Probability and Statistics for Engineering and the Sciences

9th Edition

ISBN: 9781305251809

Author: Jay L. Devore

Publisher: Cengage Learning

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Textbook Question

Chapter 2, Problem 96SE

According to the article “Optimization of Distribution Parameters for Estimating Probability of Crack Detection” (J. of Aircraft, 2009: 2090–2097), the following “Palmberg” equation is commonly used to determine the probability P_d(c) of detecting a crack of size c in an aircraft structure:

P d ( c ) = ( c / c * ) β 1 + ( c / c * ) β

where c* is the crack size that corresponds to a .5 detection probability (and thus is an assessment of the quality of the inspection process).

a. Verify that P_d (c*) = .5
b. What is P_d (2c*) when β = 4?
c. Suppose an inspector inspects two different panels, one with a crack size of c* and the other with a crack size of 2c*. Again assuming β = 4 and also that the results of the two inspections are independent of one another, what is the probability that exactly one of the two cracks will be detected?
d. What happens to P_d (c) as → ∞?

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Chapter 2 Solutions

Probability and Statistics for Engineering and the Sciences

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Solution Summary: The author explains the Palmberg equation to determine the probability of detecting a crack of size c in an aircraft structure.

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Chapter 2 Solutions