True positive / false positive probability

[rogenfx]

Homework Statement

in screening for breast cancer, the probability that a mammogram will give a postive result, given that the in dividual has breast cancer is 0.95 (true positive). However the probability of a false positive is 0.1

(a) in a particular population of women, the likelihood of breast cancer is 0.025. obtain the probability that , if test is positive, the woman has breast cancer.

(b) If the result of the test is postitive, another independent test will be done on the same woman. For the same population as in part(a)

(i) what is the probability that both tests are positive?
(ii) if both are positive, what is the probability that the woman does NOT have cancer? what is the probability that she DOES have cancer?

(c) we now look at subset of population-- those women whose mother is known to have cancer.The likelihood of cancer in these women is 0.12. obtain the probability that a positive result indicates that the woman has cancer if her mother had cancer

Homework Equations

The Attempt at a Solution

i am totally have no idea about this cos the example usually talks about 1 test only... not 2 tests... if someone can help that would be nice :)

 

[statdad]

Can you do the calculations for a single test? If so: there are two situations in which a person can have two positive tests

have breast cancer and 1st test positive and 2nd test positive

no breast cancer and 1st test positive and 2nd test positive

see what you can do with this.

 

[rogenfx]

this is my answer...

A)
let B be event of a patiet having breast cancer
let X be event of a mammogram giving a positive result

P(X) = P(X|B)*P(B) + P(X| not B) * P (not B) = 0.95*0.025 + 0.1*0.975 = 0.12125

P(B|X) = {P(X|B) * P(B)} / P(X) = 0.95 * 0.025 / 0.12125 = 19/97


B)
i) P(both tests are positives)

P(B|X)*P(X|B) + P(notB | X) * P(X | notB)
this is where I am not sure...
is it become like this then ?

P(X|B) * P(B) / P(X) * P(X|B) + P(X|notB) * P(notB) / P(X) * P(X|notB)


and how about ii?

ii) should be
P(notB | both results positives)
= P(both tests are positive and notB) / P(both test are positives)