Deck 5: Probability Distributions

Question 1

Provide an appropriate response.
Sampling without replacement involves dependent events, so this would not be considered a binomial experiment. Explain the circumstances under which sampling without replacement could be considered independent and, thus, binomial.

Answer 1

When the sample size is no more than 5% of the total population size.

Question 2

Provide an appropriate response.

-Describe the differences in the Poisson and the binomial distribution.

Answer 2

The Poisson computes probabilities for occurrences of events over some interval.
The Poisson distribution is affected only by the mean µ, whereas the binomial is affected by sample size n and probability p.
The Poisson distribution has discrete values from 1, 2, 3, . . . with no upper limit. A binomial distribution has discrete values from 1, 2, 3, to n; that is, the upper limit of values is n.

Question 3

Provide an appropriate response.

-Identify each of the variables in the Binomial Probability Formula. $P ( x ) = \frac { n ! } { ( n - x ) ! x ! } \cdot p ^ { x } \cdot q ^ { n - x }$
Also, explain what the fraction $\frac { n ! } { ( n - x ) ! x ! }$ computes.

Answer 3

n is the fixed number of trials, x is the number of successes, p is the probability of success in one of the n trials, and q is the probability of failure in one of the n trials. The fraction determines the number of different orders of x successes out of n trials.

Question 4

Identify the given random variable

-The number of freshmen in the required course, English 101

A)Discrete
B)Continuous

Question 5

Provide an appropriate response.
Describe the Poisson distribution and give an example of a random variable with a Poisson distribution.

Question 6

Provide an appropriate response.

-Suppose an event has a 80% chance of success. Show a probability distribution for the number of successes in 8 trials. Find the mean and SD. Create an interval of ±2 SD about the mean. Would exactly 4 successes out of 8 be an unusual occurrence? Justify your answer in terms of the Range Rule of Thumb.

Question 7

Identify the given random variable

-The number of oil spills occurring off the Alaskan coast

A)Discrete
B)Continuous

Question 8

Provide an appropriate response.
A game is said to be "fair" if the expected value for winnings is 0, that is, in the long run, the player can expect to win 0. Consider the following game. The game costs $1 to play and the winnings are $5 for red, $3 for blue, $2 for yellow, and nothing for white. The following probabilities apply. What are your expected winnings? Does the game favor the player or the owner?

Question 9

Provide an appropriate response.
List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Question 10

Provide an appropriate response.
List the four requirements of a Poisson distribution.

Question 11

Provide an appropriate response.
Previously we learned to find the three important characteristics of data: the measure of central tendency, the measure of variation, and the nature of the distribution, We can find the same three characteristics for a binomial distribution. Given a binomial distribution with p = 0.4 and n = 8, find the three characteristics.

Question 12

Provide an appropriate response.
Compare the probability histogram for the expected sum with the actual results. What do you conclude about the dice results displayed in the Actual Sum of Two Dice histogram?

Question 13

Provide an appropriate response.

-Suppose a mathematician computed the expected value of winnings for a person playing each of seven different games in a casino. What would you expect to be true for all expected values for these seven games?

Question 14

Identify the given random variable

-The pH level in a shampoo

A)Discrete
B)Continuous

Question 15

Identify the given random variable

-The number of phone calls between New York and California on Thanksgiving day

A)Discrete
B)Continuous

Question 16

Provide an appropriate response.
Do probability distributions measure what did happen or what will probably happen? How do we use probability distributions to make decisions?

Question 17

Identify the given random variable

-The cost of a randomly selected orange

A)Discrete
B)Continuous

Question 18

Identify the given random variable

-The height of a randomly selected student

A)Discrete
B)Continuous

Question 19

Provide an appropriate response.

-List the two requirements for a probability histogram. Discuss the relationship between the sum of the probabilities in a probability distribution and the total area represented by the bars in a probability histogram.

Question 20

Provide an appropriate response.
List the three methods for finding binomial probabilities in the table below, and then complete the table to discuss the advantages and disadvantages of each.

Question 21

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.
In a certain town, 70% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults)who have a college degree.

