Before You Begin

Question #41726

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the feasible region in the first quadrant ( $x \ge 0, y \ge 0$ ) satisfying the system of inequalities: $-x - 2y \ge -6$ $2x + y \le 6$ $3x + 4y \le 24$ Let $M$ be the maximum value of the objective function $Z = x + 2y$ over the region $S$ . A biased coin, whose probability of showing heads is $p = \frac{M}{15}$ , is tossed 5 times. Let $X$ be the number of heads obtained. Find the probability that the coin shows heads at least 3 times.

Options

A.
$\frac{992}{3125}$
B.
$\frac{2944}{3125}$
C.
$\frac{2133}{3125}$
D.
$\frac{272}{3125}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Binomial Distribution Tail-Ends|Multiple Optimal Solutions|Sign Flipping Rule|Redundant Constraints

Why This Question Is Tricky

Common Mistakes

This question tests the integration of Linear Programming with Binomial Distribution tail-ends.

The primary trap lies in the first inequality: students often forget to flip the inequality sign when dividing by a negative number, leading to an incorrect feasible region.

Furthermore, the third constraint is redundant.

Failing to identify the correct active constraints and flipping the sign leads to testing the false vertex $(0,6)$ , which yields an incorrect maximum $M=12$ and a trap probability $p=4/5$ , perfectly matching option B.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Binomial Distribution Tail-Ends
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free