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

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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

I do not think the value changes how many you tested and result in positive.

If the numbers in the example have changed from 1 % to 3 % etc and false positive 9.6% to 15 % etc ( I considered as just examples for my explanation the numbers may be any value either less or more ).

The value of 7.76% varies.

If we have conducted 1000 times and results in positive and the above number remains same the value of probability 7.76% do not change.

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