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Question #40696

Difficulty: 8/10 | 🟢 Free

The Question

Let $\{I_n\}_{n=1}^{\infty}$ be a sequence of independent and identically distributed indicator random variables such that $P(I_n = 1) = p$ and $P(I_n = 0) = 1 - p$ , where $p \in (0, 1)$ . A random variable $S$ is defined by the infinite series $S = \sum_{n=1}^{\infty} I_n p^n$ . If the expected value of $S$ is given by $E[S] = \frac{1}{6}$ , then the probability that $S \ge \frac{10}{27}$ is equal to:

Options

A.
$\frac{5}{27}$
B.
$\frac{1}{9}$
C.
$\frac{4}{27}$
D.
$\frac{2}{9}$

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

Selection of GP Terms|Infinite GP Convergence|Indicator Function Logic|Combinatorial Probability

Why This Question Is Tricky

Common Mistakes

This question fuses infinite geometric series convergence with combinatorial probability and indicator functions.

The primary trap lies in the student's handling of the boundary condition $S \ge \frac{10}{27}$ .

A common mistake is to assume $S$ follows a continuous uniform distribution (which is only true if $p=1/2$ ) and attempt to integrate, or to incorrectly evaluate the maximum residual sum of the geometric series $\sum_{k=m}^{\infty} p^k$ , leading to erroneous inclusion of sequences like $I_1=1, I_2=0, I_3=0$ .

Furthermore, failing to reject the negative root of the expectation quadratic $6p^2+p-1=0$ due to ignoring the domain constraint $p \in (0,1)$ will derail the solution immediately.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Selection of GP Terms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free