To determine the probability that a product is fake given a positive test result, we use Bayes' theorem. The test has a false positive rate (fpr) defined as the probability the test is positive when the product is fake (i.e., sensitivity), and a false negative rate (fnr) defined as the probability the test is negative when the product is genuine (i.e., specificity). The prior probability that a product is fake is ( p ). The probability that a product is fake given a positive test result, ( P(\text{Fake} \mid \text{Positive}) ), is calculated as:

[ P(\text{Fake} \mid \text{Positive}) = \frac{\text{fpr} \times p}{\text{fpr} \times p + (1 - \text{fnr}) \times (1 - p)} ]

Explanation:

1. Numerator: (\text{fpr} \times p)

   • This represents the joint probability that the product is fake and the test is positive.

2. Denominator: (\text{fpr} \times p + (1 - \text{fnr}) \times (1 - p))

   • This is the total probability of a positive test result, considering both fake and genuine products:
     • (\text{fpr} \times p): Probability the test is positive when the product is fake.
     • ((1 - \text{fnr}) \times (1 - p)): Probability the test is positive when the product is genuine (since (1 - \text{fnr}) is the false positive rate in standard terms).

Example Calculation:

Suppose:

• Prior probability of fake product, ( p = 0.1 )
• False positive rate (fpr) = 0.9 (test correctly identifies fake products 90% of the time)
• False negative rate (fnr) = 0.8 (test correctly identifies genuine products 80% of the time)

Then: [ P(\text{Fake} \mid \text{Positive}) = \frac{0.9 \times 0.1}{0.9 \times 0.1 + (1 - 0.8) \times (1 - 0.1)} = \frac{0.09}{0.09 + 0.2 \times 0.9} = \frac{0.09}{0.09 + 0.18} = \frac{0.09}{0.27} \approx 0.333 ]

Thus, there is a 3% chance the product is fake given a positive test result.

Key Insight:

Even with a high sensitivity (fpr) and specificity (fnr), the prior probability ( p ) significantly impacts the result. If fake products are rare (( p ) is small), a positive test is more likely due to false positives from genuine products. Always consider the base rate (prior probability) when interpreting test results.