ST 560 Module 5

Find the probability that the entire system works properly.The primary air exchange system on a proposed spacecraft has four separate components (call them A, B, C, and D) that all must work properly for the system to operate well. Assume that the probability of any one component working is independent of the other components. It has been shown that the probabilities of each component working are P(A) = 0.95, P(B) = 0.90, P(C) = 0.99, and P(D) = 0.90. a.0.7618 b.0.6561 c.0.8145 d.0.2382

a.0.7618 P(all) = 0.95 * 0.90 * 0.99 * 0.90 =0.7618

The following represents the probability distribution for the daily demand of computers at a local store. Demand Probability 0 0.1 1 0.2 2 0.3 3 0.2 4 0.2 What is the expected daily demand? a.2.2 b.2.0 c.4.0 d.1.0

a.2.2 µ = (0*0.1) + (1*0.2) + (2*0.3) + (3*0.2) + (4*0.2) = 2.2

The United States Senate has 100 members. Suppose there are 54 Republicans and 46 Democrats. A committee of 15 Senators is selected at random. What is the expected number of Republicans on this committee? a.E(X) = 8.1 b.E(X) = 7.1 c.E(X) = 9.0 d.E(X) = 6.7

a.E(X) = 8.1 15 * 0.54 = 8.1

The random variable x is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two-time periods of an equal length. It is known that the mean number of occurrences in ten minutes is 5.3. a.Poisson b.binomial c.hypergeometric d.normal

a.Poisson

In the textile industry, a manufacturer is interested in the number of blemishes or flaws occurring in each 100 feet of material. The probability distribution that models this situation is the a.Poisson distribution. b.binomial distribution. c.uniform distribution. d.normal distribution.

a.Poisson distribution.

Which of the following is not a characteristic of an experiment where the binomial probability distribution is applicable? a.The trials are dependent upon one another. b.There are exactly two outcomes are possible on each trial. c.The probabilities of the outcomes do not change from one trial to another. d.The experiment has a sequence of n identical trials.

a.The trials are dependent upon one another.

Which of the following statements about discrete distributions is not true? a.When calculating probabilities for discrete distributions, P(X<1) and the P(X≤1) will result in the same value. b.Discrete distributions must be finite. c.The sum of the probabilities in a discrete distribution must equal one. d.In discrete distributions, the random variable can begin with X = 0.

a.When calculating probabilities for discrete distributions, P(X<1) and the P(X≤1) will result in the same value.

The random variable x is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two-time periods of an equal length. It is known that the mean number of occurrences in ten minutes is 5.3. The appropriate probability distribution for the random variable is a.discrete. b.continuous. c.either discrete or continuous depending on how the interval is defined. d.None of these alternatives is correct.

a.discrete.

A measure of the average value of a random variable is called a(n) a.expected value. b.standard deviation. c.coefficient of variation. d.variance.

a.expected value.

To compute the probability that in a random sample of n elements, selected without replacement, we will obtain x successes, we would use the a.hypergeometric probability distribution. b.Poisson probability distribution. c.exponential probability distribution. d.binomial probability distribution.

a.hypergeometric probability distribution.

The expected value for a binomial distribution is a.np. b.np(1-np). c.np(1-p). d.np(n-1).

a.np.

Four percent of the customers of a mortgage company default on their payments. A sample of five customers is selected. What is the probability that exactly two customers in the sample will default on their payments? a.0.9588 b.0.0142 c.0.7408 d.0.2592

b.0.0142 Binomial probability fxn f(x) = ( n x ) pˣ (1-p)ⁿ⁻ˣ f(2) = ( 5 2 ) (0.04)² * (0.96)³ f(2) = 5!/2!3! (0.0016) * (0.884736) f(2) = 0.021233664 ???

The accident rate in a factory is 4 per month. What is the probability that there will be 6 accidents in a particular month? a.0.1105 b.0.1042 c.0.1558 d.0.0995

b.0.1042 Poisson Probability Distribution f(x) = ( µˣ e^(⁻µ) ) / x! f(6) = 4⁶ * e⁻⁴ / 6! f(6) = (4096*0.01831563889)/(6*5*4*3*2*1) =0.1042

Roth is a computer-consulting firm. The number of new clients that they have obtained each month has ranged from 0 to 6. The number of new clients has the probability distribution that is shown below. # New Clients Probability 0 0.05 1 0.10 2 0.15 3 0.35 4 0.20 5 0.10 6 0.05 What is the variance of the number of new clients? a.3.05 b.2.05 c.21 d.1.431

b.2.05 µ = 3.05 (calculated) 𝜎² = Σ(x-µ)² f(x) x x-µ (x-µ)² f(x) (x-µ)² f(x) 0 -3.05 9.3025 0.05 0.465125 1 -2.05 4.2025 0.10 0.42025 2 -1.05 1.1025 0.15 0.165375 3 -0.05 0.0025 0.35 0.000875 4 0.95 0.9025 0.20 0.1805 5 1.95 3.8025 0.10 0.38025 6 2.95 8.7025 0.05 0.435125 Σ(x-µ)² f(x) = 𝜎² = 2.0475

