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Question #40700

Difficulty: 8/10 | 🟢 Free

The Question

Let $X$ be a random variable taking values in $\{1, 2, 3\}$ with probabilities $p_1, p_2, p_3$ respectively. It is given that $p_1, p_2, p_3$ form a strictly decreasing Harmonic Progression and their product is $\frac{3}{128}$ . An experiment consists of 4 independent observations of $X$ . Let $E$ be the event that the value 2 is observed exactly once, and $F$ be the event that the value 2 is observed at least once. The conditional probability $P(E|F)$ is:

Options

A.
$\frac{108}{175}$
B.
$\frac{27}{64}$
C.
$\frac{81}{175}$
D.
$\frac{135}{256}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Harmonic Mean Constraints|Axiomatic Probability Constraints|Complementary Counting

Why This Question Is Tricky

Common Mistakes

The core trap lies in the intersection of Harmonic Progressions and axiomatic probability constraints.

Students often set up the cubic equation for the probabilities but fail to recognize that the sum of the HP terms equaling 1 imposes a strict discriminant bound ( $S \ge 2/3 \implies p_2 \le 1/3$ ) for the probabilities to remain real numbers.

Without checking this axiomatic constraint, they might select the extraneous root $x = (3+\sqrt{21})/8 \approx 0.94$ , which is in $[0,1]$ and satisfies the algebraic product, but yields complex probabilities for $p_1$ and $p_3$ , completely derailing the complementary counting phase.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Harmonic Mean Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free