Question

Breast cancer is the most common non-skin malignancy among women in the United States, and it is the second leading cause of death from cancer among women (lung cancer ranks Örst). A mammogram is a diagnostic procedure designed to quickly detect breast cancer. Such a test can exhibit two kinds of errors: One error is known as a false positive: The test incorrectly indicates cancer in a healthy woman. The other error (which is more serious error in this situation) is a false negative: The test fails to detect cancer.
The probabilities of these errors are established in clinical trials. In a clinical trial, the test is applied to two groups of people, some known to be healthy and others known to have cancer (through an established test that is more complicated, time consuming, and expensive than the proposed new method). Among women with breast cancer, the probability that mammography detects the cancer is 0.85. Among women without breast cancer, the chance for a negative result is 0.925.
If a mammogram indicates that a 55-year-old Caucasian woman tests positive for breast cancer, then what is the probability that she in fact has breast cancer? This problem requires conditional proba- bilities where we are told that
Pr(Test negative j No cancer) = 0:925 Pr(Test positive j Cancer) = 0:85
We need one more probability; the overall presence of this cancer in the population: In this case, Pr(Cancer) = 0.005 in Caucasian women of this age. The problem is to Önd Pr(Cancer j Test positive)

Answer 1

Answer:

Breast cancer is the most common non-skin malignancy among women in the United States, and it is the second leading cause of death from cancer among women (lung cancer ranks Örst). A mammogram is a diagnostic procedure designed to quickly detect breast cancer. Such a test can exhibit two kinds of errors: One error is known as a false positive: The test incorrectly indicates cancer in a healthy woman. The other error (which is more serious error in this situation) is a false negative: The test fails to detect cancer.
The probabilities of these errors are established in clinical trials. In a clinical trial, the test is applied to two groups of people, some known to be healthy and others known to have cancer (through an established test that is more complicated, time consuming, and expensive than the proposed new method). Among women with breast cancer, the probability that mammography detects the cancer is 0.85. Among women without breast cancer, the chance for a negative result is 0.925.
If a mammogram indicates that a 55-year-old Caucasian woman tests positive for breast cancer, then what is the probability that she in fact has breast cancer? This problem requires conditional proba- bilities where we are told that
Pr(Test negative / No cancer) = 0:925 Pr(Test positive /Cancer) = 0:85
We need one more probability; the overall presence of this cancer in the population: In this case, Pr(Cancer) = 0.005 in Caucasian women of this age.

The problem is to Find Pr(Cancer / Test positive)

If there are 10000 persons, we are given the following frequencies:

	cancer	No cancer	Total
test +ve	42.5	746.25	788.75
test -ve	7.5	9203.75	9211.25
Total	50	9950	10000

Pr(Cancer / Test positive) = 42.5/788.75

=0.053883

If we use Bayes, Theorem,

Pr(Cancer / Test positive)

	Cancer(0.5%)	No cancer(99.5%)	Total
test +ve	True positive= 0.005*0.85 =0.00425	False positive=0.074625	0.078875
test -ve	False negative =0.00075	True negative =0.995*0.925 =0.920375	0.921125
Total	0.005	0.995	1

Pr(Cancer / Test positive) = 0.00425/0.078875

=0.053883