Hello, alexmahone!

A horizontal flat square board 6 inches width is ruled with a grid of fine lines,

2 inches apart and has a vertical rim like a carrom board, all around the edge.

If a coin of diameter 2/3 inches is tossed on to the board,

what is the probability that it rests without crossing a line?

Each square is 2 inches by 2 inches; there are 9 such squares.

The coin has a radius of $\displaystyle \tfrac{1}{3}$ inches.

To avoid all lines, its center must be more than $\displaystyle \tfrac{1}{3}$ inch from any line.

The center of the coin can be in this shaded region: Code:

: - - - - 2 - - - - :
- *---+-----------+---* -
: | | | | 1/3
: + - + - - - - - + - * -
: | |:::::::::::| | :
: | |:::::::::::| | :
2 | |:::::::::::| | 4/3
: | |:::::::::::| | :
: | |:::::::::::| | :
: + - + - - - - - + - * -
: | | | | 1/3
- *---+-----------+---* -
1/3 4/3 1/3

The shaded area is: .$\displaystyle \left(\frac{4}{3}\right)^2 \:=\:\frac{16}{9}$ inē.

In the nine squares, the shaded area is: .$\displaystyle 9 \times\frac{16}{9} \:=\:16$ inē.

The total area of the board is: .$\displaystyle 6^2 \:=\:36$ inē.

Therefore, the probability is: .$\displaystyle \frac{16}{36}\;=\;\frac{4}{9}$