Work problems in mod 1 ch 2 exercises. See "NOTES" attachment for instructions. Show all work.

2-14

A student taking Management Science 301 at East Haven University will receive one of the five possible grades fro the course: A, B, C, D, or F. The distribution of the grades over the past two years is as follows:

Grade Number of Students

A 80

B 75

C 90

D 30

F 25

Total 300

If this past distribution is a good indicator of future grades, what is the probability of a student receiving a C in the course?

2-16

An urn contains 8 red chips, 10 green chips, and 2 white chips. A chip is drawn and replaced, and then a second chip is drawn. What is the probability of

(a) a white chip on the first draw?

(b) a white chip on the first draw and a red on the second?

(c) two green chips being drawn?

(d) a red chip on the second, given that a white chip was drawn on the first?

2-19

The Springfield Kings, a professional basketball team, has won 12 of its last 20 games and is expected to continue winning at the same percentage rate. The team's ticket manager is anxious to attract a large crowd to tomorrow's game but believes that depends on how well the Kings perform tonight against the Galveston Comets. He assesses the probability of drawing a large crowd to be 0.90 should the team win tonight. What is the probability that the team wins tonight and there will be a large crowd at tomorrow's game?

2-23

Ace Machine Works estimates that the probability its lathe tool is properly adjusted is 0.8. When the lathe is properly adjusted, there is a 0.9 probability that the parts produced pass inspection. If the lathe is out of adjustment, however, the probability of a good part being produced is only 0.2. A part randomly chosen is inspected and found to be acceptable. At this point, what is the posterior probability that the lathe tool is properly adjusted?

2-26

The Northside Rifle team has two markspersons, Dick and Sally. Dick hits a bull's-eye 90% of the time and Sally hits a bull's-eye 95% of the time.

(a) What is the probability that either Dick or Sally or both will hit the bull's-eye if each takes one shot?

(b) What is the probability that Dick and Sally will hit both the bull's-eye?

(c) Did you make any assumptions in answering the preceding questions? If you answered yes, do you think you are justified in making the assumption(s)?

2-28

Compute the probability of "loaded die, given that a 3 was rolled," as shown in Example 7, this time using the general form of Baye's theorem from Equation 2-7.

2-31

What are the expected value and variance of the following probability distribution?

RANDOM VARIABLE X PROBABILITY

1 0.05

2 0.05

3 0.10

4 0.10

5 0.15

6 0.15

7 0.25

8 0.15

2-34

If 10% of all disk drives produced on an assembly line are defective, what is the probability there will be exactly one defect in a random sample of 5 of these? What is the probability that there will be no defects in a random sample of 5?

2-37

An industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars is able to maintain fairly constant temperatures. The temperature range of the oven follows a normal distribution with a mean of 450 degrees F and a standard deviation of 25 degrees F. Leslie Larsen, president of the factory, is concerned about the large number of defective cores that have been produced in the past several months. If the oven gets hotter than 475 degrees F. the core is defective. What is the probability that the oven will cause a core to be defective? What is the probability that the temperature of the oven will range from 460 degrees to 470 degrees F?

2-39

Susan Williams has been the production manager of Medical Suppliers, Inc., for the past 17 years. Medical Suppliers, Inc., is a producer of bandages and arm slings. During the past 5 years, the demand for No-Stick bandages has been fairly constant. On the average, sales have been about 87,000 packages of No-Stick. Susan has reason to believe that the distribution of No-Stick follows a normal curve, with a standard deviation of 4,000 packages. What is the probability that sales will be less than 81,000 packages?

2-43

Patients arrive at the emergency room of Costa Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

(a) Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.

(b) What is the sum of these probabilities, and why is the number less than 1?

Student 041201

Module 1, Ch 2, Exercises

2-14

A student taking Management Science 301 at East Haven University will

receive one of the five possible grades fro the course: A, B, C, D, or

F. The distribution of the grades over the past two years is as

follows:

Grade Number of Students

A 80

B 75

C 90

D 30

F 25

Total 300

If this past distribution is a good indicator of future grades, what is

the probability of a student receiving a C in the course?

2-16

An urn contains 8 red chips, 10 green chips, and 2 white chips. A chip

is drawn and replaced, and then a second chip is drawn. What is the

probability of

(a) a white chip on the first draw?

(b) a white chip on the first draw and a red on the second?

