Here’s how I think about these problems. You know that the woman tested positive. There are two kinds of women who test positive: women who actually have breast cancer and correctly test positive (the true positives), and women who don’t have breast cancer but mistakenly test positive (the false positives). How big are these two groups?
First, the true positives: women who actually have breast cancer and correctly test positive. 1% of women actually have breast cancer, and 80% of them test positive, so 0.8% of women are in that group (.01 x .8).
Then, the false positives: women who don’t have breast cancer but mistakenly test positive. 99% of women don’t have breast cancer, and 9.6% of them mistakenly test positive, so 9.5% of women are in that group (.99 x .096).
What you care about is the relative size of the two groups. For every 0.8 women who have cancer and test positive, 9.5 women don’t have cancer but still test positive. That’s a 0.8:9.5 ratio; if you want to turn it into a percentage it’s 0.8/10.3 = 7.8% of the women who test positive are ones that actually have cancer.
So instead of using the whole formula, I think through the problem and do three simple calculations along the way. If you only need to calculate a rough estimate, you can do this even quicker with less calculating. Glancing at the numbers, about 1% of women are in the true positive group, and about 10% of women are in the false positive group, so about 1⁄11 of women who test positive have cancer (9%). That’s pretty close to the actual answer of 7.8%.
Here’s how I think about these problems. You know that the woman tested positive. There are two kinds of women who test positive: women who actually have breast cancer and correctly test positive (the true positives), and women who don’t have breast cancer but mistakenly test positive (the false positives). How big are these two groups?
First, the true positives: women who actually have breast cancer and correctly test positive. 1% of women actually have breast cancer, and 80% of them test positive, so 0.8% of women are in that group (.01 x .8).
Then, the false positives: women who don’t have breast cancer but mistakenly test positive. 99% of women don’t have breast cancer, and 9.6% of them mistakenly test positive, so 9.5% of women are in that group (.99 x .096).
What you care about is the relative size of the two groups. For every 0.8 women who have cancer and test positive, 9.5 women don’t have cancer but still test positive. That’s a 0.8:9.5 ratio; if you want to turn it into a percentage it’s 0.8/10.3 = 7.8% of the women who test positive are ones that actually have cancer.
So instead of using the whole formula, I think through the problem and do three simple calculations along the way. If you only need to calculate a rough estimate, you can do this even quicker with less calculating. Glancing at the numbers, about 1% of women are in the true positive group, and about 10% of women are in the false positive group, so about 1⁄11 of women who test positive have cancer (9%). That’s pretty close to the actual answer of 7.8%.