Question 22

Find the mean of the given probability distribution.
The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.5470, 0.3562, 0.0870, 0.0094, and 0.0004, respectively. Round answer to the nearest hundredth.

A)2.00
B)0.56
C)1.11
D)0.46

Question 23

Identify the given random variable

-The braking time of a car

A)Discrete
B)Continuous

Question 24

Solve the problem.

-In a certain town, 40% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults)who have a college degree. Find the standard deviation for the probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.1296 \\\hline 1 & 0.3456 \\\hline 2 & 0.3456 \\\hline 3 & 0.1536 \\\hline 4 & 0.0256\end{array}$$

A)0.96
B)1.12
C)1.88
D)0.98

Question 25

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.
If a person is randomly selected from a certain town, the probability distribution for the number, x, of siblings is as described in the accompanying table.

Question 26

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.

Question 27

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.
A police department reports that the probabilities that 0, 1, 2, 3, and 4 car thefts will be reported in a given day are 0.202, 0.323, 0.258, 0.138, and 0.055, respectively.

Question 28

Solve the problem.

-Find the standard deviation for the given probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.37 \\\hline 1 & 0.05 \\\hline 2 & 0.13 \\\hline 3 & 0.25 \\\hline 4 & 0.20\end{array}$$

A)2.45
B)2.56
C)1.60
D)1.71

Question 29

Find the mean of the given probability distribution.
A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.50, 0.38, 0.11, and 0.01, respectively.

A)0.63
B)0.25
C)1.50
D)1.13

Question 30

Find the mean of the given probability distribution.

-The number of golf balls ordered by customers of a pro shop has the following probability distribution. $$\begin{array} { r | r | r | r | r | r } \mathrm { x } & 3 & 6 & 9 & 12 & 15 \\\hline \mathrm { p } ( \mathrm { x } ) & 0.14 & 0.11 & 0.36 & 0.29 & 0.10\end{array}$$

A)8.22
B)9
C)6.63
D)9.3

Question 31

Find the mean of the given probability distribution.

- $$\begin{array}{l}\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.19 \\1 & 0.37 \\2 & 0.16 \\3 & 0.26 \\4 & 0.02\end{array}\\\end{array}$$

A) 1.45
B)1.64
C)1.55
D)1.74

Question 32

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.

Question 33

Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied.

Question 34

Find the mean of the given probability distribution.

-The accompanying table shows the probability distribution for x, the number that shows up when a loaded die is rolled. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 1 & 0.13 \\2 & 0.12 \\3 & 0.16 \\4 & 0.13 \\5 & 0.15 \\6 & 0.31\end{array}$$

A)0.17
B)3.50
C)3.85
D)3.98

Question 35

Solve the problem.

-Find the variance for the given probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.17 \\\hline 1 & 0.28 \\\hline 2 & 0.05 \\\hline 3 & 0.15 \\\hline 4 & 0.35\end{array}$$

A)2.63
B)7.43
C)2.69
D)2.46

Question 36

Find the mean of the given probability distribution.

-In a certain town, 30% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults)who have a college degree. $$\begin{array} { c | r } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 0 & 0.2401 \\1 & 0.4116 \\2 & 0.2646 \\3 & 0.0756 \\4 & 0.0081\end{array}$$

A)1.10
B)2.00
C)1.44
D)1.20

Question 37

Find the mean of the given probability distribution.

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

A)3.60
B)3.40
C)3.50
D)3.35

Question 38

Identify the given random variable

-The number of field goals kicked in a football game

A)Discrete
B)Continuous

Question 39

Solve the problem.

-Find the variance for the given probability distribution. $$\begin{array} { c | c } x & P ( x ) \\\hline 0 & 0.05 \\\hline 2 & 0.17 \\\hline 4 & 0.43 \\\hline 6 & 0.35\end{array}$$

A)1.56
B)2.44
C)2.85
D)1.69

Question 40

Solve the problem.

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

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

Question 41

Solve the problem.
Suppose you pay $3.00 to roll a fair die with the understanding that you will get back $5.00 for rolling a 3 or a 6, nothing otherwise. What is your expected value?

