Deck 8: Probability Distributions

Question 1

One application of the binomial distribution formula or one number in Table V (in your textbook) is always the answer to what kind of question?

A) exactly
B) at most
C) at least
D) between

Answer 1

exactly

Question 2

To calculate the probability of obtaining three aces in eight draws of a card with replacement from an ordinary deck, we would use the

A) hypergeometric distribution.
B) Poisson distribution.
C) binomial distribution.
D) multinomial distribution.

Answer 2

binomial distribution.

Question 3

To calculate the probability of obtaining three aces in eight draws of a card without replacement, the required distribution is the

A) hypergeometric distribution.
B) Poisson distribution.
C) binomial distribution.
D) multinomial distribution.

Answer 3

hypergeometric distribution.

Question 4

If a binomial distribution problem has $\mathrm{p}=0.003$ and $\mathrm{n}=1000$ , then the answer should be approximated by using

A) Table V (in your textbook).
B) the Poisson distribution.
C) the hypergeometric distribution.
D) the multinomial distribution.

Question 5

One application of the Poisson distribution formula is always the answer to what kind of question?

A) exactly
B) at most
C) at least
D) between

Question 6

A die is rolled 10 times. The probability of obtaining from 4 to 7 threes should be determined using the formula for the

A) hypergeometric distribution.
B) Poisson distribution.
C) binomial distribution.
D) multinomial distribution.

Question 7

A service station receives an average of six customers per hour. The probability that there will be three arrivals in a given hour can be found by using the

A) binomial distribution.
B) Poisson distribution.
C) hypergeometric distribution.
D) multinomial distribution.

Question 8

A multiple choice test consists of seven questions with five choices each. If a student guesses on all questions the probability that a student gets exactly 3 correct answers is given by

A) $\left(\begin{array}{l}7 \\ 3\end{array}\right)\left(\frac{3}{5}\right)^{3}\left(\frac{2}{5}\right)^{4}$
B) $\left(\begin{array}{l}7 \\ 5\end{array}\right)\left(\frac{1}{3}\right)^{5}\left(\frac{2}{3}\right)^{2}$
C) $\left(\begin{array}{l}7 \\ 5\end{array}\right)\left(\frac{1}{5}\right)^{3}\left(\frac{4}{5}\right)^{4}$
D) $\left(\begin{array}{l}7 \\ 3\end{array}\right)\left(\frac{1}{5}\right)^{3}\left(\frac{4}{5}\right)^{4}$

Question 9

We want to determine the probability of obtaining at most 4 successful operations in 10 independent surgical operations where the probability of success is the same for each operation. The appropriate formula to be used is

A) the binomial formula.
B) the mean of the binomial distribution.
C) the hypergeometric formula.
D) the mean of the hypergeometric distribution.

Question 10

Which of the following formulas, when applicable, gives exact answers to probability questions involving the binomial distribution?

A) the binomial formula
B) Table V
C) the hypergeometric formula
D) the Poisson formula

Question 11

Which of the following can be probability distributions? Justify your answer.
a. $f(1)=0.40, f(2)=0.20, f(3)=0.50$ where the random variable can take on only the values 1,2 , and 3 .
b. $f(x)=x / 10$ for $x=0,1,2,3,4$ .
c. $f(1)=0.25, f(2)=0.30, f(3)=0.15, f(4)=0.15$ where the random variable can take on only the values $1,2,3$ and 4 .
d. $f(x)=\frac{x-3}{7}$ for $x=0,1,2,3,4,5$

Question 12

Experience has shown that $20 \%$ of a certain airline's flights leaving Green Field airport have to be delayed due to weather conditions. Determine the probability that among twenty flights at Green Field airport at most three will have to be delayed due to weather conditions. Round to the nearest thousandth.

Question 13

A study shows that $90 \%$ of the tomatoes shipped out by a large tomato grower are ripe and ready to eat. Find the probability that among 50 tomatoes shipped out by this grower all are ripe and ready to eat. Round to the nearest thousandth.

