Fraudulent Product Reviews

Solution & Explanation

To solve this problem, we need to determine the probability that a review is actually fraudulent given that the algorithm has flagged it as fake. This is a classic case of applying Bayes' Theorem. Let's break down the problem step-by-step:

Step 1: Define the Events

• F: The event that a review is fraudulent.
• L: The event that a review is legitimate.
• D: The event that a review is detected as fake by the algorithm.

Step 2: Given Probabilities

• P(F) = 0.02 (2% of reviews are fraudulent)
• P(L) = 0.98 (98% of reviews are legitimate)
• P(D|F) = 0.95 (95% probability the algorithm correctly flags a fraudulent review)
• P(D|L) = 0.10 (10% probability the algorithm incorrectly flags a legitimate review as fake)

Step 3: Bayes' Theorem

Bayes' Theorem allows us to find the probability of an event based on prior knowledge of conditions related to the event. The theorem is stated as:

$P(F|D) = \frac{P(D|F) \cdot P(F)}{P(D)}$

Where:

• P(F|D): Probability that a review is fraudulent given it is detected as fake.
• P(D): Total probability that a review is detected as fake.

Step 4: Calculate P(D)

The total probability that a review is detected as fake, P(D), can be calculated using the law of total probability:

$P(D) = P(D|F) \cdot P(F) + P(D|L) \cdot P(L)$

Substitute the known values:

$P(D) = 0.95 \cdot 0.02 + 0.10 \cdot 0.98$ $P(D) = 0.019 + 0.098$ $P(D) = 0.117$

Step 5: Calculate P(F|D)

Now, substitute the values into Bayes' Theorem:

$P(F|D) = \frac{0.95 \cdot 0.02}{0.117}$ $P(F|D) = \frac{0.019}{0.117}$ $P(F|D) \approx 0.162$

Conclusion

The probability that a review is actually fraudulent when the algorithm flags it as fake is approximately 16.2%. This means that even when the algorithm identifies a review as fake, there is still a significant chance that the review could be legitimate due to the lower prevalence of fraudulent reviews and the algorithm's false positive rate.