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:
Step-by-Step Solution
Key Concept: For a square inscribed in a circle of radius a, the diagonal equals 2a (diameter), making the side length a√2 and area 2a². Geometric probability requires comparing this area (2a²) to the circle's area (πa²) to find p₁ = 2/π and p₂ = (π-2)/π, then verify algebraic relationships.
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