Applying the Concepts 6–4 Mountain Climbing Safety Assume one of your favorite activities is mountain climbing. When you go mountain climbing, you have several safety devices to keep you from falling. You notice that attached to one of your safety hooks is a reliability rating of 97%. You estimate that throughout the next year you will be using this device about 100 times. Answer the following questions. 1. Does a reliability rating of 97% mean that there is a 97% chance that the device will not fail any of the 100 times? 2. What is the probability of at least one failure? 3. What is the complement of this event ? 4. Can this be considered a binomial experiment? 5. Can you use the binomial probability formula? Why or why not? 6. Find the probability of at least two failures. 7. Can you use a normal distribution to accurately approximate the binomial distribution? Explain why or why not. 8. Is correction for continuity needed? 9. How much safer would it be to use a second safety hook independent of the first? See page 368 for the answers.

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Answer 1

Textbook Question

Chapter 6.4, Problem 1AC

Applying the Concepts 6–4

Mountain Climbing Safety

Assume one of your favorite activities is mountain climbing. When you go mountain climbing, you have several safety devices to keep you from falling. You notice that attached to one of your safety hooks is a reliability rating of 97%. You estimate that throughout the next year you will be using this device about 100 times. Answer the following questions.

1. Does a reliability rating of 97% mean that there is a 97% chance that the device will not fail any of the 100 times?

2. What is the probability of at least one failure?

3. What is the complement of this event?

4. Can this be considered a binomial experiment?

5. Can you use the binomial probability formula? Why or why not?

6. Find the probability of at least two failures.

7. Can you use a normal distribution to accurately approximate the binomial distribution? Explain why or why not.

8. Is correction for continuity needed?

9. How much safer would it be to use a second safety hook independent of the first?

See page 368 for the answers.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

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Answer 2

Textbook Question

Chapter 6.4, Problem 1AC

Applying the Concepts 6–4

Mountain Climbing Safety

Assume one of your favorite activities is mountain climbing. When you go mountain climbing, you have several safety devices to keep you from falling. You notice that attached to one of your safety hooks is a reliability rating of 97%. You estimate that throughout the next year you will be using this device about 100 times. Answer the following questions.

1. Does a reliability rating of 97% mean that there is a 97% chance that the device will not fail any of the 100 times?

2. What is the probability of at least one failure?

3. What is the complement of this event?

4. Can this be considered a binomial experiment?

5. Can you use the binomial probability formula? Why or why not?

6. Find the probability of at least two failures.

7. Can you use a normal distribution to accurately approximate the binomial distribution? Explain why or why not.

8. Is correction for continuity needed?

9. How much safer would it be to use a second safety hook independent of the first?

See page 368 for the answers.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

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Answer 3

Textbook Question

Chapter 6.4, Problem 1AC

Applying the Concepts 6–4

Mountain Climbing Safety

Assume one of your favorite activities is mountain climbing. When you go mountain climbing, you have several safety devices to keep you from falling. You notice that attached to one of your safety hooks is a reliability rating of 97%. You estimate that throughout the next year you will be using this device about 100 times. Answer the following questions.

1. Does a reliability rating of 97% mean that there is a 97% chance that the device will not fail any of the 100 times?

2. What is the probability of at least one failure?

3. What is the complement of this event?

4. Can this be considered a binomial experiment?

5. Can you use the binomial probability formula? Why or why not?

6. Find the probability of at least two failures.

7. Can you use a normal distribution to accurately approximate the binomial distribution? Explain why or why not.

8. Is correction for continuity needed?

9. How much safer would it be to use a second safety hook independent of the first?

See page 368 for the answers.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

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Answer 4

Textbook Question

Chapter 6.4, Problem 1AC

Applying the Concepts 6–4

Mountain Climbing Safety

Assume one of your favorite activities is mountain climbing. When you go mountain climbing, you have several safety devices to keep you from falling. You notice that attached to one of your safety hooks is a reliability rating of 97%. You estimate that throughout the next year you will be using this device about 100 times. Answer the following questions.

1. Does a reliability rating of 97% mean that there is a 97% chance that the device will not fail any of the 100 times?

2. What is the probability of at least one failure?

3. What is the complement of this event?

4. Can this be considered a binomial experiment?

5. Can you use the binomial probability formula? Why or why not?

6. Find the probability of at least two failures.

7. Can you use a normal distribution to accurately approximate the binomial distribution? Explain why or why not.

8. Is correction for continuity needed?

9. How much safer would it be to use a second safety hook independent of the first?

See page 368 for the answers.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

Answer 8

1.

Expert Solution

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer to Problem 1AC

No.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

Answer 9

1.

Expert Solution

Answer 10

1.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To check: If reliability of 97% means that there is a 97% chance that the device will not fail any of the 100 items.

Answer 13

Answer to Problem 1AC

No.

Answer 14

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculations:

The random variable X denotes the number of times device fails out of 100 times.

Since reliability rate 97% means that, on average the device will not fail for 97% of time. Therefore, reliability of 97% doesn’t means that there is a 97% chance that the device will not fail any of the 100 items.

Answer 15

2.

Expert Solution

To determine

The probability of at least one failure.

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

Answer 16

2.

Expert Solution

Answer 17

2.

