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# Studies show that about five women in seven $approximately 71.4%$ who live to be 90 will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 25% of the time. Also, suppose that in the general population of women, the test for breast cancer is negative about 45% of the time. Let B=women develops breast cancer, P=tests positive, and N=tests negative. Suppose one woman is selected at random. Round your answer to three decimal places. Given that a woman develops breast cancer, what is the probability that she tests positive. Find P$P|B$ = 1-P$N|B$. What is the probability that a woman develops breast cancer and tests positive. Find P$B AND P$ = P$P|B$P$B$. What is the probability that a woman does not develop breast cancer. Find P$B’$ = 1-P$B$. What is the probability that a woman tests positive for breast cancer. Find P$P$=1-P$N$.

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## Answer to a math question Studies show that about five women in seven $approximately 71.4%$ who live to be 90 will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 25% of the time. Also, suppose that in the general population of women, the test for breast cancer is negative about 45% of the time. Let B=women develops breast cancer, P=tests positive, and N=tests negative. Suppose one woman is selected at random. Round your answer to three decimal places. Given that a woman develops breast cancer, what is the probability that she tests positive. Find P$P|B$ = 1-P$N|B$. What is the probability that a woman develops breast cancer and tests positive. Find P$B AND P$ = P$P|B$P$B$. What is the probability that a woman does not develop breast cancer. Find P$B’$ = 1-P$B$. What is the probability that a woman tests positive for breast cancer. Find P$P$=1-P$N$.

Fred
4.4
Let's define the events:

B : Woman develops breast cancer
P : Test result is positive
N : Test result is negative

Given probabilities:
P$B$ = 5/7
P$N|B$ = 0.25
P$N$ = 0.45

1. Probability a woman who develops breast cancer tests positive: P$P|B$ = 1 - P$N|B$
P$P|B$ = 1 - P$N|B$ = 1 - 0.25 = 0.75

2. Probability a woman develops breast cancer and tests positive: P$B and P$ = P$P|B$ \cdot P$B$
P$B and P$ = P$P|B$ \cdot P$B$ = 0.75 \cdot 5/7 = 0.5357

3. Probability a woman does not develop breast cancer: P$B'$ = 1 - P$B$
P$B'$ = 1 - P$B$ = 1 - 5/7 = 2/7 \approx 0.286

4. Probability a woman tests positive for breast cancer: P$P$ = 1 - P$N$
P$P$ = 1 - P$N$ = 1 - 0.45 = 0.55

1. P$P|B$ = 0.75
2. P$B and P$ = 0.5357
3. P$B'$ = 0.286
4. P$P$ = 0.55
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