Trouble with Bayes Theorem? (The actual math is confusing)

But when I try to do the same with percentages, it all gets sort of screwy. I take the 80 percent of women (.8) divided by that same 80 percent (.8) plus 9.5 percent of women without cancer who test postive for it (.095). So I get .8/.8+.095=89%.

Let's look at those numbers closely. In particular the numbers on the denominator (80% + 9.5%). What are those percentages of?

• 80% of women with breast cancer
• 9.5% of women without cancer

When we add 80% + 9.5% and get 89.5% what is that 89.5% of? Nothing that makes sense because those are two completely different units. It's like adding 80% of an apple with 10% of an orange. We end up with 90% of an appleorange which just looks silly.

I recommend taking a look at An Intuitive Explanation of Eliezer Yudkowsky’s Intuitive Explanation of Bayes’ Theorem.

You're forgetting the "base rate" in your calculation: the actual rate of cancer in the population. What you should really be taking the ratio of is (the fraction of all women that have cancer and test positive) / (the fraction of all women that test positive, whether or not they have cancer). In percentages, that's

(80% of the 1% of women who have cancer, who correctly test positive) = 0.8 * 0.01.

divided by

(80% of the 1% of women who have cancer, who correctly test positive) together with (9.6% of the 99% of women who don't have cancer, who test positive anyway) = 0.8 0.01 + 0.096 0.99.

So the ratio is (0.8 0.01) / (0.8 0.01 + 0.096 * 0.99), and that does equal 0.078.

80/80+950 (or 80/1030=.078)

In mathematical language this is written 80/(80+950). This is because when you're doing expressions you do multiplication and division before addition but parentheses before either. 80/80+950 actually equals 951. This isn't what is giving you the problem right now but it actually would be if you used an advanced calculator.

Does this help make it intuitive? I haven't seen it explained this way.

After going through all this trouble, did you find math less boring than before?

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%.

I absolutely understand this situation. Been there, done that, got the Post-It note which explains the concept in Rubix-code and lives permanently in my notebook.

You've got your p(can+|mam+)=(p[mam+|can+]p[can+])/p(mam+). In numbers, that's .8(eighty percent of 1%,) divided by 9.5(9.5% of 100%) plus that same .8. What you're doing, I think, is omitting to recall that the eighty percent of women with cancer who test positive compromise .8% of all the women who get tested.

Also, it shook me up like nobody's business that the percentage of women without cancer who tested positive kept switching between problems, between 9.5 and 9.6! (I'm easily shaken.)

The error is pretty basic, but it is not stupid because you asked for help. The only stupid question is the one which is not asked. This is a common statement, but a true one and I mean it.

Owen has your answer for you here but something I noticed was a part of your math that could tell you the idea was wrong before you checked the final percentage: When you were adding the percentages, you ended up with .895 or 89.5% chance of getting a positive result from the test (you added all the positive odds, with the thinking that they're independent and referring to the same group). But it's fairly clear that more than 10.1% will get negative results, so the addition of those probabilities can't be right (This is more useful to way to check if you know the negative rate but not the quantity amount).