Before You Begin

Question #41028

Difficulty: 8/10 | 🟢 Free

The Question

Let $A, B,$ and $C$ be three independent events. A matrix $M$ is defined as: $M = \begin{bmatrix} 1 & P(A) & P(B) & P(C) \\ 1 & 1 & P(B) & P(C) \\ 1 & P(A) & 1 & P(C) \\ 1 & P(A) & P(B) & 1 \end{bmatrix}$ Let $N$ be the matrix obtained by swapping the 2nd and 3rd columns of $M$ . If $3\det(N) + 5P(A' \cap B' \cap C') = \frac{1}{29}$ , find the probability that at least one of the events $A, B,$ or $C$ occurs.

Options

A.
$\frac{57}{58}$
B.
$\frac{231}{232}$
C.
$\frac{1}{58}$
D.
$\frac{1}{232}$

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

Complementary Counting|Independent vs Disjoint Events|Minors and Cofactors Sign Convention|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of determinant properties and probability (complementary counting and independence).

The primary trap lies in the column exchange parity: students frequently forget that swapping two columns multiplies the determinant by -1.

If this parity flip is ignored, the equation yields $8P(A' \cap B' \cap C') = \frac{1}{29}$ , leading to the trap answer $\frac{231}{232}$ .

Furthermore, the matrix rank and structure obfuscate the direct calculation of $P(A' \cap B' \cap C')$ , rewarding students who efficiently use row operations to reveal the diagonalized form of the complementary probabilities.

Quick Facts About This Question

Question Information

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