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Question #41804

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be two independent events associated with a random experiment. The probabilities of these events are given by: $P(A) = \frac{1}{3^{20}} \sum_{r=0}^{10} \binom{20}{2r} 2^{2r}$ $P(B) = \frac{1}{2^{20}} \sum_{r=0}^{20} \binom{20}{r} \sin^2\left(\frac{r\pi}{2}\right)$ Find the value of $\lfloor 4 \cdot P(A \cup B) \rfloor$ , where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .

Options

A.
2
B.
3
C.
4
D.
5

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

Independent vs Disjoint Events|Properties of Binomial Coefficients|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to evaluate complex binomial sums and apply the correct probability rules for independent events.

The primary trap lies in confusing 'independent' events with 'disjoint' (mutually exclusive) events.

If a student incorrectly assumes $P(A \cup B) = P(A) + P(B)$ , they will calculate $P(A \cup B) = 1 + \frac{1}{2 \cdot 3^{20}}$ .

Multiplying by 4 yields $4 + \frac{2}{3^{20}}$ , which results in a Greatest Integer Function (GIF) value of 4.

The GIF acts as a strict precision check, ensuring that this conceptual error leads to a definitively incorrect integer option, penalizing the conflation of independence and disjointness.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Independent vs Disjoint Events
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free