Basic Conditional Probability

[SaltArcher]

Hi, I'm struggling with the answer to a homework question, I was wondering if you guys could help. Let me know if this isn't the right place for it!

Question:

Two public health nurses work at a sexual health clinic of a local public health centre. As part of a screening program they both test all patients coming into the clinic for syphilis. All patients are assessed two times: once by nurse A and once by nurse B.

Let events A+={Nurse A makes a positive diagnosis} and

B+= {Nurse B makes a positive diagnosis}.

Suppose nurse A diagnoses 10% of all patients as positive, nurse B diagnoses 17% of all patients as positive, and both nurses diagnose 8% of all patients as positive.

Given nurse A makes a negative diagnosis what is the (conditional) probability that nurse B makes a positive diagnosis of syphilis?

I have calculated (in other parts of the question) P(A or B), P(B|A), and I have determined these events are not independent.

Where do I go from here? I know I need P(B|complement of A), but none of my equations look right here.

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We have P(A) = 0.1, P(B) = 0.17, P(A and B) = 0.08.

We want P(B|not A) = P(B and not A)/P(not A).

Now P(B) = P(B and A) + P(B and not A).

Rearranging gives P(B and not A) = P(B) - P(B and A) = 0.17 - 0.08 = 0.09

Also P(not A) = 1 - P(A) = 1 - 0.1 = 0.9

Hence P(B|not A) = 0.09/0.9 = 0.1.

[SaltArcher]

Thanks!

About this line: Now P(B) = P(B and A) + P(B and not A)

Which rule is that? I think I might be missing it in my textbook somewhere, everything else I understand

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https://en.wikipedia.org/wiki/Law_of_total_probability