Question 1

We record the blood types (O,A,B,or AB)found in a group of 100 people.Assume that the people are unrelated to each other.

Accepted Answer

A) Yes. 
B) No.The chance of getting a particular blood group changes from one person to the next. 
C) No.More than two outcomes are possible. 
D) No,400 is more than 10% of the population. 
E) No.The chance of getting a particular blood group depends on the blood groups already recorded. 
A) Yes. 
B) No.The chance of getting a particular blood group changes from one person to the next. 
C) No.More than two outcomes are possible. 
D) No,400 is more than 10% of the population. 
E) No.The chance of getting a particular blood group depends on the blood groups already recorded.

Question 2

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 1022 g and a standard deviation of 11 g.The amount of rice that Sue uses in a week has a mean of 240 g and a standard deviation of 26 g.Find the mean and standard deviation of the amount of rice remaining in the packet after a week.

Accepted Answer

A) μ = 1262 g,σ = 37 g 
B) μ = 782 g,σ = 23.56 g 
C) μ = 782 g,σ = 28.23 g 
D) μ = 782 g,σ = 15 g 
E) μ = 1262 g,σ = 28.23 g 
A) μ = 1262 g,σ = 37 g 
B) μ = 782 g,σ = 23.56 g 
C) μ = 782 g,σ = 28.23 g 
D) μ = 782 g,σ = 15 g 
E) μ = 1262 g,σ = 28.23 g

Question 3

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.

Accepted Answer

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

Question 4

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.

Accepted Answer

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

Question 5

A tennis player makes a successful first serve 65% of the time.If she serves 90 times in a match,what is the probability that she makes no more than 50 first serves? Assume that each serve is independent of the others.

Accepted Answer

A) 0.337 
B) 0.970 
C) 0.018 
D) 0.030 
E) 0.663 
A) 0.337 
B) 0.970 
C) 0.018 
D) 0.030 
E) 0.663

Question 6

In one city,the probability that a person will pass his or her driving test on the first attempt is 0.68.11 people are selected at random from among those taking their driving test for the first time.What is the probability that among these 11 people,the number passing the test is between 2 and 4 inclusive?

Accepted Answer

A) 0.0057 
B) 0.0345 
C) 0.0299 
D) 0.0251 
E) 0.0308 
A) 0.0057 
B) 0.0345 
C) 0.0299 
D) 0.0251 
E) 0.0308

Question 7

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.

Accepted Answer

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

Question 8

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.

Accepted Answer

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)  $$\begin{array} { l | l r r r } 
\mathrm { X } & 12 & 19 & 20 & 21 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 1 / 4 & 1 / 4 & 1 / 4 & 1 / 4
\end{array}$$ 
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 9

The probability that a car will have a flat tire while driving through a certain tunnel is 0.00008.Use the Poisson approximation to the binomial distribution to find the probability that among 16,000 cars passing through this tunnel,at least one will have a flat tire.

Accepted Answer

A) 0.7220 
B) 0.6441 
C) 0.2780 
D) 1.6462 
E) 0.3559 
A) 0.7220 
B) 0.6441 
C) 0.2780 
D) 1.6462 
E) 0.3559

Question 10

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.

Accepted Answer

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)  $$\begin{array} { l | l l l } 
\text { Rainy days } & 0 & 1 & 2 \
\hline \text { P(Rainy days) } & 0.35 & 0.4 & 0.25
\end{array}$$ 
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 11

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.

Accepted Answer

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

Question 12

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?

Accepted Answer

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

Question 13

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.

Accepted Answer

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)  $$\begin{array} { l | c c c c } 
\text { Children } & 1 & 2 & 3 & 4 \
\hline \text { P(Children) } & 0.25 & 0.25 & 0.25 & 0.25
\end{array}$$ 
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 14

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

Accepted Answer

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

Question 15

A car insurance company has determined that 7% of all drivers were involved in a car accident last year.If 11 drivers are randomly selected,what is the probability that at least 3 were involved in a car accident last year?

Accepted Answer

A) 0.0317 
B) 0.4871 
C) 0.0695 
D) 0.9683 
E) 0.9630 
A) 0.0317 
B) 0.4871 
C) 0.0695 
D) 0.9683 
E) 0.9630

Question 16

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.

Accepted Answer

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

Question 17

An airline estimates that 97% of people booked on their flights actually show up.If the airline books 71 people on a flight for which the maximum number is 69,what is the probability that the number of people who show up will exceed the capacity of the plane?

