Assume that there is a test for cancer that is 90% accurate for both those who do and do not have cancer.

probability

Assume that there is a test for cancer that is 90% accurate for both those who do and do not have cancer. Also assume that 5% of the population has cancer. If an individual takes this test and tests positive for cancer, find the probability that they really do have cancer?

  • For this problem, I did a tree diagram with population on top then split it into 5% cancer and 95% no cancer. Then I split the 5% into 90% positive test and 10% negative test. Finally I split the 95% into 10% positive test and 90% negative. However, I feel like I'm off somewhere and I do not know how to continue from here. Any help would be much appreciated.

Best Answer

You are correct, there are 4 cases:

  • $5\%$ cancer
    1. $90\%$ positive test (total weight $0.05 \cdot 0.9 = 0.045$)
    2. $10\%$ negative test (total weight $0.05 \cdot 0.1 = 0.005$)
  • $95\%$ no cancer
    1. $10\%$ positive test (total weight $0.95 \cdot 0.1 = 0.095$)
    2. $90\%$ negative test (total weight $0.95 \cdot 0.9 = 0.855$)

Now, when a person tests positive for cancer, this is possible in one of two ways:

  • he really had cancer and tested positive, with weight $0.045$
  • he didn't have cancer and tested positive, with weight $0.095$

So what is the total probability of him actually being sick?

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