KM unit 7

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New Jersey Institute Of Technology *

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650

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Mechanical Engineering

Date

Feb 20, 2024

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docx

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4

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An independent research organization is trying to estimate the probability that an accident at a nuclear power plant will result in radiation leakage. The types of accidents possible at the plant are, fire hazards, mechanical failure, or human error. The research organization also knows that two or more types of accidents cannot occur simultaneously. According to the studies carried out by the organization, the probability of a radiation leak in case of a fire is 20%, the probability of a radiation leak in case of a mechanical 50%, and the probability of a radiation leak in case of a human error is 10%. The studies also showed the following; The probability of a radiation leak occurring simultaneously with fire is 0.1%. The probability of a radiation leak occurring simultaneously with a mechanical failure is 0.15%. The probability of a radiation leak occurring simultaneously with a human error is 0.12%. On the basis of the information available, answer the questions below: 1.1 What are the probabilities of a fire, a mechanical failure, and a human error respectively? Probability = Number of ways the event can occur/ The total number of outcomes The probability of a fire = 20/(20+50+10) = 0.25% By mechanical failure = 50/80 = 0.625% By human error = 10/80 = 0.125% The probability of a fire occurring is 25%, a mechanical failure is 62.5%, and a human error is 12.5%. 1.2 What is the probability of a radiation leak? The probability of a radiation leak occurring simultaneously with a-
Fire: 0.1%. Mechanical failure: 0.15%. Human error: 0.12% The probability of radiation leakage = 0.1+0.15+0.12 = 0.37% 1.3 Suppose there has been a radiation leak in the reactor for which the definite cause is not known. What is the probability that it has been caused by: a) a fire? b) a mechanical failure? c) a human error? To find the probability of cause of radiation leak= The probability of a radiation leak occurring simultaneously due to an event/ the probability of a radiation leak The probability of radiation leak occurred due to: Fire = 0.1/0.37 = 0.27% Mechanical failure = 0.15/0.37 = 0.40% Human error = 0.12/0.37 = 0.32%
the probability that a randomly selected student scores between 65 and 87 is 0.9554. the passing cut-off so that 75% of the students clear the exam is 82.789. 2.1 The probability that a randomly chosen student gets a grade below 85 is 0.9332 To find this we need to find cumulative probability up to the value of 85 in a normal distribution with a mean of 77 and a standard deviation of 8.5. This represents the area under the curve to the left of 85. Using a standard normal distribution table or a statistical software, we can find the corresponding cumulative probability. Let's denote it as P(X < 85). By looking up the value in the table or using the software, we find that P(X < 85) is approximately 0.9332. Therefore, the probability that a randomly chosen student gets a grade below 85 is 0.9332 or 93.32%. 4.2 To find the probability that a randomly selected student scores between 65 and 87, we need to calculate the area under the normal distribution curve between these two values. We can find the cumulative probabilities P(X < 65) and P(X < 87), and then subtract them to obtain the desired probability. Using the standard normal distribution table or a statistical software, we find that P(X < 65) is approximately 0.0159 and P(X < 87) is approximately 0.9713. Therefore, the probability that a randomly selected student scores between 65 and 87 is P(65 < X < 87) = P(X < 87) - P(X < 65) = 0.9713 - 0.0159 = 0.9554 or 95.54%. 4.3 To determine the passing cut-off so that 75% of the students clear the exam, we need to find the score that corresponds to the 75th percentile in the normal distribution. Using the standard normal distribution table or a statistical software, we can find the z-score that corresponds to the 75th percentile, which is approximately 0.674. We can then use the formula z = (X - μ) / σ to find the corresponding value of X, where X is the desired cut-off score, μ is the mean, and σ is the standard deviation. Solving for X, we have X = z * σ + μ = 0.674 * 8.5 + 77 = 82.789.
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Therefore, the passing cut-off should be set at approximately 82.789 so that 75% of the students clear the exam. Learn more about probability here: brainly.com/question/31828911