1. **Problem statement:** We are given that 90% of emails are spam and 10% are legitimate. We want to find probabilities related to the first legitimate email found in a sequence of checked emails. 2. **Relevant formula:** This is a geometric probability problem where the probability of success (finding a legitimate email) is $p=0.1$ and failure (spam) is $q=1-p=0.9$. The probability that the first success occurs on the $k$-th trial is given by: $$P(X=k) = q^{k-1} p$$ 3. **Compute the probability that the first legitimate email is the 7th message:** $$P(X=7) = 0.9^{6} \times 0.1 = 0.531441 \times 0.1 = 0.0531$$ 4. **Compute the probability that the first legitimate email is the 7th or 8th message:** $$P(X=7 \text{ or } X=8) = P(X=7) + P(X=8)$$ $$= 0.9^{6} \times 0.1 + 0.9^{7} \times 0.1 = 0.0531 + 0.0478 = 0.1009$$ 5. **Compute the probability that the first legitimate email is among the first 7 messages:** This is the cumulative probability: $$P(X \leq 7) = 1 - P(X > 7) = 1 - q^{7} = 1 - 0.9^{7} = 1 - 0.4783 = 0.5217$$ 6. **Expected number of messages to check before finding a legitimate email:** For a geometric distribution, the expected value is: $$E(X) = \frac{1}{p} = \frac{1}{0.1} = 10.0$$ **Final answers:** - Probability first legitimate email is 7th: $0.0531$ - Probability first legitimate email is 7th or 8th: $0.1009$ - Probability first legitimate email is among first 7: $0.5217$ - Expected number of messages to check: $10.0$ messages