Given 5 points inside a square of side 10 cm, what is the probability that
at least two of the points are within 8cm of each other?
Divide the square into four 5cm subsquares. By the pigeon hole principle at least two of the points must be in the same square.
The largest possible distance between two points in a 5cm square is 5*sqrt(2) = 7.071 cm. This is less than 8cm.
Therefore it is not possible to place five points in a 10cm square and have all the points at least 8cm from each other. In other words, the probability is one that there is a pair of points which are within 8cm of each other.
Solution
