1. **State the problem:** We have 3 email accounts with different probabilities for receiving messages and spam rates. We want to find the probability that a randomly selected message is spam. 2. **Given data:** - Probability message is from account 1: $P(A_1) = 0.7$ - Probability message is from account 2: $P(A_2) = 0.2$ - Probability message is from account 3: $P(A_3) = 0.1$ - Probability message is spam given account 1: $P(S|A_1) = 0.01$ - Probability message is spam given account 2: $P(S|A_2) = 0.02$ - Probability message is spam given account 3: $P(S|A_3) = 0.05$ 3. **Calculate the total probability of spam $P(S)$:** $$ P(S) = P(S|A_1)P(A_1) + P(S|A_2)P(A_2) + P(S|A_3)P(A_3) $$ 4. **Substitute the values:** $$ P(S) = (0.01)(0.7) + (0.02)(0.2) + (0.05)(0.1) = 0.007 + 0.004 + 0.005 $$ 5. **Sum the probabilities:** $$ P(S) = 0.007 + 0.004 + 0.005 = 0.016 $$ 6. **Final answer:** The probability that a randomly selected message is spam is $0.016$, or 1.6%.