\begin{tabular}{|l|l|} Q.3 & \( \begin{array}{l}\text { It is estimated that } 50 \% \text { of emails are spam emails. Some software has been applied } \\ \text { to filter these spam emails before they reach your inbox. A certain brand of } \\ \text { software claims that it can detect } 99 \% \text { of spam emails, and the probability for a } \\ \text { false positive (a non-spam email detected as spam) is } 5 \% \text {. Now if an email is } \\ \text { detected as spam, then what is the probability that it is in fact a non-spam email? }\end{array} \) \\ \hline\end{tabular}

To calculate the probability that an email detected as spam is actually a non-spam email, we can use Bayes' theorem. Here's a brief breakdown: Let \( S \) be the event that an email is spam, and \( N \) be the event that it's non-spam. Given: - \( P(S) = 0.5 \) (50% of emails are spam) - \( P(N) = 0.5 \) (50% of emails are non-spam) - The probability of detecting spam is \( P(\text{Detected as Spam} | S) = 0.99 \). - The probability of a false positive is \( P(\text{Detected as Spam} | N) = 0.05 \). Using Bayes' theorem: \[ P(N | \text{Detected as Spam}) = \frac{P(\text{Detected as Spam} | N) \cdot P(N)}{P(\text{Detected as Spam})} \] First, we need to find \( P(\text{Detected as Spam}) \): \[ P(\text{Detected as Spam}) = P(\text{Detected as Spam} | S) \cdot P(S) + P(\text{Detected as Spam} | N) \cdot P(N) \] \[ P(\text{Detected as Spam}) = (0.99 \cdot 0.5) + (0.05 \cdot 0.5) = 0.495 + 0.025 = 0.52 \] Now we can find \( P(N | \text{Detected as Spam}) \): \[ P(N | \text{Detected as Spam}) = \frac{0.05 \cdot 0.5}{0.52} = \frac{0.025}{0.52} \approx 0.048 \] Thus, the probability that an email detected as spam is actually a non-spam email is approximately 0.048, or 4.8%. So, the spam filters aren't perfect, but they help keep your inbox cleaner! To add a bit of fun, did you know that the first recognized spam email was sent way back in 1978? It was a message meant to promote a digital system and inadvertently sparked a marketing phenomenon! Fast forward to today, and we're battling millions of spam emails daily—thank goodness for filters! Speaking of filters, it's super helpful to regularly update and configure your email settings to minimize the clutter! Make sure you're marking emails correctly, as this trains your spam filter to become more accurate over time—think of it as teaching your computer to understand your preferences better. Always double-check the spam folder before deleting, too; you might find a hidden gem!