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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%
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B
8%
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C
10.2%
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D
12.2%
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Solution

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(A)=0.01 and P(AC)=0.99,P(X|A)=0.9 and P(XAC)=0.08

Using Bayes' theorem, we haveP(A|X) = P(AX)P(AX) + P(ACX)P(A|X) = P(A).P(X|A)P(A).P(X|A) + P(AC).P(XAC)P(A|X) = 0.01×0.90.01×0.9 + 0.99×0.08P(A|X) = 0.0090.009 + 0.0792P(A|X) = 0.0090.0882P(A|X) = 0.102Therefore, the probability of a woman having cancer given she has a positive test result is 10.2%.







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