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Chapter 3: Q.3.10 (page 107)

Two percent of women age 45 who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammographies. Eight percent of the women who do not have breast cancer will also have positive mammographies. Given that a woman has positive mammography, what is the probability she has breast cancer?

Short Answer

Expert verified

The probability that the woman has breast cancer is0.1867

Step by step solution

01

Step 1:Given Information

Given that two percent of women age 45 who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammographies. Eight percent of the women who do not have breast cancer will also have positive mammographies.

02

Explanation

B=Breast cancer

C=Positive mammographies

P(B)=2%=0.02

P(C鈭�B)=90%=0.9

PC鈭�Bc=8%=0.08

Use the complement rule:

PAc=P(not鈦�A)=1鈭�P(A)

PBc=1鈭�P(B)=1鈭�0.02=0.98

03

Explanation of Bayes's Theorem

Use the equation

PAi鈭�B=PB鈭�AiPAi鈭�j=1k鈥�PB鈭�AjPAj

P(B鈭�C)=P(C鈭�B)P(B)P(C鈭�B)P(B)+PC鈭�BcPBc

Substitute the value,

=0.9脳0.020.9脳0.02+0.08脳0.98

=0.0180.018+0.0784

=0.0180.0964
=180964

We get,

=45241

localid="1648546616788" =0.1867

04

Final Answer

The probability she has breast cancer is45241=0.1867.

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