Question 14

Check whether the condition for the binomial approximation to the hypergeometric distribution is satisfied if a $=150, \mathrm{~b}=270$ , and $\mathrm{n}=21$ .

Question 15

Check whether the condition for the binomial approximation to the hypergeometric distribution is satisfied if a $=230, \mathrm{~b}=290$ , and $\mathrm{n}=27$ .

Question 16

Among 18 job applicants, six are actually qualified to do the job. Assume seven applicants are randomly selected to be hired.}

-Find the probability that in this sample from two to four applicants are qualified.

Question 17

Among 18 job applicants, six are actually qualified to do the job. Assume seven applicants are randomly selected to be hired.}

-Find the probability that in this sample exactly 3 applicants are qualified.

Question 18

Among 18 job applicants, six are actually qualified to do the job. Assume seven applicants are randomly selected to be hired.}

-Find the mean in the situation above.

Question 19

Among 18 job applicants, six are actually qualified to do the job. Assume seven applicants are randomly selected to be hired.}

-Find the variance in the situation above.

Question 20

Among 18 job applicants, six are actually qualified to do the job. Assume seven applicants are randomly selected to be hired.}

-Find the standard deviation in the situation above.

Question 21

A company produces stereo components. The probability of the company producing a defective component is 0.003 .
-In the situation above, if 1000 components are produced, find the probability that exactly 5 components are defective.

Question 22

A company produces stereo components. The probability of the company producing a defective component is 0.003 .
-In the situation above, if 1000 components are produced, find the probability that at least 3 components are defective.

Question 23

A company produces stereo components. The probability of the company producing a defective component is 0.003 .
-In the situation above, if 1000 components are produced, find the probability that at most 4 components are defective.

Question 24

A company produces stereo components. The probability of the company producing a defective component is 0.003 .
-In the situation above, if 1000 components are produced, find the probability that from 4 to 6 components are defective.

Question 25

A test has 12 multiple-choice questions with four answer choices for each question. Assume a student guesses on all questions.}

-In the situation above, find the probability that she gets exactly five correct answers.

Question 26

A test has 12 multiple-choice questions with four answer choices for each question. Assume a student guesses on all questions.}

-In the situation above, find the probability that she gets at least three correct answers.

Question 27

A test has 12 multiple-choice questions with four answer choices for each question. Assume a student guesses on all questions.}
-In the situation above, find the mean.

Question 28

A test has 12 multiple-choice questions with four answer choices for each question. Assume a student guesses on all questions.}
-In the situation above, find the variance.

Question 29

A test has 12 multiple-choice questions with four answer choices for each question. Assume a student guesses on all questions.}
-In the situation above, find the standard deviation.

Question 30

It has been found that $40 \%$ of the employees who complete a sequence of executive seminars go on to become vice presidents. Assume that 10 graduates of the program are randomly selected.
-In the situation above, find the probability that exactly 5 become vice presidents.

Question 31

It has been found that $40 \%$ of the employees who complete a sequence of executive seminars go on to become vice presidents. Assume that 10 graduates of the program are randomly selected.
-In the situation above, find the probability that no one becomes a vice president.

Question 32

It has been found that $40 \%$ of the employees who complete a sequence of executive seminars go on to become vice presidents. Assume that 10 graduates of the program are randomly selected.
-In the situation above, find the probability that at least three become vice presidents.

Question 33

It has been found that $40 \%$ of the employees who complete a sequence of executive seminars go on to become vice presidents. Assume that 10 graduates of the program are randomly selected.
-In the situation above, find the probability that at least two become vice presidents.

Question 34

It has been found that $40 \%$ of the employees who complete a sequence of executive seminars go on to become vice presidents. Assume that 10 graduates of the program are randomly selected.
-In the situation above, find the probability that from two to four become vice presidents.

