The email says "free," is it spam?

Question

At your inbox, 40\% of arriving email is spam and 60\% is legitimate. The word "free" shows up in 60\% of spam messages but only 5\% of legitimate ones. A new email contains the word "free." What is the probability the email is spam?

Accepted Answer

Imagine 1{,}000 incoming emails: 400 spam and 600 legitimate. Count how many contain "free." - Spam with "free": 400 	imes 0.60 = 240. - Legitimate with "free": 600 	imes 0.05 = 30. Of the 240 + 30 = 270 emails that say "free," 240 are spam: P(	ext{spam} \mid 	ext{"free"}) = \frac{240}{270} = \frac{8}{9} \approx 0.889. The word is far more common in spam, so seeing it lifts your belief from the 40\% base rate up to about 89\%. Takeaway. Bayes combines the base rate (how common spam is) with the likelihood (how telling the word is); a word that is much more frequent in spam swings the verd…

The email says "free," is it spam?

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