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’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. :)