  $$\begin{array} { l | c | c | c } 
\text { Profit } & \$ 400 & - \$ 9,600 & - \$ 1,600 \
\hline \text { P(profit) } & 0.9987 & 0.0003 & 0.001
\end{array}$$ 
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 17

Sunita's job is to provide technical support to computer users.Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 4.9 calls per hour.What is the probability that in the next 30 minutes there will be 2 or more calls?

Accepted Answer

A) 0.7886 
B) 0.2590 
C) 0.0894 
D) 0.7023 
E) 0.9561 
A) 0.7886 
B) 0.2590 
C) 0.0894 
D) 0.7023 
E) 0.9561

Question 18

A tennis player makes a successful first serve 64% of the time.If she serves 6 times,what is the probability that she gets no more than 3 first serves in? Assume that each serve is independent of the others.

Accepted Answer

A) 0.2446 
B) 0.6268 
C) 0.1286 
D) 0.8714 
E) 0.3732 
A) 0.2446 
B) 0.6268 
C) 0.1286 
D) 0.8714 
E) 0.3732

Question 19

The amount of money that Maria earns in a week is a random variable,X,with a mean of $\$ 800$ and a standard deviation of $\$ 20$ The amount of money that Daniel earns in a week is a random variable,Y,with a mean of $\$ 900$ and a standard deviation of $\$ 30$ The total,X +Y,of Maria's weekly income and Daniel's weekly income is a random variable with a mean of $\$ 800 + \$ 900 = \$ 1700$ and a standard deviation of $\sqrt { 20 ^ { 2 } + 30 ^ { 2 } } \approx \$ 36$ The calculation of the standard deviation requires the assumption that the incomes are independent of one another.Which of the following examples describes a situation in which that assumption may be violated? A: Maria and Daniel are married to each other.
B: Maria and Daniel work for the same company.
C: Maria and Daniel work in the same business.
D: Maria and Daniel were born in the same year.

Accepted Answer

A) A and B 
B) A,B,C,and D 
C) B,C,and D 
D) B and C 
E) A,B,and C 
A) A and B 
B) A,B,C,and D 
C) B,C,and D 
D) B and C 
E) A,B,and C

Question 20

If Judy,who forgot to study for the test,guesses on all questions,what is the probability that she will answer exactly 3 questions correctly?

Accepted Answer

A) 0.7664 
B) 0.0156 
C) 0.2336 
D) 0.0028 
E) 0.3003 
A) 0.7664 
B) 0.0156 
C) 0.2336 
D) 0.0028 
E) 0.3003

Exam 14: Random Variables

If a student guesses on each question,what is the probability that the student will answer at least 6 questions correctly.

An airline,believing that 6% of passengers fail to show up for flights,overbooks.Suppose a plane will hold 320 passengers and the airline sells 335 seats.What is the probability the airline will not have enough seats and will have to bump someone?

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.

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. Mean SD 30 9 50 6

Suppose a computer chip manufacturer rejects 3% of the chips produced because they fail presale testing.If you test 3 chips,what is the probability that none of the chips fail?

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.

Suppose that 10% of people are left handed.If 5 people are selected at random,what is the probability that exactly 2 of them are left handed?

0 1 2 (=) 0.7 0.2 0.1

Sunita's job is to provide technical support to computer users.Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 4.8 calls per hour.What is the probability that in the next 15 minutes there will be no calls?

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.

Sunita's job is to provide technical support to computer users.Suppose the arrival of calls can be modeled by a Poisson distribution with a mean of 4.9 calls per hour.What is the probability that in the next 30 minutes there will be 2 or more calls?

A tennis player makes a successful first serve 64% of the time.If she serves 6 times,what is the probability that she gets no more than 3 first serves in? Assume that each serve is independent of the others.

If Judy,who forgot to study for the test,guesses on all questions,what is the probability that she will answer exactly 3 questions correctly?

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