Question

Consider the Cartesian plane R² and let X denote the subset of points for which both coordinates are integers. A coin of diameter 1/2 is tossed randomly into the plane. The probability P that the coin covers a point of X satisfies

(A) = π/16

(B) P < π/3

(C) P > π/30

(D) P = 1/4

Answer 1

Answer is (A) P = π/16

See Fig.

Let S denote the set of points inside a square with corners (x, y), (x, y + 1), (x + 1, y), (x + 1, y + 1), x and y are integers. 

Clearly, each of the four points belong to the set X. 

Let P denote the set of points in S with distance less than 1/4 from any corner point. P consists of four quarter circles each of radius 1/4.

A coin, whose centre falls in S, will cover a point of X if and only if its centre falls in P.

Hence,

required probability,