This from 2009:
Yudkowsky poses the following canonical problem:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies.
A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
The frightening thing is that only 15% of doctors get this right. And they’re off by a lot. That is, the average answer is in the range of 80% while the correct answer is 7.8%. Apparently, there is something about the way we think about the problem that makes 7.8% hard to accept, and Yudkowsky does a great job of walking you through the logic in painfully small steps.
Let’s try something similar here…
What Do We Know & What Does It Imply?
We have three pieces of information:
1% of sample are TRUE (that is, they have cancer)
80% of sample who are TRUE will test TRUE
9.6% of sample who are FALSE will test TRUE.
On the face of it, we should guess that the percentage of women who test TRUE who actually are TRUE (test positive and actually have cancer) is pretty small based on two facts provided: the actual percentage of women from the sample who are TRUE (regardless of testing) is only 1%, and the test has a false positive for 9.6% of those tested.
So, to solve this:
1) Assume we have a sample of 1000 women (I use 1000 to reduce the amount I have to talk about fractional people, but I don’t use 10,000 as I get lost in the zeros).
2) We know that the reality is that of the 1,000 women, 10 will have cancer (1%).
990 = no cancer
10 = cancer
3) Of the 10 who have cancer, 8 will test positive
8 out 1000 women tested will test True and are True
4) Of the 990 with no cancer 9.6% will also test positive = 990 * .096 = 95.04.
95.04 women out of 1,000 will test True but are False.
5) The total number testing true is 8 + 95.04 = 103.04.
Of these, 8 actually have Cancer.
6) So the value for tests positive (103.04) versus is positive (8) is 8/103.4 or 0.773 or 7.8%
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Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.
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Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.
Right now it seems like Expression Engine is the preferred blogging platform out there right now. (from what I’ve read) Is that what you’re using on your blog?
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Aw, this was an incredibly nice post. Spending some time and actual effort to produce a very good article… but what can I say… I put things off a whole lot and don’t seem to get anything done.
Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.
At this time it sounds like Drupal is the best blogging platform available right now. (from what I’ve read) Is that what you are using on your blog?
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