1. So I was presented with this elementary statistics problem...

Originally Posted by Problem
Approximately 1% of women aged 40-50 have breast cancer.
A woman with breast cancer has a 90% chance of a positive test from a mammogram,
while a woman without breast cancer has a 10% percent chance of a false positive result.
Using the simple statement of Baye's theorem, find the probability that a woman has breast cancer given she receive a positive result.
Okay. So Baye's theorem states .

Let denote a woman with breast cancer, a woman without, a positive result, and " - " negative.

The problem information directly implies...

Now, I have to find the probability of .

So if a woman receives a positive result from the mammogram, there's a 9 in 108 chance that she actually has breast cancer, or approximately 8.33%. Apparently, there's also a 91.7% chance that she comes positive but actually doesn't have it (yes, I also computed as well as the other two outcomes). So two of the four outcome probabilities seem anti-intuitive, namely and . 8.3% seems... unreliable at best. But maybe the whole point of this problem is to show an unexpectancy. (Monty Hall, anyone?)

Are my calculations correct? Are the statistics shaky or is a more detailed statement of Baye's theorem needed? Thanks in advance.

2.

3. Your calculations sem correct. The results may sound paradoxal, but IMO, they are not.

The probability to get a positive test is 10.8%, but, in most cases (9.9%), this result corresponds to an healthy person. So, in most cases, a positive result is false.
But a positive test is a start point for further examinations.

4. Hello,

Should not be written in the numerator 0.01 x 0.90, i.e. likelihood that a person is a women with cancer and likelihood that is a positive test given she really has cancer?

5. Yes, ivanf. I had the right thing in mind and wrote the correct result, but typo'd the function. My bad.

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