Probability that only one component is defective?

Question

A component has a probability of 0.05 to have a certain defect (a) and 0.01 to have another defect (b). The two defects are independent. I have to find the probability that the component has only one defect knowing that is defective.

I've done this:

     _      _    _   

p(D)=1-p(D)=1-p(a)*p(b)=1-0.95*0.99=1-0.9405=0.0595

I started calculating the probability to have only one defect and I'd use the conditional probability with D when I had the result:

p([a/(a^b)]U[b/(a^b)])=p[a/(a^b)]+p[b/(a^b)]-p[[a/(a^b)]^[b/(a^b)]
I don't know how to continue.

Answer 1

As a concrete example (you could/should use percentages when doing a problem like this in general), suppose there are 10,000 components:

• We know 500 have defect 1

• We know 100 have defect 2

• Because the defects are independent, we have .05*.01*10000 = 5 components with both defects.

If you know the component is defective, it is one of the 600 total defective components.

Of these defective components, only 5 have both defects, so 595 have only one defect.

Thus, the probability of only one defect is 595/600 or about 99.17%

Suggestion: avoid using formulas blindly, and consider doing a step-by-step analysis, at least while you're still learning.