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Question #41839

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{x \in \mathbb{Z} : -20 \le x \le 20, x \neq 0\}$ . An integer $x$ is chosen at random from $S$ with equal probability. Consider the infinite geometric series: $P = 1 + \left(\frac{8}{x}\right) + \left(\frac{8}{x}\right)^2 + \left(\frac{8}{x}\right)^3 + \dots$ What is the probability that the series converges to a finite sum $P$ such that $P < 2$ ?

Options

A.
$\frac{2}{5}$
B.
$\frac{1}{10}$
C.
$\frac{31}{40}$
D.
$\frac{19}{40}$

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

Equiprobable Outcomes|Infinite GP Convergence|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of probability, infinite geometric series convergence, and rational inequalities.

The primary trap is the 'Cross Multiplication Trap'.

When solving the inequality $\frac{x}{x-8} < 2$ , students frequently cross-multiply to get $x < 2x - 16 \implies x > 16$ , implicitly and incorrectly assuming that the denominator $x - 8$ is always positive.

This error causes them to completely miss the negative solution set $x \in (-\infty, -8)$ , leading them to find only 4 valid outcomes and incorrectly select the trap option 1/10.

A secondary trap involves forgetting to apply the strict convergence condition $|r| < 1$ , which would lead to including extraneous solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Equiprobable Outcomes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free