Deck 11: Discrete Probability Distributions

Question 1

Which of the following is a condition for the binomial distribution?

A)There are a fixed number of independent observations, n.
B)Each observation may have only two possible outcomes: success or failure.
C)The probability of success p is the same for all possible observations.
D)All of these choices are correct.

Answer 1

D

Question 2

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
Which distribution does X follow?

A)Normal distribution with mean 5 and variance 1/33
B)binomial distribution with n = 5 and p = 1/33
C)binomial distribution with n = 5 and p = 32/33
D)Normal distribution with mean 0 and variance 1

Answer 2

C

Question 3

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
What are the approximate mean and standard deviation of the distribution of X?

A)Mean = 0.1515 and standard deviation = 0.1469
B)Mean = 4.8485 and standard deviation = 0.3833
C)Mean = 0.1515 and standard deviation = 0.3833
D)Mean = 4.8485 and standard deviation = 0.1469

Answer 3

B

Question 4

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
What is the probability that two of the five births do not result in defects?

A)0.0003
B)0.0008
C)0.0084
D)0.3125

Question 5

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
What is the probability that at least one of the births results in a defect?

A)0.0000
B)0.1426
C)0.8574
D)1.0000

Question 6

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
A researcher suggests using 500 births instead of only 5. In this case, which distribution is a reasonable approximation of X?

A)N(500,1/33)
B)N(484.85, 1/33)
C)N(500, 3.83)
D)N(484.85, 3.83)

Question 7

The proportion of U.S. births that result in a birth defect is approximately 1/33 according to the Centers for Disease Control and Prevention (CDC). A local hospital randomly selects five births and lets the random variable X count the number not resulting in a defect. Assume the births are independent.
If 500 births were observed rather than only 5, what is the approximate probability that at least 490 do not result in birth defects?

A)0.0000
B)0.0901
C)0.6371
D)0.9105

Question 8

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy.
Which distribution does X follow?

A)Poisson distribution
B)Binomial distribution
C)Conditional distribution
D)Uniform distribution

Question 9

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy.
What are the approximate values for the mean and standard deviation of the distribution of X?

A)Mean = 0.037 and standard deviation = 14.8
B)Mean = 14.8 and standard deviation = 3.775
C)Mean = 400 and standard deviation = 0.037
D)Mean = 14.25 and standard deviation = 14.2524

Question 10

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy.
What is the probability that none of the 400 employees has a food allergy?

A)Nearly 1
B)0.0370
C)0.0025
D)Nearly 0

Question 11

Under which conditions can the Normal approximation to the binomial distribution be used?

A)n > 30
B)np ≥10 and np(1-p) ≥10
C)np ≥10 or np(1-p) ≥10
D)None of these choices is correct.

Question 12

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy.
What is a reasonable approximation of the distribution of X?

A)N(400, 0.037)
B)N(14.8, 3.775)
C)N(400, 0.0025)
D)N(14.2524, 0.0025)

Question 13

According to the National Institute of Allergy and Infectious Diseases, approximately 3.7% of U.S. adults have a food allergy. A large company plans a lunch reception for its 400 employees. Assume that employees are independent. Let the random variable X be the number of company employees who have a food allergy.
Using a Normal approximation, what is the probability that at least 20 of the 400 employees have a food allergy?

A)0.000
B)0.050
C)0.084
D)0.644

Question 14

The Poisson distribution describes the count X of random independent events over a fixed period of time or space.

Question 15

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York.
Which distribution does X follow?

A)Poisson distribution with mean 29
B)Poisson distribution with mean 14.5
C)Binomial distribution with mean 0.145
D)Binomial distribution with mean 2.9

Question 16

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York.
Rounded to four decimal places, what is the probability that none of the toddlers in the sample is obese?

Question 17

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York.
Rounded to four decimal places, what is the probability P(X ≤ 1)?

Question 18

State-wide surveys indicate that 14.5% of toddlers in New York are obese. Let the random variable X be the number of toddlers who are obese in a random sample of 20 toddlers from the state of New York.
In this context, the probability P(X ≤ 1) is equal to which of these probabilities?

A)P(X = 1)
B)P(X < 1)
C)P(X < 2)
D)P(X ≤ 2)

Question 19

According to the National Institute of Allergy and Infectious Diseases, approximately 7% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 8 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.
Which distribution does X follow?

A)Normal distribution with mean 8 and variance 0.07
B)Binomial distribution with n = 8 and p = 0.07
C)Binomial distribution with n = 8 and p = 0.93
D)Poisson distribution with mean and variance 0.07

Question 20

According to the National Institute of Allergy and Infectious Diseases, approximately 7% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 8 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.
What are the approximate values of the mean and standard deviation of the distribution of X?

A)Mean = 0.07 and standard deviation = 0.0049
B)Mean = 0.56 and standard deviation = 0.3136
C)Mean = 0.07 and standard deviation = 0.2646
D)Mean = 0.56 and standard deviation = 0.7217

Question 21

According to the National Institute of Allergy and Infectious Diseases, approximately 7% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 8 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.
What is the probability that one of the eight children has a food allergy?

A)0.8965
B)0.3370
C)0.1250
D)0.0700

Question 22

According to the National Institute of Allergy and Infectious Diseases, approximately 7% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 8 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.
What is the probability that at least one of the eight children has a food allergy?

A)0.8965
B)0.5596
C)0.4404
D)0.3370

Question 23

According to the National Institute of Allergy and Infectious Diseases, approximately 7% of U.S. children 4 years of age or younger have a food allergy. A day care program has capacity for 8 children in that age range. Assume that the children attending the day care program are independent. Let the random variable X be the number of children in this day care who have a food allergy.
What is the probability P(X < 2)?

