Ken's Question from Yahoo Answers: Probability Question For Verification?

CaptainBlack

Well-known member
Question:
"A point $${\rm{P}}(x,y)$$ is chosen at random in a unit disc, centred at $$(0,0)$$.

The probability required is that the point chosen is such that both $$| x -y| \lt 1$$ and $$|x+y| \lt 1$$ .

Is the answer $$2/\pi$$ or $$1-2/\pi$$?

Thank you."

The region defined by the inequalities $$|x-y| \lt 1$$ and $$|x+y| \lt 1$$ is an inscribed square to the circle, which has side $$\sqrt{2}$$ and hence area $$2$$. The area of the circle is $$\pi$$, so the probability that a point sampled uniformly on the unit disc satisfies the inequalities is the ratio of these two area: $$2/\pi$$.
To convince yourself that the required region is the interior of the square rather than the exterior consider the point $$(0,0)$$, does it satisfy the inequalities. It it does then you want the interior of the square rather than the exterior.