Solution
Coin on the Tiles
Show the problem again
A floor is covered with square tiles of side 1, separated by grout lines of negligible width. A coin of diameter 1/3 is tossed onto the floor so that its centre is uniformly distributed over the tile it lands on. What is the probability that the coin lies entirely inside a single tile, touching no grout line? Give a reduced fraction.
Worked solution
The answer is 4/9. The coin misses every line exactly when its centre is at distance more than its radius, 1/6, from all four sides of its tile. Those centres form a smaller concentric square of side 1 − 2·(1/6) = 2/3. Its area relative to the tile is (2/3)² = 4/9. The general lesson: a coin of diameter d on unit tiles stays clear with probability (1 − d)², which falls off much faster than intuition suggests. A coin one third the width of a tile still touches a line more than half the time.
Source: Classic carnival-game exercise in geometric probability, in wide circulation; no traceable original source. Statement written for AxiomIQ.