Consider the Cartesian plane `R^(2)` and let `X` denote the subset of points for which both coordinates are integer. A coin of diameter `1//2` is tossed randomly onto the plane. The probability `p` that the coin covers a point of `X`

Options

A. `0.2`
B. `0.8`
C. `1.2`
D. None of these

Answer

A

Explanation

`(a)` Let `S` denote the set of points inside a square with corner `(a,b)`, `(a,b+1))`, `(a+1,b)`, `(a+1,b+1)`
Let `P` denotes the set of points in `S` with distance less than `1//4` from any corner point.
Thus a coin whose centre falls in `S` will cover a point of `X` if and only if its centre falls in a point of `P`
`:.` Required probability `=("Area of " P)/("Area of" S)=(pi(1//4)^(2))/(1)=(pi)/(16)=0.2`