To battle against spam, Bob installs two anti-spam programs. An email arrives, which is either legitimate (event L) or spam (event Lc), and which program j marks as legitimate (event M) or marks as spam (event Mc) for j ∈ {1,2} . Assume that 10% of Bob’s email is legitimate and that the two programs are each “90% accurate” in the sense that P(M|L)=P(Mc|LC)=9/10. Also assume that given whether an email is spam, the two programs’ outputs are conditionally independent. Find the probability that the email is legitimate, given that the 1st  program marks it as legitimate (simplify). 

We are tasked with finding the probability that an email is legitimate given that the first anti-spam program marks it as legitimate, i.e., we need to compute \( P(L \mid M_1) \), where:

- \( L \) is the event that the email is legitimate.
- \( L^c \) is the event that the email is spam.
- \( M_1 \) is the event that the first program marks the email as legitimate.
- \( M_1^c \) is the event that the first program marks the email as spam.

### Given Information:
- \( P(L) = 0.1 \) (10% of emails are legitimate).
- \( P(L^c) = 0.9 \) (90% of emails are spam).
- \( P(M_1 \mid L) = 0.9 \) (Program 1 correctly marks legitimate emails as legitimate with 90% accuracy).
- \( P(M_1 \mid L^c) = 0.1 \) (Program 1 incorrectly marks spam emails as legitimate 10% of the time).

### Applying Bayes' Theorem

To compute \( P(L \mid M_1) \), we use Bayes' theorem:

\[
P(L \mid M_1) = \frac{P(M_1 \mid L) P(L)}{P(M_1)}
\]

Now, we need to calculate \( P(M_1) \), which is the total probability that Program 1 marks an email as legitimate. This can be found using the law of total probability:

\[
P(M_1) = P(M_1 \mid L)P(L) + P(M_1 \mid L^c)P(L^c)
\]

Substituting the known values:

\[
P(M_1) = (0.9)(0.1) + (0.1)(0.9)
\]
\[
P(M_1) = 0.09 + 0.09 = 0.18
\]

### Calculating \( P(L \mid M_1) \)

Now, using Bayes' theorem:

\[
P(L \mid M_1) = \frac{(0.9)(0.1)}{0.18}
\]
\[
P(L \mid M_1) = \frac{0.09}{0.18} = 0.5
\]

Thus, the probability that the email is legitimate given that the first program marks it as legitimate is \( \boxed{0.5} \).

Would you like further details on this solution or any clarification?

Here are 5 related questions to consider:
1. What is the probability that the email is spam given that Program 1 marks it as legitimate?
2. How does the accuracy of Program 2 affect the overall spam detection system?
3. What would be the result if Program 1 had a different accuracy level?
4. How can the conditional independence assumption of the programs impact the system's performance?
5. How would adding a third program affect the probability calculations?

**Tip**: Bayes' theorem is powerful when updating probabilities based on new information. It’s frequently used in spam filtering, diagnostics, and other decision-making problems.

To battle against spam, Bob installs two anti-spam programs. An email arrives, which is either legitimate (event L) or spam (event Lc), and which program j marks as legitimate (event M) or marks as spam (event Mc) for j ∈ {1,2}. Assume that 10% of Bob’s email is legitimate and that the two programs are each “90% accurate” in the sense that P(M|L) = P(Mc|Lc) = 9/10. Also assume that given whether an email is spam, the two programs’ outputs are conditionally independent. Find the probability that the email is legitimate, given that the 1st program marks it as legitimate (simplify).

Discover how to apply Bayes' Theorem to find the probability that an email is legitimate, given that an anti-spam program marks it as legitimate. This example uses conditional probability, the law of total probability, and real-world data for spam filtering. Suitable for advanced high school or undergraduate students.

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