There are 13 Democrats, 12 Republicans and 8 Independents sitting in a room. Eight of these people will be selected to serve on a special committee. What is the probability that exactly five of the committee members will be Democrats? a.P(X)= 0.10783 b.P(X)= 0.10567 c.P(X)= 0.11000 d.P(X)= 0.09957

b.P(X)= 0.10567 Hyperprobability Distribution Fxn f(x) = ( r x ) ( N - r n - x ) / ( N n ) x = # successes r = # elements labeled success in pop. n = # trials N = # elements in pop. 0.10567297 (work on paper)

A medicine is known to produce side effects in 1 in 5 patients taking it. Suppose a doctor prescribes the medicine to 4 unrelated patients. What is the probability that none of the patients will develop side effects? a.P(X)=0.25 b.P(X)=0.4096 c.P(X)=0.2 d.P(X)=0.8

b.P(X)=0.4096 Binomial probability fxn f(x) = ( n x ) pˣ (1-p)ⁿ⁻ˣ f(0) = ( 4 0 ) 0.2⁰ * 0.8⁴ f(0) = 4!/0!4! * 1 * 0.4096 f(0) = 1 * 1 * 0.4096 = 0.4096

Which of the following is a required condition for a discrete probability function? a. f(x) < 0 for all values of x. b. f(x) = 1 for all values of x. c. Σf(x) = 1 for all values of x. d. Σf(x) = 0 for all values of x.

c. Σf(x) = 1 for all values of x.

An insurance company charges $800 annually for car insurance. The policy specifies that the company will pay $1000 for a minor accident and $5000 for a major accident. If the probability of a motorist having a minor accident during the year is 0.2, and of having a major accident, 0.05, how much can the insurance company expect to make on the policy? (Expected profit) a.$300 b.$250 c.$350 d.$200

c.$350 The expected payout for an accident is 0.2(1000) + 0.05(5000) = $450 The company charges $800. $800 - $450 = $350

The primary air exchange system on a proposed spacecraft has four separate components (call them A, B, C, and D) that all must work properly for the system to operate well. Assume that the probability of any one component working is independent of the other components. It has been shown that the probabilities of each component working are P(A) = 0.95, P(B) = 0.90, P(C) = 0.99, and P(D) = 0.90. What is the probability that at least one of the four components will work properly? a.0.761805 b.0.238195 c.0.999995 d.0.000005

c.0.999995 P(at least one) = 1 - P(none) P(none) = (0.005) * (0.1) * (0.01) * (0.1) = 0.000005 P(at least one) = 1 - 0.000005 = 0.999995

Random variable x has the probability function:​ f(x) = x/6 for x = 1, 2 or 3 What is the expected value of a.2.00 b.0.50 c.2.33 d.0.33

c.2.33

Which of the following are true statements? I. By the law of large numbers, the mean of a random variable will get closer and closer to a specific value. II. The SD of a random variable is never negative. III. The SD of a random variable is 0 only if the random variable takes a lone single value. a.II and III b.I and III c.I, II, and III d.I and II

c.I, II, and III

A department store has determined in connection with its inventory control that the demand for a certain CD player averages 4 per day. If the store stocks 5 of these items on a particular day, what is the probability that demand will exceed supply? a.P(X)=0.2034 b.P(X)=0.1904 c.P(X)=0.2150 d.P(X)=0.1885

c.P(X)=0.2150 Poisson Probability Distribution f(x) = ( µˣ e^(⁻µ) ) / x!

The variance Var(x) for the binomial distribution is a.n(1-p). b.np(1-np). c.np(1-p). d.np(n-1).

c.np(1-p)

