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Question #41814

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all positive integers $n$ for which there exists a positive integer $m$ satisfying the system of linear inequalities: $3n - 2m \ge 14$ $n + m \le 6$ A biased coin, whose probability of getting heads is $p = \max_{x > 0} \left( \frac{x^2}{x^4 + 4} \right)$ , is tossed $n$ times for each $n \in S$ . Let $P_n$ be the probability of getting exactly 2 heads in $n$ tosses. Find the value of $4096 \sum_{n \in S} P_n$ .

Options

A.
0
B.
1080
C.
1215
D.
2295

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

Bernoulli Trials Logic|Arithmetic Mean Inequality|System of Inequalities

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to rigorously evaluate the domain of discrete parameters before applying probability formulas.

The primary trap lies in the system of linear inequalities bounding the number of trials $n$ .

Students commonly relax the strict 'positive integer' constraint on the auxiliary variable $m$ , incorrectly assuming $m \ge 0$ or $m \in \mathbb{R}$ , which yields false valid values for $n$ (like 5 or 6) and leads to calculating a non-zero binomial probability.

By strictly enforcing $m, n \in \mathbb{Z}^+$ , the system yields an empty set for $n$ , nullifying the Bernoulli trials and making the sum of probabilities exactly zero.

Quick Facts About This Question

Question Information

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