Joint & Conditional Probability
Your spam filter flags an email holding `free money` and calls it almost certainly spam. Ask the mirror question, how often spam even contains `free money`, and that number comes back small: the same evidence reads two different ways.
- ▸The same email, the same phrase: swap which number you divide by and the answer swings from near-certain to unlikely.
- ▸A test that's
90%accurate can still be wrong most times it fires, when the thing it hunts is rare. - ▸Assuming two events don't interact turned
1 in 8500into1 in 73 millionin court, jailing an innocent woman.
A spam filter sorts a morning's 1,000 emails. 400 are spam, so before you look at anything else, a random one is spam 40% of the time. Now filter down to just the emails containing the phrase free money, maybe 120 of them. Inside that smaller pile, 108 are spam. The chance jumped to 90%.
Nothing about any single email changed. What changed is the pile you are dividing by. The first 40% divided the spam count by all 1,000 emails; the second 90% divided a spam count by only the 120 that contain the phrase. Shrink the world you are standing in, and the same event carries a new probability.
That is the whole move behind everything here: pick a smaller population, count inside it, divide by its size. The number you get answers a sharper question, not "how common is spam?" but "how common is spam among emails that say free money?"