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 at certain positions. 2. **Relevant Formula:** This is a geometric distribution 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: $$P(X=k) = q^{k-1} p$$ 3. **Step 1: Probability first legitimate email is the 7th message** $$P(X=7) = 0.9^{6} \times 0.1$$ Calculate: $$0.9^{6} = 0.531441$$ So, $$P(X=7) = 0.531441 \times 0.1 = 0.0531$$ 4. **Step 2: Probability first legitimate email is the 7th or 8th message** $$P(X=7 \text{ or } X=8) = P(X=7) + P(X=8)$$ Calculate $P(X=8)$: $$P(X=8) = 0.9^{7} \times 0.1 = 0.4782969 \times 0.1 = 0.0478$$ Sum: $$0.0531 + 0.0478 = 0.1009$$ 5. **Step 3: Probability first legitimate email is among first 7 messages** This is the cumulative probability: $$P(X \leq 7) = 1 - P(X > 7) = 1 - q^{7} = 1 - 0.9^{7}$$ Calculate: $$0.9^{7} = 0.4782969$$ So, $$P(X \leq 7) = 1 - 0.4783 = 0.5217$$ 6. **Step 4: Expected number of messages to check before finding a legitimate email** For geometric distribution, expected value is: $$E(X) = \frac{1}{p} = \frac{1}{0.1} = 10$$ **Final answers:** - $P(X=7) = 0.0531$ - $P(X=7 \text{ or } 8) = 0.1009$ - $P(X \leq 7) = 0.5217$ - Expected messages to check = 10.0