Question 35

The number of monthly breakdowns of a conveyor belt at a local factory is a random variable having the Poisson distribution with $\lambda=2.8$ .
-Find the probability that the conveyor belt will function for a month without a breakdown. Round to the nearest thousandth.

Question 36

The number of monthly breakdowns of a conveyor belt at a local factory is a random variable having the Poisson distribution with $\lambda=2.8$ .
-Find the probability that the conveyor belt will function for a month with one breakdown. Round to the nearest thousandth.

Question 37

The number of monthly breakdowns of a conveyor belt at a local factory is a random variable having the Poisson distribution with $\lambda=2.8$ .
-Find the probability that the conveyor belt will function for a month with two breakdowns. Round to the nearest thousandth.

Question 38

Three companies share the entire market for a particular product. Company A has a $50 \%$ share, company B has a $\mathbf{3 0} \%$ } share, and company $\mathrm{C}$ has a $20 \%$ .
-If 12 people who buy the product are selected at random, find the probability that five buy the product from company A, four from company B, and three from company C.

Question 39

According to the Mendelian theory of heredity, if plants with long violet petals are crossbred with plants with short white petals, the probabilities of getting a plant that has long violet petals, short violet petals, long white petals, or short white petals are, respectively, $\frac{9}{16}, \frac{3}{16}, \frac{3}{16}$ , and $\frac{1}{16}$ . What is the probability that among ten plants thus obtained there will be four with long violet petals, three with short violet petals, two with long white petals, and one with short white petals? If necessary, round to the nearest thousandth.

Question 40

The probabilities that a customer entering a particular bookstore buys 0, 1, 2, 3, 4, or 5 books are 0.30, 0.20, 0.20, 0.15, 0.10, and 0.05 respectively.
-For the probability distribution above, find the mean.

Question 41

The probabilities that a customer entering a particular bookstore buys 0, 1, 2, 3, 4, or 5 books are 0.30, 0.20, 0.20, 0.15, 0.10, and 0.05 respectively.
-For the probability distribution above, find the variance.

Question 42

The probabilities that a customer entering a particular bookstore buys 0, 1, 2, 3, 4, or 5 books are 0.30, 0.20, 0.20, 0.15, 0.10, and 0.05 respectively.
-For the probability distribution above, find the standard deviation.

Question 43

The probabilities of getting $0,1,2$ , or 3 heads in three flips of a balanced coin are, respectively, $\frac{1}{8}, \frac{3}{8}, \frac{3}{8}$ , and $\frac{1}{8}$ .
Use the formulas that define $\mu$ and $\sigma^{2}$ to find the mean and the standard deviation of this probability distribution. If necessary, round to the nearest thousandth.

Question 44

Find the mean and standard deviation of the number of tails obtained in 576 flips of a balanced coin using the fact that $\mathrm{x}$ is a binomial random variable. If necessary, round to the nearest thousandth.

Question 45

Find the mean and standard deviation of the number of defectives in a sample of 700 parts made by a machine, when the probability is 0.05 that any one of the parts is defective. Use the fact that $\mathrm{x}$ is a binomial random variable. If necessary, round to the nearest thousandth.

Question 46

The daily number of people who examine cars in a particular automobile showroom is a random variable with $\mu=80$ and $\sigma=6$ .
-According to Chebyshev's theorem, with what probability can we claim that between 68 and 92 people will look at cars in the showroom on a particular day?

Question 47

The daily number of people who examine cars in a particular automobile showroom is a random variable with $\mu=80$ and $\sigma=6$ .
-Give the interval such that there is at least $35 / 36$ probability that the number of people who look at cars will be in that interval.

Question 48

The editor of a particular women's magazine claims that the magazine is read by $60 \%$ of the female students on a college campus.

-If the above claim is true, find the probability that in a random sample of 10 female students more than two read the magazine.

Question 49

The editor of a particular women's magazine claims that the magazine is read by $60 \%$ of the female students on a college campus.

