To determine the risk of a woman having a cancerous tumor given a positive mammogram, we need to consider both the sensitivity of the mammogram and the prevalence of cancer in the population.
Given:
- Prevalence of malignant breast cancer tumors: 1% (0.01)
- Sensitivity of the mammogram: 85% (0.85)
Let's break down the calculation step-by-step:
1. Out of 10,000 mammograms, we expect 100 women to have cancer (10,000 x 0.01 = 100) and the remaining 9,900 women to have benign tumors or nothing detected.
2. The mammogram correctly identifies a cancerous tumor 85% of the time. This means that out of the 100 women with cancer, 85 will receive a positive mammogram result (100 x 0.85 = 85).
3. However, the mammogram can also produce false positive results, indicating cancer when there is actually none. To determine the number of false positives, we need to subtract the true positives from the total number of positive results. Since we know there are 85 true positives, the remaining positive results must be false positives. In this case, there are 15 false positives (100 - 85 = 15).
4. Now, we need to determine the overall number of positive mammogram results. This is the sum of the true positives and false positives. So, there are a total of 100 positive mammograms (85 true positives + 15 false positives = 100).
5. Finally, to calculate the risk of a woman having a cancerous tumor given a positive mammogram, we divide the number of true positives by the total number of positive mammograms and convert it into a percentage. So, the risk is 85% (85 true positives / 100 positive mammograms = 0.85 or 85%).
Therefore, if a woman receives a positive mammogram result, her risk of having a cancerous tumor is 85%. However, it's important to note that this risk is based on the sensitivity of the mammogram and the prevalence of cancer in the population. Additional tests and evaluations may be necessary to confirm the diagnosis.