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Question #41111

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B,$ and $C$ be three events in a probability space such that $P(A) = P(B) = P(C) > 0$ . Let $I_E$ denote the indicator function of an event $E$ . If the relation $|I_A - I_C| = |I_A - I_B| + |I_B - I_C|$ holds almost surely, and $P(A|B) = P(B|C) = \frac{3}{4}$ , find the value of $P(A|C)$ .

Options

A.
$\frac{1}{2}$
B.
$\frac{9}{16}$
C.
$\frac{5}{8}$
D.
$\frac{3}{8}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Transitivity of Identity|Indicator Function Logic|Sample Space Partitioning

Why This Question Is Tricky

Common Mistakes

This question targets the 'Conditioning Fallacy' and false transitivity in probability.

Students often incorrectly assume that conditional probabilities behave transitively like a Markov chain (i.e., assuming $P(A|C) = P(A|B)P(B|C) = 9/16$ ), especially when events are highly correlated.

By mapping the $L^1$ distance of indicator functions to the probabilities of symmetric differences, the problem enforces a strict set-theoretic geometry ( $A \cap C \subseteq B \subseteq A \cup C$ ).

This forces the student to abandon heuristic probability rules and rigorously apply expectation operators and set algebra to find the exact intersection probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Transitivity of Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free