Which one of these variables is a continuous random variable? a.the number of women taller than 68 inches in a random sample of 5 women b.the number of tattoos a randomly selected person has c.the time is takes a randomly selected student to complete an exam d.the number of correct guesses on this part of the multiple choice test

c.the time is takes a randomly selected student to complete an exam

The average annual incomes of high school and college graduates in a Midwestern town are $21,000 and $35,000, respectively. If a company hires only personnel with at least a high school diploma and 20% of its employees have been through college, what is the mean income of the company employees? a.$28,000 b.$27,110 c.$32,200 d.$23,800

d.$23,800 µ = (21,000*0.8) + (35,000*0.2) = 23,800

The telephone sales department of a certain store receives an average of 24 calls per hour. What is the probability that between 10:00 a.m. and 10:05 a.m. there will be 3 calls? a.0.1874 b.0.1853 c.0.1902 d.0.1804

d.0.1804 E(x) = 24 calls / 60 min = 0.4 / minute E(5) = 2 calls Poisson Probability Distribution f(x) = ( µˣ e^(⁻µ) ) / x! f(3) = ( 2³ * e⁻² ) / 3! f(3) = (8 * 0.1353352832) / (3*2*1) f(3) = 0.1804

Find the probability that a family of five children will have exactly three boys. a.0.8125 b.0.6875 c.0.1875 d.0.3125

d.0.3125 Binomial probability distribution f(x) = ( n x ) pˣ (1-p)ⁿ⁻ˣ f(3) = ( 5 3 ) (0.5)³ (0.5)² f(3) = 5!/3!2! (0.03125) f(3) = 5*4/2 (0.03125) = 10 * 0.03125 = 0.3125

A continuous random variable may assume a.only fractional values in an interval or collection of intervals. b.only integer values in an interval or collection of intervals. c.only the positive integer values in an interval. d.any value in an interval or collection of intervals.

d.any value in an interval or collection of intervals.

If the probability of a basketball player scoring on any shot is 0.75, what is the probability that she will score on at most 5 of her next 6 shots? a.P(X)=0.1780 b.P(X)=0.8220 c.P(X)=0.4661 d.P(X)=0.3560

b.P(X)=0.8220 Binomial probability fxn f(x) = ( n x ) pˣ (1-p)ⁿ⁻ˣ f(5) = ( 6 5 ) (0.75)⁵ (0.25)¹ f(5) = 6!/5!1! (0.05932617188) f(5) = 6 * 0.05932617188 = 0.3559570313 f(4) = (6 4 ) (0.75)⁴ (0.25)² f(4) = (6!/4!2!) (0.31640625)(0.0625) f(4) = 15 * (0.01977539062) = 0.2966308593 f(3) = ( 6 3 ) (0.75)³) (0.25)³ f(3) = 6!/3!3! (0.421875) * (0.015625) f(3) = (6*5*4)/(3*2) * 0.006591796875 f(3) = 20 * 0.006591796875 = 0.1318359375 f(2) = ( 6 2 ) (0.75)² (0.25)⁴ f(2) = 6!/2!4! (0.5625) *(0.00390625) f(2) = 15 *0.002197265625 = 0.03295898438 f(1) = ( 6 1 ) (0.75)¹ * (0.25)⁵ f(1) = 6!/1!5! (0.75) * (0.0009765625) f(1) = 6* 0.000732421875 = 0.00439453125 P(5 at most) = f(5) + f(4) + f(3) + f(2) + f(1) = 0.3559570313 + 0.2966308593 + 0.1318359375 + 0.03295898438 + 0.00439453125 = 0.8218

When dealing with the number of occurrences of an event over a specified interval of time or space, the appropriate probability distribution is a a.normal distribution. b.Poisson distribution. c.hypergeometric probability distribution. d.binomial distribution.

b.Poisson distribution.

Which of the following is not true of discrete probability distributions? a.The value of the standard deviation can be less than, equal to, or greater than the value of the mean. b.The graph of the distribution always exhibits symmetry. c.Each of the probability in the distribution must be greater than or equal to 0. d.The sum of the probabilities is 1.

b.The graph of the distribution always exhibits symmetry.

Which of the following is a characteristic of an experiment where the binomial probability distribution is applicable? a.The trials are dependent upon one another. b.There are exactly two outcomes are possible on each trial. c.The probabilities of the outcomes changes from one trial. d.The experiment has at least two possible outcomes.

b.There are exactly two outcomes are possible on each trial.

The binomial probability distribution is used with a.a uniform random variable. b.a discrete random variable. c.an intermittent random variable. d.a continuous random variable.

b.a discrete random variable.

An experiment consists of determining the speed of automobiles on a highway by the use of radar equipment. The random variable in this experiment is a a.complex random variable. b.continuous random variable. c.categorical random variable. d.discrete random variable.

b.continuous random variable.

The Poisson probability distribution is a a.uniform probability distribution. b.discrete probability distribution. c.normal probability distribution. d.continuous probability distribution.

b.discrete probability distribution.

The key difference between the binomial and hypergeometric distribution is that with the hypergeometric distribution a.the trials are independent of each other. b.the probability of success changes from trial to trial. c.the random variable is continuous. d.the probability of success must be less than 0.5.

b.the probability of success changes from trial to trial.