An email server receives messages, out of which 20 % are spam. The server's spam filter correctly identifies and flags 90 % of the spam emails. However, it also incorrectly flags 5 % of the non-spam emails as spam. If an email is chosen at random and is found to be flagged as spam, the probability that it is actually a spam email is :

Step-by-step solution

Let S be the event that an incoming email is spam, and S' be the event that it is not spam. Let F be the event that an email is flagged as spam. From the given information: P(S) = 0.20 P(S') = 0.80 The conditional probabilities of an email being flagged are: P(F|S) = 0.90 P(F|S') = 0.05 We need to find the probability that a flagged email is actually spam, which is P(S|F) . Using Bayes' theorem: P(S|F) = P(S)P(F|S) P(S)P(F|S) + P(S')P(F|S') Substitute the values into the formula: P(S|F) = 0.20 0.90 0.20 0.90 + 0.80