A coin is randomly thrown on an infinite square grid. What is the probability that coin does not touch the grid lines?

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A coin is randomly thrown on an infinite square grid. What is the probability that the coin does not touch the grid lines?
Diameter of coin : 1cm
Width of square grid : 2cm

posted Sep 30, 2018

Coin when flipped and allowed to land on an infinite grid it should always be closer to one of the infinite squares or fall on the grid itself (to be equidistant from 2 or 4 squares).
So to find the probability that the coin won't land on the grid lines we have to find the area in a square where if the coin's centre lands the coin doesn't touch the grid lines.
This will happen if the centre of the coin stays at 1 radius distance away from the borders of the square. This means the allowable area in the square where the event favourable to the required probabity happens = (2 - 1)^2 = 1 cm^2
Required Probability = 1 cm^2 / 4 cm^2 = 0.25
ie., there is only a 25% chance that the coin will not land on the grid lines of an infinite grid.