June 2024: Dropped Coin

A coin of radius $r$ is dropped randomly onto a grid of parallel lines spaced $1$ unit apart (where $r < 1/2$ ). What is the probability that the coin crosses a line?

Solution

This is Buffon’s Needle problem adapted for a coin. The probability is:

P = \frac{2r}{1} = 2r

For a penny (radius $\approx 0.75$ cm) on a grid with $2.5$ cm spacing:

P = \frac{2 \times 0.75}{2.5} = 0.6