A)0.9853
B)0.8965
C)0.3370
D)0.0888

Question 24

For which of the following counts would a binomial probability model be reasonable?

A)The number of emergency room visits in a one-hour period at one hospital
B)The number of individuals with a pacemaker who go through airport security in one day
C)The number of individuals who take a cholesterol-lowering drug in a random sample of 100 medical records
D)All of these choices are correct.

Question 25

People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 7.2% of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X denote the number of universal donors among them.
What is the mean of X?

A)7.2
B)1.44
C)1.34
D)0.072

Question 26

People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 7.2% of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X denote the number of universal donors among them.
What is the standard deviation of X?

A)1.16
B)1.20
C)1.34
D)1.44

Question 27

People with type O-negative blood are universal donors whose blood can safely be given to anyone. Only 7.2% of the population has O-negative blood. A mobile blood center is visited by 20 donors in the afternoon. Let X denote the number of universal donors among them.
What is the probability that X is at least 2?

A)0.224
B)0.257
C)0.427
D)0.743

Question 28

Assuming that both parents are heterozygous for a recessive gene, the probability that a single offspring develops the recessive trait is 1/4. Assuming separate births are independent, the probability that exactly three out of four offspring develop the trait is given by

A) $4 \left( \frac { 1 } { 4 } \right) ^ { 3 } \left( \frac { 3 } { 4 } \right) ^ { 1 }$
B) $\left( \frac { 1 } { 4 } \right) ^ { 3 } \left( \frac { 3 } { 4 } \right) ^ { 1 }$
C) $4 \left( \frac { 1 } { 4 } \right) ^ { 1 } \left( \frac { 3 } { 4 } \right) ^ { 3 }$
D) $\left( \frac { 1 } { 4 } \right) ^ { 1 } \left( \frac { 3 } { 4 } \right) ^ { 3 }$

Question 29

Assuming a parent is heterozygous for a particular gene, the likelihood of passing on the dominant allele is 50%-the same as the likelihood of passing on the recessive allele. Assuming that separate births are independent, what is the probability that exactly five out of the next ten children born receive the recessive allele?

A)0.1667
B)0.2461
C)0.3125
D)0.5000

Question 30

The African tigerfish can jump out of water to catch barn swallows flying low over a lake. A study found that, during summer time, 20 birds were caught per day on average by tigerfish in Schroda Dam Lake. Let X be the number of birds caught in flight by tigerfish in Schroda Dam Lake in a day. Assuming that catches are random and independent, we use a Poisson distribution with a mean of 20 to describe the probability distribution of X.
What is the probability that X is exactly 20?

A)0.050
B)0.089
C)0.500
D)1.000

Question 31

The African tigerfish can jump out of water to catch barn swallows flying low over a lake. A study found that, during summer time, 20 birds were caught per day on average by tigerfish in Schroda Dam Lake. Let X be the number of birds caught in flight by tigerfish in Schroda Dam Lake in a day. Assuming that catches are random and independent, we use a Poisson distribution with a mean of 20 to describe the probability distribution of X.
The probability P(X < 20) = 0.470. What is the probability P(X ≤ 20)?

A)0.470
B)0.500
C)0.530
D)0.559

Question 32

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25.
What is the standard deviation of the distribution of X?

A)0.0625
B)0.2500
C)0.5000
D)It depends on the number of patients sampled.

Question 33

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25.
What is the probability of observing someone with exactly zero distinct species?

A)0.0000
B)0.2212
C)0.6065
D)0.7788

Question 34

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25.
What is the probability of observing exactly two distinct species?

A)0.0000
B)0.0243
C)0.4631
D)0.9978

Question 35

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25.
What is the probability of observing fewer than three distinct species?

A)0.0022
B)0.9978
C)0.9999
D)1.0000

Question 36

Patients with infections are randomly selected and the number of distinct species belonging to the Spirochaete phylum (responsible for illnesses such as Lyme disease and syphilis) are recorded. Let X count the number of distinct species found. It is speculated that X follows a Poisson distribution with a mean of 0.25.
What is the probability of observing a nonzero number of distinct species?

A)0.2212
B)0.2500
C)0.5000
D)1.0000

Question 37

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
Which distribution does X follow?

A)Poisson distribution
B)Binomial distribution
C)Conditional distribution
D)Uniform distribution

Question 38

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
What is the standard deviation of the distribution of X?

A)23.04
B)4.8
C)2.19
D)It depends on the number of emergency arrivals sampled.

Question 39

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
What is the probability of observing exactly zero emergency arrival?

A)Nearly 0
B)0.0082
C)0.1118
D)0.8230

Question 40

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
What is the probability of observing exactly one emergency arrival?

A)Nearly 0
B)0.0082
C)0.0395
D)0.0477

Question 41

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
What is the probability of observing at least one emergency arrival?

A)0.0477
B)0.9523
C)0.9918
D)Nearly 1

Question 42

On weekend nights, a large urban hospital has an average of 4.8 emergency arrivals per hour. Let X be the number of arrivals per hour on a weekend night at this hospital. Assume that successive arrivals are random and independent.
What is the probability P(X < 3)?

A)Nearly 0
B)0.1425
C)0.1517
D)0.2942

Question 43

Which of the following would be appropriately modeled by a Poisson distribution?

A)The number of males born in the next seven randomly selected births
B)The number of lesions on a fish exposed to toxic waste
C)The weights of babies born to mothers who regularly smoked during pregnancy
D)The cholesterol levels of adult males before enrolling in a medical drug trial