(c) two green chips being drawn?

(d) a red chip on the second, given that a white chip was drawn on the

first?

2-19

The Springfield Kings, a professional basketball team, has won 12 of its

last 20 games and is expected to continue winning at the same percentage

rate. The teamâs ticket manager is anxious to attract a large crowd

to tomorrowâs game but believes that depends on how well the Kings

perform tonight against the Galveston Comets. He assesses the

probability of drawing a large crowd to be 0.90 should the team win

tonight. What is the probability that the team wins tonight and there

will be a large crowd at tomorrowâs game?

2-23

Ace Machine Works estimates that the probability its lathe tool is

properly adjusted is 0.8. When the lathe is properly adjusted, there is

a 0.9 probability that the parts produced pass inspection. If the lathe

is out of adjustment, however, the probability of a good part being

produced is only 0.2. A part randomly chosen is inspected and found to

be acceptable. At this point, what is the posterior probability that

the lathe tool is properly adjusted?

2-26

The Northside Rifle team has two markspersons, Dick and Sally. Dick

hits a bullâs-eye 90% of the time and Sally hits a bullâs-eye 95% of

the time.

(a) What is the probability that either Dick or Sally or both will hit

the bullâs-eye if each takes one shot?

(b) What is the probability that Dick and Sally will hit both the

bullâs-eye?

(c) Did you make any assumptions in answering the preceding questions?

If you answered yes, do you think you are justified in making the

assumption(s)?

2-28

Compute the probability of âloaded die, given that a 3 was rolled,â

as shown in Example 7, this time using the general form of Bayeâs

theorem from Equation 2-7.

2-31

What are the expected value and variance of the following probability

distribution?

RANDOM VARIABLE X PROBABILITY

0.05

0.05

0.10

0.10

0.15

0.15

0.25

0.15

2-34

If 10% of all disk drives produced on an assembly line are defective,

what is the probability there will be exactly one defect in a random

sample of 5 of these? What is the probability that there will be no

defects in a random sample of 5?

2-37

An industrial oven used to cure sand cores for a factory manufacturing

engine blocks for small cars is able to maintain fairly constant

temperatures. The temperature range of the oven follows a normal

distribution with a mean of 450 degrees F and a standard deviation of 25

degrees F. Leslie Larsen, president of the factory, is concerned about

the large number of defective cores that have been produced in the past

several months. If the oven gets hotter than 475 degrees F. the core is

defective. What is the probability that the oven will cause a core to

be defective? What is the probability that the temperature of the

oven will range from 460 degrees to 470 degrees F?

2-39

Susan Williams has been the production manager of Medical Suppliers,

Inc., for the past 17 years. Medical Suppliers, Inc., is a producer of

bandages and arm slings. During the past 5 years, the demand for

No-Stick bandages has been fairly constant. On the average, sales have

been about 87,000 packages of No-Stick. Susan has reason to believe

that the distribution of No-Stick follows a normal curve, with a

standard deviation of 4,000 packages. What is the probability that

sales will be less than 81,000 packages?

2-43

Patients arrive at the emergency room of Costa Valley Hospital at an

average of 5 per day. The demand for emergency room treatment at Costa

Valley follows a Poisson distribution.

(a) Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4,

and 5 arrivals per day.

(b) What is the sum of these probabilities, and why is the number less

than 1?

Notes:

Â· â|â doesn't mean divide when written as P(A | B). The notation

means the âprobability of A occurring given that B has occurred.â

Â· P(R), P(W) etc. are symbolsânot functions. P(R) is the

âprobability of the event R occurring.â

2-16

Â· b) These are independent events--each drawn chip is replaced before

the next chip is drawn. Therefore, the white chip has nothing to do with

drawing the red chip--the probability is the marginal probability of the

second event. Use equation 2-4 at the bottom of page 27

2-19

Â· There are two parts to this problem:

o P(W) â probability of winning

o P(L|W) â probability of large crowd tomorrow night given that the

team wins tonight.

Youâre given P(L|W): P(L|W) = 0.90

And P(WL) = P(L|W) * P(W)

2-23 Bayes' Theorem (for posterior probabilities) involves conditional

probabilities. (Posterior probabilities are discussed in section 2.6.)

Bayes' Theorem:

Â§ What the problem is asking for is P(adjusted | pass) or P(A | B).

(note positions of A & B.)

