1.  3/4
  2.  1/4
  3.  1/3

Take x and y (each between 0 and 1) to represent the positions of the breaks, as shown. Then if we use x and y as the co-ordinates of a point in a unit square, we can illustrate ranges of values as regions within the square, and probabilities are represented by areas. The diagram represents the solution for Q.2. Because the longest piece is less than 0.5, each piece must be less than 0.5. So if x < y we have: x < 0.5 AND y - x < 0.5 AND 1 - y < 0.5 and if y < x we have: y < 0.5 AND x - y < 0.5 AND 1 - x < 0.5 The red regions represent this area, and the total area is 1/4.