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Question #41058

Difficulty: 8/10 | 🟢 Free

The Question

A real number $x$ is chosen uniformly at random from the interval $[1, N)$ , where $N$ is an integer. The mapping $f: [1, N) \to \{0, 1, 2, 3\}$ defined by $f(x) = [\log_2 x]$ is surjective. Let $A$ be the event that $f(x)$ is an even integer, and $B$ be the event that $\{ \log_2 x \} \ge \frac{1}{2}$ , where $[\cdot]$ and $\{\cdot\}$ denote the greatest integer and fractional part functions respectively. If $A$ and $B$ are independent events, what is the probability of the event $A \cup B$ ?

Options

A.
$\frac{5 - 2\sqrt{2}}{3}$
B.
$\frac{4 - \sqrt{2}}{3}$
C.
$\frac{7 - 3\sqrt{2}}{3}$
D.
$\frac{5 - \sqrt{2}}{3}$

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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

Mutually Exclusive vs Independent|Surjective Mapping (Onto)|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of continuous probability, mutually exclusive interval partitioning via the Greatest Integer Function (GIF), and event independence.

The primary trap is conflating the discrete jumps of the GIF with continuous probability, leading students to incorrectly partition the sample space or miscalculate the lengths of valid intervals.

Furthermore, the surjectivity condition acts as a strict constraint on the sample space boundary $N$ .

Students often mistakenly assume that events defined by the same underlying function (here, $\log_2 x$ ) must be dependent, ignoring the specific fractal-like symmetry of the fractional part over powers of 2 that uniquely creates independence at $N=16$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Mutually Exclusive vs Independent
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free