Question

For two events A and B, if \(P(A) = P(A/B) = \frac{1}{4}\) and \(P(B/A) = \frac{1}{2}\), then which of the following is not true?

• A. A and B are independent
• B. \(P(comp(A)/B) = \frac{3}{4}\)
• C. \(P(comp(B)/comp(A)) = \frac{1}{2}\)
• D. None of these

Hint:

When calculating conditional probabilities, always check if events are independent, as this simplifies calculations.

Answer 1

The Correct Option is D

We are given: \[ P(B/A) = \frac{1}{2} \quad \Rightarrow \quad P(B \cap A) = \frac{1}{8} \] \[ P(A/B) = \frac{1}{4} \quad \Rightarrow \quad P(A \cap B) = \frac{1}{4} \times P(B) \] Since \(P(A \cap B) = P(B \cap A)\), we can confirm that A and B are independent events. Now for the other probabilities: \[ P(A') = 1 - P(A), \quad P(B') = 1 - P(B) \] Thus, all the probabilities check out. Therefore, all the options are correct.

Answer 2

Given two events A and B, we have:
1. Probabilities:

• \(P(A) = \frac{1}{4}\)
• \(P(A/B) = \frac{1}{4}\)
• \(P(B/A) = \frac{1}{2}\)

2. Determine independence:
Two events A and B are independent if \(P(A \cap B) = P(A) \cdot P(B)\). For \(P(A \cap B)\), use the conditional probability formula:
\(P(A/B) = \frac{P(A \cap B)}{P(B)}\)
Thus, \(P(A \cap B) = P(A/B) \cdot P(B)\). Given \(P(A/B) = \frac{1}{4}\), we get:
\(P(A \cap B) = \frac{1}{4} \cdot P(B)\)
Now use \(P(B/A) = \frac{P(A \cap B)}{P(A)}\):
\(\frac{1}{2} = \frac{P(A \cap B)}{\frac{1}{4}}\)
which gives \(P(A \cap B) = \frac{1}{8}\). Therefore:
\(P(B) = \frac{1}{2}\)
Check independence:
Since \(P(A \cap B) = \frac{1}{8}\) and \(P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8}\), A and B are independent.

3. Other conditions:

• \(P(comp(A)/B) = 1 - P(A/B) = 1 - \frac{1}{4} = \frac{3}{4}\)
• \(P(comp(B)/comp(A)) = 1 - P(B/A) = 1 - \frac{1}{2} = \frac{1}{2}\)

Conclusion: All statements are consistent based on the given probabilities. Thus, the correct answer is:
None of these