A stick is broken at two points, chosen at random. If the length of the stick is 6 then what is the probability that all three resulting pieces are shorter than 2 units?
To solve the problem, we can use the following steps:
1. Let x be the length of the shortest piece, y be the length of the middle piece, and z be the length of the longest piece.
2. Since the stick has length 6, we have x + y + z = 6.
3. We want to find the probability that all three resulting pieces are shorter than 2 units, which means x < 2, y < 2, and z < 2.
4. Substituting z = 6 - x - y and rearranging, we get x + y < 4.
5. This inequality defines a triangular region in the plane, where x and y are the coordinates. The vertices of the triangle are (0,0), (0,4), and (4,0).
6. The total area of the rectangle that contains the triangle is 6 x 6 = 36. Therefore, the probability that the stick is broken in a way that all three resulting pieces are shorter than 2 units is equal to the area of the triangle divided by the area of the rectangle.
7. The area of the triangle is (1/2) x 4 x 4 = 8, so the probability is 8/36 = 2/9.
Therefore, the probability that all three resulting pieces are shorter than 2 units is 2/9.
