Probability Consider the situation discussed in Example 4.3 in the text. The wom

Question

Probability

Consider the situation discussed in Example 4.3 in the text. The woman in question tested positive and her posterior probability of having breast cancer was calculated to be 7.76%. If she decides to go for another test and she tests positive a second time, what is the probability of her having breast cancer? (And what if a third, fourth, fifth test was positive, what would the corresponding probabilities be? Of course, this is an unlikely scenario since each test exposes her to X-rays and a consequent risk of actually causing cancer.) (Try by calculation, and then check using CondProb.xls.)

Example 4.3 Breast Screening About 1% of women who participate in routine breast screening have breast cancer. About 80% of those with breast cancer get a positive test result and 9.6% without breast cancer also get a positive result. A woman gets a positive test result. What is the probability that she has breast cancer? The women who have a positive mammography results and actually have breast cancer amount to 80% of 1%, or 0.8%, of all the women who were tested. The women who do not have breast cancer, but still show a (false) positive mammography results, amount to 9.6% of 99%, or 9.504%, of all the women who were tested. Therefore, total positive mammography results amount to 0.8% 9.504%, or 10.304%, of all the women who were tested. Of this per- centage, the women who actually have cancer (0.8%) are in the distinct minority. The probability of a woman who has a positive test result actually having cancer, PAIB), is 0.8/10.304 0.0776 or 7.76% We can see this by using a Venn diagram and taking a population of 100,000 (Fig. 4.7). About 1% of these have breast cancer, i.e., A 1,000. Now 80% of (continued)

Explanation / Answer

Test results of the second test is independent of the first test results. Hence there is no impact on the probability of having a breast cancer provided test is positive.

Hence probability that a woman has breast cancer if her test is positive is 0.0776 thought the test is done second time.

Similarly, if the test is done for 3rd and 4th time, probability remains same of 0.0776

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