Accepted Answer

A) 0.6324 
B) 0.3676 
C) 0.6410 
D) 0.1150 
E) 0.2526 
A) 0.6324 
B) 0.3676 
C) 0.6410 
D) 0.1150 
E) 0.2526

Question 18

The amount of money that Maria earns in a week is a random variable,X,with a mean of $\$ 900$ and a standard deviation of $\sigma _ { \mathrm { m } }$ .The amount of money that Daniel earns in a week is a random variable,Y,with a mean of $\$ 800$ and a standard deviation of $\sigma _ { \mathrm { d } }$ .The difference, $X - Y,$ between Maria's weekly income and Daniel's weekly income is a random variable with a mean of $\$ 900 - \$ 800 = \$ 100$ If the incomes are independent of one another,which of the following shows the correct method for calculating the standard deviation of the random variable $X - Y ?$

Accepted Answer

A) SD(X - Y)= Var(X - Y)= Var(X)+ Var(Y)= $\sigma _ { \mathrm { m } } ^ { 2 }$ + $\sigma _ { \mathrm { d } } ^ { 2 }$ 
B) SD(X - Y)= $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) + \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } + \sigma _ { \mathrm { d } } ^ { 2 } }$ 
C) SD(X - Y)= SD(X)+ SD(Y)= $\sigma _ { \mathrm { m } }$ + $\sigma _ { \mathrm { d } }$ 
D) SD(X - Y)= SD(X)- SD(Y)= $\sigma _ { \mathrm { m } }$ - $\sigma _ { \mathrm { d } }$ 
E) SD(X - Y)= $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) - \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } - \sigma _ { \mathrm { d } } ^ { 2 } }$ 
A) SD(X - Y)= Var(X - Y)= Var(X)+ Var(Y)= $\sigma _ { \mathrm { m } } ^ { 2 }$ + $\sigma _ { \mathrm { d } } ^ { 2 }$ 
B) SD(X - Y)= $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) + \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } + \sigma _ { \mathrm { d } } ^ { 2 } }$ 
C) SD(X - Y)= SD(X)+ SD(Y)= $\sigma _ { \mathrm { m } }$ + $\sigma _ { \mathrm { d } }$ 
D) SD(X - Y)= SD(X)- SD(Y)= $\sigma _ { \mathrm { m } }$ - $\sigma _ { \mathrm { d } }$ 
E) SD(X - Y)= $\sqrt { \operatorname { Var } ( X - Y ) }$ = $\sqrt { \operatorname { Var } ( X ) - \operatorname { Var } ( Y ) }$
= $\sqrt { \sigma _ { \mathrm { m } } ^ { 2 } - \sigma _ { \mathrm { d } } ^ { 2 } }$

Question 19

At a furniture factory,tables must be assembled,finished,and packaged before they can be shipped to stores.Based on past experience,the manager finds that the means and standard deviations (in minutes)of the times for each phase are as shown in the table: $$\begin{array} { l | l | l } 
\text { Phase } & \text { Mean } & \text { SD } \
\hline \text { Assembly } & 26.5 & 2.6 \
\text { Finishing } & 35.5 & 3.2 \
\text { Packaging } & 16.0 & 2.3
\end{array}$$ Find the probability that a table can be prepared for shipping in less than 67.61 minutes.Assume that the times for each phase are independent and that the times for each phase follow a Normal model.

Accepted Answer

A) 0.014 
B) 0.023 
C) 0.986 
D) 0.045 
E) 0.955 
A) 0.014 
B) 0.023 
C) 0.986 
D) 0.045 
E) 0.955

Question 20

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.

Accepted Answer

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

Exam 14: Random Variables

We record the blood types (O,A,B,or AB)found in a group of 100 people.Assume that the people are unrelated to each other.

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.

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 tennis player makes a successful first serve 65% of the time.If she serves 90 times in a match,what is the probability that she makes no more than 50 first serves? Assume that each serve is independent of the others.

The probability that a car will have a flat tire while driving through a certain tunnel is 0.00008.Use the Poisson approximation to the binomial distribution to find the probability that among 16,000 cars passing through this tunnel,at least one will have a flat tire.

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.

Jo is a hairstylist.The probability model below describes the number of clients that she may see in a day. Number of clients 0 1 2 3 4 5 Probability 0.1 0.1 0.2 0.25 0.2 0.15 What is the expected value of the number of clients that Jo sees per day?

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.

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable x1+x2+x3x _ { 1 } + x _ { 2 } + x _ { 3 }x1​+x2​+x3​ Mean SD 12 2 50 11

A car insurance company has determined that 7% of all drivers were involved in a car accident last year.If 11 drivers are randomly selected,what is the probability that at least 3 were involved in a car accident last year?

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.

An airline estimates that 97% of people booked on their flights actually show up.If the airline books 71 people on a flight for which the maximum number is 69,what is the probability that the number of people who show up will exceed the capacity of the plane?

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.

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Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $x _ { 1 } + x _ { 2 } + x _ { 3 }$ Mean SD 12 2 50 11