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 Pd(c) of detecting a crack of size c in an aircraft structure:
Pd(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 Pd (c*) = .5
b. What is Pd (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 Pd(c) as β → ∞?