Â· Need to pay attention to statements in the problemâsome of the

probabilities are conditional (as needed for posterior probability):

a. Probability that the lathe is properly adjusted is 0.8 P(adjusted) =

0.8 This means that P(not adjusted) = 0.2

b. When the lathe is properly adjusted, there's a 90% probability it

will pass inspection. This is represented: P(pass | adjusted) = 0.90

c. If the lathe is out of adjustment, the probability that is good part

is being produce is only 0.2. P(pass | not adjusted) = 0.2

= "not adjusted.

Â§ Then replace the variables with values from the problem.

2-26

Â· First you must determine whether the events are mutually exclusive or

not and independent or not.

Â· See the "Law of Addition for Events that Are Not Mutually Exclusive"

and 2.4 "Statistically Independent Events" on pages 26-28.

o For the "eitherâ part, see equation 2-3, Figure 2.2.

o For the "both" part, see equation 2-4, Figure 2.2

Â· Note: The answer in the back of text is incorrect. The correct answer

= 0.855.

2-34

Â· Use appropriate table.

2-37

Â· First Part:

o The problem asks for the probability that the oven will get hotter

than 475 degrees.

o The standard normal table gives the probability that the oven will be

less than 475 degrees

o Since the area under the curve = 1 for the standard normal curve, you

must subtract the value obtained from the table from 1.

Â· Second Part:

o See p-48, half-way down the page starting with, âOne final

exampleâ¦.â through the end of the page, including Figure 2.14.

Â· After you obtained Z = â¦. standard deviations, go to the table and

retrieve a valueâthat value is a probability, not Z.

2-43

Â· Poisson Distribution equation:

= 0.0067. That holds for all P(X).

Â· Calculate for P(X=0), P(X=1), P(X=2), P(X=3), P(X=4), and P(X=5).

Notes:

1. 0! = 1 and 1! = 1 by definition. (See bottom of page 39.)

2. Any number to the power of zero = 1

3. Any number to the power of 1 = the starting number itself.

For X= 0, substitute the values into the equation:

Do this for X = 1, ... , 5. Show all work.

Â· b) Hint: The sum of all probabilities is supposed to equal 1, so why

don't these probabilities add up to one? Are there other possibilities

for arrivals?

2-14

A student taking Management Science 301 at East Haven University will receive one of the five possible grades fro the course: A, B, C, D, or F. The distribution of the grades over the past two years is as follows:

Grade Number of Students

A 80

B 75

C 90

D 30

F 25

Total 300

If this past distribution is a good indicator of future grades, what is the probability of a student receiving a C in the course?

2-16

An urn contains 8 red chips, 10 green chips, and 2 white chips. A chip is drawn and replaced, and then a second chip is drawn. What is the probability of

(a) a white chip on the first draw?

(b) a white chip on the first draw and a red on the second?

(c) two green chips being drawn?

(d) a red chip on the second, given that a white chip was drawn on the first?

2-19

The Springfield Kings, a professional basketball team, has won 12 of its last 20 games and is expected to continue winning at the same percentage rate. The team's ticket manager is anxious to attract a large crowd to tomorrow's game but believes that depends on how well the Kings perform tonight against the Galveston Comets. He assesses the probability of drawing a large crowd to be 0.90 should the team win tonight. What is the probability that the team wins tonight and there will be a large crowd at tomorrow's game?

2-23

Ace Machine Works estimates that the probability its lathe tool is properly adjusted is 0.8. When the lathe is properly adjusted, there is a 0.9 probability that the parts produced pass inspection. If the lathe is out of adjustment, however, the probability of a good part being produced is only 0.2. A part randomly chosen is inspected and found to be acceptable. At this point, what is the posterior probability that the lathe tool is properly adjusted?

2-26

The Northside Rifle team has two markspersons, Dick and Sally. Dick hits a bull's-eye 90% of the time and Sally hits a bull's-eye 95% of the time.

(a) What is the probability that either Dick or Sally or both will hit the bull's-eye if each takes one shot?

(b) What is the probability that Dick and Sally will hit both the bull's-eye?

(c) Did you make any assumptions in answering the preceding questions? If you answered yes, do you think you are justified in making the assumption(s)?

2-28

Compute the probability of "loaded die, given that a 3 was rolled," as shown in Example 7, this time using the general form of Baye's theorem from Equation 2-7.

