How can we solve this using a Venn diagaram?
12 out of 1000 women at age forty who participate in routine screening have breast cancer. 880 out of 1000 women with breast cancer will get positive mammographies. 90 out of 1000 women without breast cancer will also get positive mammographies. If 100 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?
To solve this problem using a Venn diagram, we can divide the information into three parts:
1. 1000 women at age forty who participate in routine screening
2. Women with breast cancer
3. Women without breast cancer
We can draw a rectangle to represent the 1000 women in the age group. Then, we can draw one circle inside the rectangle to represent women with breast cancer and another circle outside the rectangle to represent women without breast cancer.
Since 12 out of 1000 women at age forty who participate in routine screening have breast cancer, we can label the circle representing women with breast cancer as "12". Then, we know that 880 out of these 1000 women with breast cancer will get positive mammographies, so we can label the overlap between the circle representing women with breast cancer and the rectangle as "880".
Next, we know that 90 out of 1000 women without breast cancer will also get positive mammographies. We can label the circle representing women without breast cancer as "90", and the overlap between the circle representing women without breast cancer and the rectangle as "90".
Now, we can find the fraction of women with positive mammographies who actually have breast cancer by looking at the overlap between the circle representing women with breast cancer and the positive mammographies. Since this overlap represents women with breast cancer who also have positive mammographies, we can divide it by the total number of positive mammographies.
In this case, since the total number of positive mammographies is the sum of the overlaps from both circles, which is 880 + 90 = 970, the fraction of women with positive mammographies who actually have breast cancer is 880/970.
Therefore, about 0.907 or 90.7% of women with positive mammographies will actually have breast cancer.
Remember, a Venn diagram is just one way to visually represent the information and helps with understanding the problem. The final answer can be obtained by calculating the fraction based on the numbers provided.