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A certain form of cancer is known to be found in women over 60 with probability 0.07. A blood test exists for the detection of the disease, but the test is not infallible. In fact, it is known that 10% of the time the test gives a false negative (i.e., the test incorrectly gives a negative result) and 5% of the time the test gives a false positive (i.e., incorrectly gives a positive result). If a woman over 60 is known to have taken the test and received a favorable (i.e., negative) result, what is the probability that she has the disease?

Answer :

Answer:

0.0079 is the probability that woman has cancer when the test were negative.

Step-by-step explanation:

We are given the following information in the question:

Let A be the event when the patient have cancer and B be the event when test result is positive

P(A) = 0.07

Bayes Theorem:

[tex]P(A|B') = \displaystyle\frac{P(B'|A)P(A)}{P(B'|A)P(A) + P(B'|A')P(A')}[/tex]

[tex]P(B'|A) = 10\% = 0.10\\P(B|A') = 5\% = 0.05[/tex]

We have to find the probability that she has disease when test were negative that is we have to find:

[tex]P(A|B')[/tex]

Putting all the values in the above formula, we have,

[tex]P(A|B') = \displaystyle\frac{0.10\times 0.07}{0.10\times 0.07 +(1- 0.05)\times 0.93} = 0.0079[/tex]

Thus. 0.0079 is the probability that woman has cancer when the test were negative.

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