Probability
Probability
star_batch_jee_advanced_2025
Grade 12
Question:
A square is inscribed in a circle. If $p_1$ is the probability that a randomly chosen point of the circle lies within the square and $p_2$ is the probability that the point lies outside the square, then:
p1 = p2
p1 > p2
p1 < p2
p1^2 - p2^2 = 1/3
Step-by-Step Solution
Key Concept: Compare geometric areas (inscribed square vs circle) to establish probability bounds and verify inequality constraints.
If a circle has radius $a$, the inscribed square has area $2a^2$. Setting $p_1 = \frac{2a^2}{\pi a^2} = \frac{2}{\pi}$ and $p_2 = 1 - p = \frac{\pi - 2}{\pi}$, we verify that $p_1^2 - p_2^2 = (p_1 + p_2)(p_1 - p_2) = \frac{4 - \pi}{\pi} < \frac{1}{3}$ since $4 - \pi \approx 0.86 < \frac{\pi}{3}$.
Correct Answer: 2,4