A stick is 5 units long. It is broken at two points, chosen at random. What is the probability that all three pieces are longer than 1 unit?
The probability is 4/5*4/5 = 16/25.
When we break off the first piece, the probability that the first piece will be longer than 1 unit is 3/5. (This is because we look at the stick as the interval [0,5], then we can break the stick anywhere between 1 and 4.) We are now left with a stick with a length of 4. When we break off the second piece, the probability that the second piece will be longer than 1 is 2/4 = 1/2. So the answer is 3/5*1/2 = 3/10.