A coin of diameter $1/2$ is tossed randomly onto the rectangular cartesian plane. The probability that the coin does not intersect any line whose equation is of the form $x = k$, or $y = k,k$ is integer, is:
Step-by-Step Solution
Key Concept: The coin has diameter 1/2 (radius 1/4), so its center must stay at least 1/4 units away from any line x=k or y=k to avoid intersection. Within a unit square [0,1]×[0,1], the safe region for the center is a smaller square of side length 1/2 centered in the middle, giving probability (1/2)²/1² = 1/4.
For adjacent lines $x = k$ and $x = k+1$ with $y = k$ and $y = k+1$ from a unit square, the favorable event occurs when a coin falls into the shaded square of side length $\frac{1}{2}$. The desired probability is $\frac{\text{area of shaded square}}{\text{area of larger square}} = \frac{(1/2)^2}{1^2} = \frac{1}{4}$.
Correct Answer: 3