Before You Begin

Question #41889

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_1, E_2, E_3, \dots$ be a countably infinite set of mutually exclusive and exhaustive events. If the probability of occurrence of event $E_k$ is given by $P(E_k) = \lambda \frac{|k - 4|}{3^k}$ for all $k \in \mathbb{N}$ , then the value of $\lambda$ is

Options

A.
$\frac{108}{137}$
B.
$\frac{4}{5}$
C.
$\frac{108}{133}$
D.
$\frac{27}{34}$

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

Exhaustive Events Concept|Sigma Notation Properties|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of probability axioms (exhaustive events summing to 1) and the evaluation of infinite arithmetico-geometric series containing a modulus.

The primary trap lies in the modulus function $|k-4|$ .

Students frequently ignore the piecewise nature of the modulus, either dropping it entirely or assuming $|k-4| = 4-k$ for all $k \in \mathbb{N}$ because the first few terms are positive.

This leads to the incorrect sum $\sum (4-k)/3^k = 5/4$ , yielding the trap answer $4/5$ .

Correct execution requires splitting the domain at the non-differentiable turning point $k=4$ and evaluating the resulting finite and infinite series separately.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Exhaustive Events Concept
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free