Suppose that we have found that the word “Rolex” occurs in 250 of 2000 messages known to be spam and in 5 of 1000 messages known not to be spam. Estimate the probability that an incoming message containing the word “Rolex” is spam, assuming that it is equally likely that an incoming message is spam or not spam.

Let \(S\) be the event that the message is spam and \(R\) be the event that the message contains the word “Rolex”. We are interested in estimating \(P(S\given R)\). By Bayes’ rule, we have,

\[\begin{align*} P(S\given R) &= \frac{P(R\given S)P(S)}{P(R\given S)P(S) + P(R\given \neg S)P(\neg S)} \\ &= \frac{P(R\given S)}{P(R\given S) + P(R\given \neg S)} \quad\quad(\text{since }P(S)=P(\neg S))\\ &= \frac{125/1000}{125/1000 + 5/1000} \\ &= \frac{125}{130} \approx 0.96 \end{align*}\]