Question

Given the following statistics, what is the probability that a woman has cancer if she has a positive mammogram result?
1. 1% of women have cancer.
2. 90% of women who have cancer test positive on mammograms.
3. 8% of women will have false positives.

• A. 20.5%
• B. 8%
• C. 10.2%
• D. 12.2%

Answer 1

The correct option is C 10.2%

Let us assume that the event actually having cancer is A and getting a positive test result is X.
Then, according to the question,
$P\left(A\right)=0.01andP\left(A^C\right)=0.99,P\left(X|A\right)=0.9andP\left(X|A^C\right)=0.08$

$\mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{have}P\left(A|X\right)=\frac{P\left(A\cap X\right)}{P\left(A\cap X\right)+P\left(A^C\cap X\right)}\Rightarrow P\left(A|X\right)=\frac{P\left(A\right).P\left(X|A\right)}{P\left(A\right).P\left(X|A\right)+P\left(A^C\right).P\left(X|A^C\right)}\Rightarrow P\left(A|X\right)=\frac{0.01\times 0.9}{0.01\times 0.9+0.99\times 0.08}\Rightarrow P\left(A|X\right)=\frac{0.009}{0.009+0.0792}\Rightarrow P\left(A|X\right)=\frac{0.009}{0.0882}\Rightarrow P\left(A|X\right)=0.102\mathrm{Therefore},\mathrm{the}\mathrm{probability}\mathrm{of}a\mathrm{woman}\mathrm{having}\mathrm{cancer}\mathrm{given}\mathrm{she}\mathrm{has}a\mathrm{positive}\mathrm{test}\mathrm{result}\mathrm{is}10.2\%.$