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Question #41082

Difficulty: 8/10 | 🟢 Free

The Question

Let $X = \{1, 2, 3, 4, 5, 6, 7\}$ and $Y = \{1, 2, 3\}$ . A function $f: X \to Y$ is chosen uniformly at random. Let $P_c$ be the conditional probability that $f$ is surjective, given that $f(1) + f(2) = c$ , where $c \in \{2, 3, 4, 5, 6\}$ . If $M$ and $m$ are the maximum and minimum values of $P_c$ respectively, then $M - m$ is equal to

Options

A.
$\frac{31}{243}$
B.
$\frac{62}{729}$
C.
$\frac{10}{81}$
D.
$\frac{32}{243}$

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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

De Morgan’s Laws in Probability|Surjective Mapping (Onto)|Arbitrary Constants vs Parameters

Why This Question Is Tricky

Common Mistakes

This question tests the application of De Morgan's Laws and the Inclusion-Exclusion Principle within a conditional probability space defined by a parameter $c$ .

The primary trap is the 'Arbitrary Constant' fallacy: students often assume the probability of surjectivity is independent of $c$ , calculating the unconditional probability ( $602/729$ ) and assuming it applies uniformly.

Furthermore, when evaluating specific cases, students frequently fail to recognize that the cardinality of the initial image set $I = \{f(1), f(2)\}$ dictates the structure of the complement events.

For $c \in \{2, 6\}$ , $I$ has only one element, forcing a full De Morgan's expansion $1 - P(A \cup B)$ for the remaining domain elements, whereas $c \in \{3, 5\}$ requires only a single complement.

Failing to adapt the set-theoretic complement to the parameter $c$ leads to incorrect bounds for $M$ and $m$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	De Morgan’s Laws in Probability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free