A)$3.00
B)-$1.33
C)-$3.00
D)$5.00

Question 42

Use the normal distribution to approximate the desired probability.
Find the probability of selecting 12 or more girls.

A)0.001
B)0.007
C)0.006
D)0.022

Question 43

Solve the problem.
The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.4521, 0.3970, 0.1307, 0.0191, and 0.0010, respectively. Find the variance for the probability distribution.

A)0.59
B)0.77
C)1.11
D)0.51

Question 44

Solve the problem.
A contractor is considering a sale that promises a profit of $23,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such)of $13,000 with a probability of 0.3. What is the expected profit?

A)$12,200
B)$25,200
C)$16,100
D)$10,000

Question 45

Assume that there is a 0.15 probability that a basketball playoff series will last four games, a 0.30 probability that it will last five games, a 0.25 probability that it will last six games, and a 0.30 probability that it will last seven games. Is it unusual for a team to win a series in 5 games?

A)Yes
B)No

Question 46

Solve the problem.
A 28-year-old man pays $57 for a one-year life insurance policy with coverage of $100,000. If the probability that he will live through the year is 0.9992, what is the expected value for the insurance policy?

A)-$56.95
B)$99,920.00
C)$23.00
D)$80.00

Question 47

Solve the problem.
The probabilities that a batch of 4 computers will contain 0, 1, 2, 3, and 4 defective computers are 0.6274, 0.3102, 0.0575, 0.0047, and 0.0001, respectively. Find the standard deviation for the probability distribution.

A)0.39
B)0.63
C)0.76
D)0.56

Question 48

Solve the problem.
In a game, you have a 1/42 probability of winning $67 and a 41/42 probability of losing $7. What is your expected value?

A)$1.60
B)$8.43
C)-$5.24
D)-$6.83

Question 49

Use the normal distribution to approximate the desired probability.
The prizes that can be won in a sweepstakes are listed below together with the chances of winning each one: $5300 (1 chance in 8600); $2500 (1 chance in 5300); $700 (1 chance in 3500); $200 (1 chance in 2800). Find the expected value of the amount won for one entry if the cost of entering is 73 cents.

A)$0.67
B)$0.63
C)$1.29
D)$200

Question 50

Solve the problem.
Suppose you buy 1 ticket for $1 out of a lottery of 1,000 tickets where the prize for the one winning ticket is to be $500. What is your expected value?

A)-$0.40
B)-$0.50
C)-$1.00
D)$0.00

Question 51

Use the normal distribution to approximate the desired probability.
Find the probability of selecting exactly 5 girls.

A)0.022
B)0.122
C)0.001
D)0.061

Question 52

Focus groups of 10 people are randomly selected to discuss products of the Famous Company. It is determined that the mean number (per group)who recognize the Famous brand name is 4.8, and the standard deviation is 0.73. Would it be unusual to randomly select 10 people and find that greater than 8 recognize the Famous brand name?

A)No
B)Yes

Question 53

Use the normal distribution to approximate the desired probability.
Find the probability of selecting exactly 8 girls.

A)0.183
B)0.000
C)0.022
D)0.122

Question 54

Solve the problem.

-In a certain town, 60% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults)who have a college degree. Find the variance 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)6.72
B)0.98
C)0.84
D)0.96

Question 55

Use the normal distribution to approximate the desired probability.
Find the probability of selecting exactly 4 girls.

A)0.022
B)0.061
C)0.122
D)0.001

Question 56

Solve the problem.

