1. **State the problem:** We want to find the probability that a message containing the word "Rolex" is spam. 2. **Given data:** - Number of spam messages: 2000 - Number of spam messages containing "Rolex": 250 - Number of non-spam messages: 1000 - Number of non-spam messages containing "Rolex": 5 - Probability that a message is spam or not spam is equal, so $P(Spam) = P(Not\ Spam) = 0.5$ 3. **Formula used:** We use Bayes' theorem to find $P(Spam|Rolex)$: $$ P(Spam|Rolex) = \frac{P(Rolex|Spam) \times P(Spam)}{P(Rolex)} $$ where $$ P(Rolex) = P(Rolex|Spam) \times P(Spam) + P(Rolex|Not\ Spam) \times P(Not\ Spam) $$ 4. **Calculate probabilities:** $$ P(Rolex|Spam) = \frac{250}{2000} = 0.125 $$ $$ P(Rolex|Not\ Spam) = \frac{5}{1000} = 0.005 $$ 5. **Calculate $P(Rolex)$:** $$ P(Rolex) = 0.125 \times 0.5 + 0.005 \times 0.5 = 0.0625 + 0.0025 = 0.065 $$ 6. **Calculate $P(Spam|Rolex)$:** $$ P(Spam|Rolex) = \frac{0.125 \times 0.5}{0.065} = \frac{0.0625}{0.065} \approx 0.9615 $$ **Final answer:** The probability that an incoming message containing the word "Rolex" is spam is approximately $0.9615$ or 96.15%.