-If the above claim is true, find the probability that in a random sample of 10 female students exactly six read the magazine.

Question 50

The editor of a particular women's magazine claims that the magazine is read by $60 \%$ of the female students on a college campus.

-If the above is true, find the probability that in a random sample of 10 female students at most two read the magazine.

Question 51

If $2 \%$ of the checks received in an all night drugstore bounce, find the probability that in the next 100 checks received by the store, the following will occur.
-At least three bounce.

Question 52

If $2 \%$ of the checks received in an all night drugstore bounce, find the probability that in the next 100 checks received by the store, the following will occur.
-At most two bounce.

Question 53

If $2 \%$ of the checks received in an all night drugstore bounce, find the probability that in the next 100 checks received by the store, the following will occur.
-Exactly 96 do not bounce.

Question 54

Among 10 videotapes that a retail store has available, four are comedies. If a customer selects three videotapes at random to buy for a friend, find the probability that: the following will occur.
-None of them are comedies.

Question 55

Among 10 videotapes that a retail store has available, four are comedies. If a customer selects three videotapes at random to buy for a friend, find the probability that: the following will occur.
-Two of them are comedies.

Question 56

Tell whether the given function can serve as the probability distribution of an appropriate random variable.

-f(x)= $\frac{1}{5}$ for $x=0,1,2,3,4$

Question 57

Tell whether the given function can serve as the probability distribution of an appropriate random variable.

-f(1)=0.25, f(2)=0.40, f(3)=0.30

Question 58

The Poisson distribution can be applied in situations which do not involve the binomial distribution.

Question 59

The hypergeometric distribution with $n=150, a=20$ , and $b=2180$ can be approximated with a Poisson distribution.

Question 60

The mean of a probability distribution is always a value that can be obtained by the random variable.

Question 61

If we want to calculate the probability of from four to six successes in a hypergeometric distribution problem, then we must use the formula three times.

Question 62

The use of the Poisson distribution requires a value $n$ which indicates a definite number of independent trials.

Question 63

In a problem involving the hypergeometric distribution, the probability of success remains the same from trial to trial.

Question 64

In a problem involving the hypergeometric distribution, the binomial distribution may sometimes be used as an approximation.

Question 65

If a distribution is known to be binomial, the most convenient formula that can be used to calculate the mean is $\mu=\sum \mathrm{xp}(\mathrm{x})$ .

Question 66

If a distribution is known to be binomial, the most convenient formula that can be used to calculate the variance is $\sigma^{2}=\sum^{2} p(x)-\mu^{2}$ .

Question 67

To determine the mean of a binomial distribution, it is necessary to know the number of successes involved in the problem.

Question 68

The standard deviation is a measure of the theoretical average value that a distribution is likely to assume for a large number of trials.

Question 69

The binomial distribution Table $\mathrm{V}$ can be used to solve problems involving the binomial distribution if $\mathrm{n}$ has the values _______ and $p$ has the values _______.

Question 70

The sum of all the values of a probability distribution must be equal to _______.

Question 71

Problems that involve the binomial distribution always assume that (1) _______, (2) _______, and (3) _______ .

Question 72

The mean of a Poisson distribution is given by _______.

Question 73

The generalization of the binomial distribution when there are _______ outcomes is called the multinomial distribution.

Question 74

The symbol $p$ in the binomial distribution formula means the probability of _______.

Question 75

According to Chebyshev's theorem, the probability that a random variable will take on a value within two standard deviations of the mean is _______ $3 / 4$ .

Question 76

If we apply Chebyshev's theorem to the 60 to 80 interval of distribution, with $\sigma=5$ , the value of $\mathrm{k}$ will be _______.

Question 77

In the hypergeometric distribution formula, the total number of trials is given by _______.

Question 78

In order to apply Chebyshev's theorem to determine the minimum fraction of data contained in a given interval, we need to know the _______ and the _______ of the distribution.