-The accompanying table shows the probability distribution for x, the number that shows up when a loaded die is rolled. Find the variance for the probability distribution. $$\begin{array} { c | c } \mathrm { x } & \mathrm { P } ( \mathrm { x } ) \\\hline 1 & 0.16 \\\hline 2 & 0.19 \\\hline 3 & 0.22 \\\hline 4 & 0.21 \\\hline 5 & 0.12 \\\hline 6 & 0.10\end{array}$$

A)12.86
B)9.62
C)2.36
D)2.03

Question 57

Assume that a researcher randomly selects 14 newborn babies and counts the number of girls selected, x. The probabilities
corresponding to the 14 possible values of x are summarized in the given table. Answer the question using the table. $$\begin{array}{l}\text { Probabilities of Girls }\\\begin{array} { c | c | c | c | c | c } x ( \text { girls } ) & P ( x ) & x ( \text { girls) } & P ( x ) & x ( \text { girls) } & P ( x ) \\\hline 0 & 0.000 & 5 & 0.122 & 10 & 0.061 \\1 & 0.001 & 6 & 0.183 & 11 & 0.022 \\2 & 0.006 & 7 & 0.209 & 12 & 0.006 \\3 & 0.022 & 8 & 0.183 & 13 & 0.001 \\4 & 0.061 & 9 & 0.122 & 14 & 0.000\end{array}\end{array}$$

-Find the probability of selecting 2 or more girls.

A)0.994
B)0.006
C)0.999
D)0.001

Question 58

Use the normal distribution to approximate the desired probability.
Find the probability of selecting 9 or more girls.

A)0.061
B)0.001
C)0.122
D)0.212

Question 59

Focus groups of 12 people are randomly selected to discuss products of the Yummy Company. It is determined that the mean number (per group)who recognize the Yummy brand name is 9.4, and the standard deviation is 0.98. Would it be unusual to randomly select 12 people and find that fewer than 5 recognize the Yummy brand name?

A)No
B)Yes

Question 60

Solve the problem.
A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.52, 0.40, 0.07, and 0.01, respectively. Find the standard deviation for the probability distribution. Round answer to the nearest hundredth.

A)0.98
B)0.88
C)0.45
D)0.67

Question 61

Assume that there is a 0.05 probability that a sports playoff series will last four games, a 0.45 probability that it will last five games, a 0.45 probability that it will last six games, and a 0.05 probability that it will last seven games. Is it unusual for a team to win a series in 7 games?

A)No
B)Yes

Question 62

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Spinning a roulette wheel 3 times, keeping track of the winning numbers.

A)Not binomial: the trials are not independent.
B)Procedure results in a binomial distribution.
C)Not binomial: there are too many trials.
D)Not binomial: there are more than two outcomes for each trial.

Question 63

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Rolling a single die 19 times, keeping track of the numbers that are rolled.

A)Not binomial: there are more than two outcomes for each trial.
B)Not binomial: the trials are not independent.
C)Not binomial: there are too many trials.
D)Procedure results in a binomial distribution.

Question 64

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Rolling a single die 46 times, keeping track of the "fives" rolled.

A)Not binomial: there are more than two outcomes for each trial.
B)Procedure results in a binomial distribution.
C)Not binomial: there are too many trials.
D)Not binomial: the trials are not independent.

Question 65

Answer the question.

-Suppose that a law enforcement group studying traffic violations determines that the accompanying table describes the probability distribution for five randomly selected people, where x is the number that have received a speeding ticket in the last 2 years. Is it unusual to find no speeders among five randomly selected people? $$\begin{array} { l | l } x & P ( x ) \\\hline 0 & 0.08 \\\hline 1 & 0.18 \\\hline 2 & 0.25 \\\hline 3 & 0.22 \\\hline 4 & 0.19 \\\hline 5 & 0.08\end{array}$$

A)Yes
B)No

Question 66

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Spinning a roulette wheel 7 times, keeping track of the occurrences of a winning number of "16".

A)Not binomial: the trials are not independent.
B)Not binomial: there are more than two outcomes for each trial.
C)Not binomial: there are too many trials.
D)Procedure results in a binomial distribution..

Question 67

Determine whether the given procedure results in a binomial distribution. If not, state the reason why.

- $$\frac { 1 } { 3 }$$ n = 10, x = 2, p =

A)0.1951
B)0.1929
C)0.2156
D)0.0028

Question 68

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Choosing 10 marbles from a box of 40 marbles (20 purple, 12 red, and 8 green)one at a time with replacement, keeping track of the colors of the marbles chosen.

