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Question #41807

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_1, E_2,$ and $E_3$ be a partition of a sample space. The prior probabilities of these events are given by: $P(E_1) = \frac{[x]}{5}, \quad P(E_2) = \frac{[x^2]}{5}, \quad P(E_3) = \frac{[x^3]}{5}$ where $x > 0$ and $[\cdot]$ denotes the greatest integer function. Let $S$ be the set of all possible values of $x$ for which this partition is valid. If $S = [a, b)$ , an observable event $A$ occurs such that the conditional probabilities are $P(A|E_1) = \frac{1}{2}$ , $P(A|E_2) = \frac{3}{4}$ , and $P(A|E_3) = \frac{1}{4}$ . Find the value of $(a^2 + b^3) P(E_2|A)$ .

Options

A.
2
B.
3
C.
4
D.
5

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Bayes' Theorem Reversal|Sample Space Partitioning|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate the Greatest Integer Function (GIF) into the foundational axioms of probability, specifically the sample space partitioning condition $\sum P(E_i) = 1$ .

The primary trap lies in the piecewise nature of the GIF; students often attempt to solve the algebraic equation $x + x^2 + x^3 = 5$ directly, ignoring the step-function boundaries, or they fail to correctly intersect the domain constraints for $[x^2]$ and $[x^3]$ .

By embedding this within a Bayes' Theorem reversal, the problem forces a dual cognitive load: rigorously establishing the valid parameter interval $S = [a, b)$ through inequality intersections, and subsequently applying the derived constant priors to compute the posterior probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bayes' Theorem Reversal
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free