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

The probability of at least one failure.

Answer 20

Answer to Problem 1AC

Probability of at least one failure is 0.95.

Answer 21

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Calculation:

The random variable X denotes the number of times device fails out of 100 times.

Therefore, probability of at least one failure is equal to 1 – Probability of zero failures.

At least one failure = 1−(No failure)100= 1− (0.97)100=1−0.04755= 0.95

Therefore, the probability of at least one failure is 0.95.

Answer 22

3.

Expert Solution

To determine

The complement of the event at least one failure.

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

Answer 23

3.

Expert Solution

Answer 24

3.

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

The complement of the event at least one failure.

Answer 27

Answer to Problem 1AC

Event of zero times failure of safety hook out of 100 trials.

Answer 28

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times.

If A denotes the event that the safety hook fails at least once. Then the complement of the event is the event that the safety hook does not fails out of 100 times used.

Answer 29

4.

Expert Solution

To determine

The given experiment is binomial experiment or not.

Answer to Problem 1AC

Given experiment is binomial experiment.

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

Answer 30

4.

Expert Solution

Answer 31

4.

Expert Solution

Answer 32

Expert Solution

Answer 33

To determine

The given experiment is binomial experiment or not.

Answer 34

Answer to Problem 1AC

Given experiment is binomial experiment.

Answer 35

Explanation of Solution

Given info:

Reliability rate is 97%.

Number of times device used is 100.

Justification:

The random variable X denotes the number of times device fails out of 100 times and there are two possible outcomes device working and device not working. Also, the reliability rate of safety hook in each time its use is constant i.e 0.97, therefore given experiment is binomial experiment of number of times the safety hooks fails out of 100 times with probability that safety hook fails in one trial is 0.03. So, given experiment is binomial experiment.

Answer 36

5.

Expert Solution

To determine

Whether the binomial formula can be used or not.

Answer to Problem 1AC

Yes, the binomial formula can be used.

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

Answer 37

5.

Expert Solution

Answer 38

5.

Expert Solution

Answer 39

Expert Solution

Answer 40

To determine

Whether the binomial formula can be used or not.

Answer 41

Answer to Problem 1AC

Yes, the binomial formula can be used.

Answer 42

Explanation of Solution

Justification:

The probability that the safety hook fails x number of times out 100 with probability that it fails in one trial being 0.03 is Cx100(0.03)x(1−0.03)100−x . Thus, the binomial formula can be used but as the number of trial is 100, the calculation may be cumbersome.

Answer 43

6.

Expert Solution

To determine

The probability of at least two failures.

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

Answer 44

6.

Expert Solution

Answer 45

6.

Expert Solution

Answer 46

Expert Solution

Answer 47

To determine

The probability of at least two failures.

Answer 48

Answer to Problem 1AC

The probability of at least two failures is 0.81.

Answer 49

Explanation of Solution

Justification:

Calculation:

Probability of at least 2 failures is,

P(X≥2)=1−P(X=0)−P(X=1)=1−C0100 (0.03)0 (1−0.03)100−0 −C1100 (0.03)1(1−0.03)100−1= 1− 0.048−0.147= 0.81

Therefore, the probability of at least 2 failures of safety hook is 0.81.

Answer 50

7.

Expert Solution

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

Answer 51

7.

Expert Solution

Answer 52

7.

Expert Solution

Answer 53

Expert Solution

Answer 54

To determine

Whether normal distribution can be used to accurate approximate the binomial distribution.

Answer 55

Answer to Problem 1AC

No, normal distribution cannot be used to accurate approximate the binomial distribution.

Answer 56

Explanation of Solution

Justification:

Conditions:

np=(100)(0.03)=3nq=(100)(0.97)=97

Here, np is less than 5.

Hence, normal distribution cannot be used to accurately approximate the binomial distribution.

Answer 57

8.

Expert Solution

To determine

Correction for continuity is needed.

Answer to Problem 1AC

No, continuity correction is not needed.

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

Answer 58

8.

Expert Solution

Answer 59

8.

Expert Solution

Answer 60

Expert Solution

Answer 61

To determine

Correction for continuity is needed.

Answer 62

Answer to Problem 1AC

No, continuity correction is not needed.

Answer 63

Explanation of Solution

Justification:

From part (7), the normal approximation is not applied. Hence, the continuity correction is not required.

Answer 64

9.

Expert Solution

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

Answer 65

9.

Expert Solution

Answer 66

9.

Expert Solution

Answer 67

Expert Solution

Answer 68

To determine

How much safer it would be to use a second safety hook independent of the first.

Answer 69

Answer to Problem 1AC

Probability of failure after second hook is 0.0009.

Answer 70

Explanation of Solution

Given Info:

Reliability rate is 97%.

Number of times device used is 100.

Second safety hook is used independent of first safety hook.

Calculation:

Reliability rate of first safety hook is 97%.

Thus,

Probability of failure if first safety hook= 1− reliability rate100= 1− 0.97= 0.03

Since, second safety hook is used independent of first safety hook, Therefore probability of failure if both safety hooks are used is

(Probability that first hook fails)(Probability that second hook fails)= (0.03)(0.03)= 0.0009

Therefore, probability of failure if both the safety hooks are used independently is 0.0009.

Answer 71

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