A)Procedure results in a binomial distribution.
B)Not binomial: there are more than two outcomes for each trial.
C)Not binomial: the trials are not independent.
D)Not binomial: there are too many trials.

Question 69

Determine whether the given procedure results in a binomial distribution. If not, state the reason why.

- $$\rho = \frac { 1 } { 6 }$$ n = 6, x = 3,

A)0.0536
B)0.0154
C)0.0322
D)0.0286

Question 70

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Rolling a single "loaded" die 48 times, keeping track of the "fives" rolled.

A)Not binomial: there are more than two outcomes for each trial.
B)Not binomial: there are too many trials.
C)Procedure results in a binomial distribution.
D)Not binomial: the trials are not independent.

Question 71

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Rolling a single "loaded" die 58 times, keeping track of the numbers that are rolled.

A)Procedure results in a binomial distribution.
B)Not binomial: there are too many trials.
C)Not binomial: the trials are not independent.
D)Not binomial: there are more than two outcomes for each trial.

Question 72

Suppose that voting in municipal elections is being studied and that the accompanying tables describes the probability distribution for four randomly selected people, where x is the number that voted in the last election. Is it unusual to find four voters among four randomly selected people? $$\begin{array} { c | c } x & P ( x ) \\\hline 0 & 0.23 \\\hline 1 & 0.32 \\\hline 2 & 0.26 \\\hline 3 & 0.15 \\\hline 4 & 0.04\end{array}$$

A)Yes
B)No

Question 73

Suppose that computer literacy among people ages 40 and older is being studied and that the accompanying tables describes the probability distribution for four randomly selected people, where x is the number that are computer literate. Is it unusual to find four computer literates among four randomly selected people? $$\begin{array} { c | c } x & P ( x ) \\\hline 0 & 0.16 \\\hline 1 & 0.25 \\\hline 2 & 0.36 \\\hline 3 & 0.15 \\\hline 4 & 0.08\end{array}$$

A)Yes
B)No

Question 74

Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
n = 5, x = 2, p = 0.70

A)0.700
B)0.198
C)0.132
D)0.464

Question 75

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Choosing 5 people (without replacement)from a group of 59 people, of which 15 are women, keeping track of the number of men chosen.

A)Not binomial: there are too many trials.
B)Not binomial: there are more than two outcomes for each trial.
C)Procedure results in a binomial distribution.
D)Not binomial: the trials are not independent.

Question 76

Determine whether the given procedure results in a binomial distribution. If not, state the reason why.

- $$n = 4 , x = 3 , p = \frac { 1 } { 6 }$$

A) $0.0154$
B) $0.0039$
C) $0.0116$
D) $0.0231$

Question 77

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Choosing 3 marbles from a box of 40 marbles (20 purple, 12 red, and 8 green)one at a time with replacement, keeping track of the number of red marbles chosen.

A)Not binomial: the trials are not independent.
B)Not binomial: there are more than two outcomes for each trial.
C)Not binomial: there are too many trials.
D)Procedure results in a binomial distribution.

Question 78

Suppose that weight of adolescents is being studied by a health organization and that the accompanying tables describes the probability distribution for three randomly selected adolescents, where x is the number who are considered morbidly obese. Is it unusual to have no obese subjects among three randomly selected adolescents? $$\begin{array} { c | c } x & P ( x ) \\\hline 0 & 0.111 \\\hline 1 & 0.215 \\\hline 2 & 0.450 \\\hline 3 & 0.224\end{array}$$

A)No
B)Yes

Question 79

Determine whether the given procedure results in a binomial distribution. If not, state the reason why.

- $n = 30 , x = 12 , p = 0.20$

A) $0.0028$
B) $0.1082$
C) $0.0064$
D) $0.0139$

Question 80

Determine whether the given procedure results in a binomial distribution. If not, state the reason why

-Choosing 8 marbles from a box of 40 marbles (20 purple, 12 red, and 8 green)one at a time without replacement, keeping track of the number of red marbles chosen.

A)Not binomial: the trials are not independent.
B)Not binomial: there are more than two outcomes for each trial.
C)Procedure results in a binomial distribution.
D)Not binomial: there are too many trials.