Thanks. I’m pretty sure I understand now. Although I’m not sure why I get the correct answer when I’m working with the actual numbers and not percentages when I do the math wrong.
But when I do the math like you wrote, I get the right answer for the precentages. So I get that part. But aren’t I ignoring the base rate in the actual numbers one? Or no?
Although I’m not sure why I get the correct answer when I’m working with the actual numbers and not percentages when I do the math wrong.
I know it now makes more sense to you now, but I want to point out that reality isn’t school, and nobody is going to take marks off for using actual numbers or ratios instead of percentages (the ‘pure’ way that the teacher prefers or what-have-you).
A calculator more reliably gets me the answer than mental arithmetic, and so I use a calculator at work even though it seems lazier than doing it in my head—in the same way, if ratios and actual numbers more reliably let you use Bayes Theorem than percentages, use actual numbers and all the people who think it’s purer to use percentages be damned.
I’m awfully glad to here that, I’m not a big fan of percentages… Real numbers just come easier to me, I suppose.
Once I figure out the formulat itself, then I feel comfortable using a calculator, but I hate using a calculator if I don’t understand the mental math to begin with.
You’re not. Remember, you’re not taking 80 of the 10,000 women in the population. You’re only taking 80 of the 100 women with breast cancer. Likewise, it’s not 9.6% of all the women, it’s 9.6% of the women who don’t have breast cancer, or 950⁄9900. The wording of the problem already took the base rates into count, so when you’re plugging the real numbers in, you are automatically taking the base rates into account. By giving you 80⁄100 and 950⁄9900, Eliezer already did the division for you.
You can think of accounting for the base rate as equivalent to using the actual numbers.
How many women have cancer and test positive? 0.8 probability 0.01 population. How many women don’t have cancer and test positive? 0.096 probability 0.99 population.
When you use the actual numbers of people, you get those numbers by using the base rate: 10,000 women total, of which 100 have cancer (that’s the base rate in action), of which 80 test positive, etc. So if you use the numbers 80 (= 0.8 0.01 10000) and 950 = (0.096 0.99 10000), you’re not ignoring the base rate. You would be ignoring the base rate if you used the numbers 8000 and 960 (80% and 9.6% of the population of 10,000, respectively), but those numbers don’t refer to any relevant groups of people.
But aren’t I ignoring the base rate in the actual numbers one? Or no?
The actual numbers in the problem were chosen in such a way to make the base rates obvious. Here is another version using real numbers where the base rates aren’t quiet so obvious, see if you can get it right:
“1 out of every 100 women at age forty who participate in routine screening have breast cancer. 80 out of every 100 women with breast cancer will get positive mammographies. 96 out of every 1,000 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?”
Thanks. I’m pretty sure I understand now. Although I’m not sure why I get the correct answer when I’m working with the actual numbers and not percentages when I do the math wrong.
But when I do the math like you wrote, I get the right answer for the precentages. So I get that part. But aren’t I ignoring the base rate in the actual numbers one? Or no?
I know it now makes more sense to you now, but I want to point out that reality isn’t school, and nobody is going to take marks off for using actual numbers or ratios instead of percentages (the ‘pure’ way that the teacher prefers or what-have-you).
A calculator more reliably gets me the answer than mental arithmetic, and so I use a calculator at work even though it seems lazier than doing it in my head—in the same way, if ratios and actual numbers more reliably let you use Bayes Theorem than percentages, use actual numbers and all the people who think it’s purer to use percentages be damned.
I’m awfully glad to here that, I’m not a big fan of percentages… Real numbers just come easier to me, I suppose.
Once I figure out the formulat itself, then I feel comfortable using a calculator, but I hate using a calculator if I don’t understand the mental math to begin with.
You’re not. Remember, you’re not taking 80 of the 10,000 women in the population. You’re only taking 80 of the 100 women with breast cancer. Likewise, it’s not 9.6% of all the women, it’s 9.6% of the women who don’t have breast cancer, or 950⁄9900. The wording of the problem already took the base rates into count, so when you’re plugging the real numbers in, you are automatically taking the base rates into account. By giving you 80⁄100 and 950⁄9900, Eliezer already did the division for you.
....Oh.
Well, thanks Owen, Swimmy. I now understand Bayes Theorem significantly more than I did a half hour ago. :)
You can think of accounting for the base rate as equivalent to using the actual numbers.
How many women have cancer and test positive? 0.8 probability 0.01 population. How many women don’t have cancer and test positive? 0.096 probability 0.99 population.
When you use the actual numbers of people, you get those numbers by using the base rate: 10,000 women total, of which 100 have cancer (that’s the base rate in action), of which 80 test positive, etc. So if you use the numbers 80 (= 0.8 0.01 10000) and 950 = (0.096 0.99 10000), you’re not ignoring the base rate. You would be ignoring the base rate if you used the numbers 8000 and 960 (80% and 9.6% of the population of 10,000, respectively), but those numbers don’t refer to any relevant groups of people.
The actual numbers in the problem were chosen in such a way to make the base rates obvious. Here is another version using real numbers where the base rates aren’t quiet so obvious, see if you can get it right:
“1 out of every 100 women at age forty who participate in routine screening have breast cancer. 80 out of every 100 women with breast cancer will get positive mammographies. 96 out of every 1,000 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?”