A point P is chosen at random on a straight segment AB of length 12 cm whose midpoint is M. What is the probability that P is nearer to M than to A and also nearer to M than to B?

Measure distances from A, so that A is at 0, M at 6 and B at 12. The point equidistant from A and M sits at 3 cm and the point equidistant from M and B sits at 9 cm, so P must lie between 3 cm and 9 cm. That is a stretch of 6 cm out of 12 cm, giving a probability of 6/12 = 1/2. For a random point on a segment the probability is a ratio of LENGTHS.