2-31

What are the expected value and variance of the following probability distribution?

RANDOM VARIABLE X PROBABILITY

1 0.05

2 0.05

3 0.10

4 0.10

5 0.15

6 0.15

7 0.25

8 0.15

2-34

If 10% of all disk drives produced on an assembly line are defective, what is the probability there will be exactly one defect in a random sample of 5 of these? What is the probability that there will be no defects in a random sample of 5?

2-37

An industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars is able to maintain fairly constant temperatures. The temperature range of the oven follows a normal distribution with a mean of 450 degrees F and a standard deviation of 25 degrees F. Leslie Larsen, president of the factory, is concerned about the large number of defective cores that have been produced in the past several months. If the oven gets hotter than 475 degrees F. the core is defective. What is the probability that the oven will cause a core to be defective? What is the probability that the temperature of the oven will range from 460 degrees to 470 degrees F?

2-39

Susan Williams has been the production manager of Medical Suppliers, Inc., for the past 17 years. Medical Suppliers, Inc., is a producer of bandages and arm slings. During the past 5 years, the demand for No-Stick bandages has been fairly constant. On the average, sales have been about 87,000 packages of No-Stick. Susan has reason to believe that the distribution of No-Stick follows a normal curve, with a standard deviation of 4,000 packages. What is the probability that sales will be less than 81,000 packages?

2-43

Patients arrive at the emergency room of Costa Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.

(a) Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.

(b) What is the sum of these probabilities, and why is the number less than 1?

Student 041201

Module 1, Ch 2, Exercises

2-14

A student taking Management Science 301 at East Haven University will

receive one of the five possible grades fro the course: A, B, C, D, or

F. The distribution of the grades over the past two years is as

follows:

Grade Number of Students

A 80

B 75

C 90

D 30

F 25

Total 300

If this past distribution is a good indicator of future grades, what is

the probability of a student receiving a C in the course?

2-16

An urn contains 8 red chips, 10 green chips, and 2 white chips. A chip

is drawn and replaced, and then a second chip is drawn. What is the

probability of

(a) a white chip on the first draw?

(b) a white chip on the first draw and a red on the second?

(c) two green chips being drawn?

(d) a red chip on the second, given that a white chip was drawn on the

first?

2-19

The Springfield Kings, a professional basketball team, has won 12 of its

last 20 games and is expected to continue winning at the same percentage

rate. The teamâs ticket manager is anxious to attract a large crowd

to tomorrowâs game but believes that depends on how well the Kings

perform tonight against the Galveston Comets. He assesses the

probability of drawing a large crowd to be 0.90 should the team win

tonight. What is the probability that the team wins tonight and there

will be a large crowd at tomorrowâs game?

2-23

Ace Machine Works estimates that the probability its lathe tool is

properly adjusted is 0.8. When the lathe is properly adjusted, there is

a 0.9 probability that the parts produced pass inspection. If the lathe

is out of adjustment, however, the probability of a good part being

produced is only 0.2. A part randomly chosen is inspected and found to

be acceptable. At this point, what is the posterior probability that

the lathe tool is properly adjusted?

2-26

The Northside Rifle team has two markspersons, Dick and Sally. Dick

hits a bullâs-eye 90% of the time and Sally hits a bullâs-eye 95% of

the time.

(a) What is the probability that either Dick or Sally or both will hit

the bullâs-eye if each takes one shot?

(b) What is the probability that Dick and Sally will hit both the

bullâs-eye?

(c) Did you make any assumptions in answering the preceding questions?

If you answered yes, do you think you are justified in making the

assumption(s)?

2-28

Compute the probability of âloaded die, given that a 3 was rolled,â

as shown in Example 7, this time using the general form of Bayeâs

theorem from Equation 2-7.

2-31

What are the expected value and variance of the following probability

distribution?

RANDOM VARIABLE X PROBABILITY

0.05

0.05

0.10

0.10

0.15

0.15

0.25

0.15

2-34

If 10% of all disk drives produced on an assembly line are defective,

what is the probability there will be exactly one defect in a random

sample of 5 of these? What is the probability that there will be no

defects in a random sample of 5?

