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Question #41718

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{1, 2, 3, 4\}$ and $S$ be the set of all functions from $A$ to $A$ . The set $S$ is partitioned into three subsets $S_1, S_2, S_3$ as follows:

$S_1$ : The set of all bijective functions. $S_2$ : The set of all functions $h \in S$ that can be expressed as $h = f \circ g$ , where $g: A \to B$ is surjective, $f: B \to A$ is injective, and $B = \{1, 2, 3\}$ . $S_3$ : The set of all remaining functions in $S$ .

A biased die is rolled to select one of the subsets, such that the probability of selecting $S_i$ is proportional to $i$ . Then, a function $h$ is chosen uniformly at random from the selected subset. Given that the chosen function satisfies $h(1) = h(2)$ , what is the probability that it was chosen from $S_2$ ?

Options

A.
$\frac{11}{56}$
B.
$\frac{3}{8}$
C.
$\frac{11}{41}$
D.
$\frac{11}{28}$

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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

Bayes' Theorem Reversal|Composite Injectivity/Surjectivity|Composition Inverse Identity Traps

Why This Question Is Tricky

Common Mistakes

This question tests the 'Composition Inverse Identity Trap' within a Bayesian probability framework.

Students often incorrectly assume a one-to-one correspondence between the pairs of functions $(f, g)$ and their composition $h = f \circ g$ .

If a student calculates $|S_2|$ by multiplying the number of surjective functions $g$ (36) by the number of injective functions $f$ (24), they obtain 864, which absurdly exceeds the total number of functions (256).

This forces the realization that the composition mapping is many-to-one (specifically 6-to-1, as the intermediate set $B$ can be permuted in $3!$ ways).

The problem then requires rigorous combinatorial counting of domain partitions to find the conditional probabilities for Bayes' Theorem, heavily penalizing any structural assumptions about inverse mappings.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bayes' Theorem Reversal
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free