2-37

An industrial oven used to cure sand cores for a factory manufacturing

engine blocks for small cars is able to maintain fairly constant

temperatures. The temperature range of the oven follows a normal

distribution with a mean of 450 degrees F and a standard deviation of 25

degrees F. Leslie Larsen, president of the factory, is concerned about

the large number of defective cores that have been produced in the past

several months. If the oven gets hotter than 475 degrees F. the core is

defective. What is the probability that the oven will cause a core to

be defective? What is the probability that the temperature of the

oven will range from 460 degrees to 470 degrees F?

2-39

Susan Williams has been the production manager of Medical Suppliers,

Inc., for the past 17 years. Medical Suppliers, Inc., is a producer of

bandages and arm slings. During the past 5 years, the demand for

No-Stick bandages has been fairly constant. On the average, sales have

been about 87,000 packages of No-Stick. Susan has reason to believe

that the distribution of No-Stick follows a normal curve, with a

standard deviation of 4,000 packages. What is the probability that

sales will be less than 81,000 packages?

2-43

Patients arrive at the emergency room of Costa Valley Hospital at an

average of 5 per day. The demand for emergency room treatment at Costa

Valley follows a Poisson distribution.

(a) Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4,

and 5 arrivals per day.

(b) What is the sum of these probabilities, and why is the number less

than 1?

Notes:

Â· â|â doesn't mean divide when written as P(A | B). The notation

means the âprobability of A occurring given that B has occurred.â

Â· P(R), P(W) etc. are symbolsânot functions. P(R) is the

âprobability of the event R occurring.â

2-16

Â· b) These are independent events--each drawn chip is replaced before

the next chip is drawn. Therefore, the white chip has nothing to do with

drawing the red chip--the probability is the marginal probability of the

second event. Use equation 2-4 at the bottom of page 27

2-19

Â· There are two parts to this problem:

o P(W) â probability of winning

o P(L|W) â probability of large crowd tomorrow night given that the

team wins tonight.

Youâre given P(L|W): P(L|W) = 0.90

And P(WL) = P(L|W) * P(W)

2-23 Bayes' Theorem (for posterior probabilities) involves conditional

probabilities. (Posterior probabilities are discussed in section 2.6.)

Bayes' Theorem:

Â§ What the problem is asking for is P(adjusted | pass) or P(A | B).

(note positions of A & B.)

Â· Need to pay attention to statements in the problemâsome of the

probabilities are conditional (as needed for posterior probability):

a. Probability that the lathe is properly adjusted is 0.8 P(adjusted) =

0.8 This means that P(not adjusted) = 0.2

b. When the lathe is properly adjusted, there's a 90% probability it

will pass inspection. This is represented: P(pass | adjusted) = 0.90

c. If the lathe is out of adjustment, the probability that is good part

is being produce is only 0.2. P(pass | not adjusted) = 0.2

= "not adjusted.

Â§ Then replace the variables with values from the problem.

2-26

Â· First you must determine whether the events are mutually exclusive or

not and independent or not.

Â· See the "Law of Addition for Events that Are Not Mutually Exclusive"

and 2.4 "Statistically Independent Events" on pages 26-28.

o For the "eitherâ part, see equation 2-3, Figure 2.2.

o For the "both" part, see equation 2-4, Figure 2.2

Â· Note: The answer in the back of text is incorrect. The correct answer

= 0.855.

2-34

Â· Use appropriate table.

2-37

Â· First Part:

o The problem asks for the probability that the oven will get hotter

than 475 degrees.

o The standard normal table gives the probability that the oven will be

less than 475 degrees

o Since the area under the curve = 1 for the standard normal curve, you

must subtract the value obtained from the table from 1.

Â· Second Part:

o See p-48, half-way down the page starting with, âOne final

exampleâ¦.â through the end of the page, including Figure 2.14.

Â· After you obtained Z = â¦. standard deviations, go to the table and

retrieve a valueâthat value is a probability, not Z.

2-43

Â· Poisson Distribution equation:

= 0.0067. That holds for all P(X).

Â· Calculate for P(X=0), P(X=1), P(X=2), P(X=3), P(X=4), and P(X=5).

Notes:

1. 0! = 1 and 1! = 1 by definition. (See bottom of page 39.)

2. Any number to the power of zero = 1

3. Any number to the power of 1 = the starting number itself.

For X= 0, substitute the values into the equation:

Do this for X = 1, ... , 5. Show all work.

Â· b) Hint: The sum of all probabilities is supposed to equal 1, so why

don't these probabilities add up to one? Are there other possibilities

for arrivals?

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