geometric probability

[Sendell]

The following is a math contest problem:
Three numbers are chosen in the interval [0,2]. what is the probability the sum is at least one and no more than two?

Is the only way to do this is with a 3d graph? If so, how? Thanks.

 

[soroban]

Hello, Sendell!

Three numbers are chosen in the interval [0,2].
What is the probability the sum is at least one and no more than two?

Is the only way to do this is with a 3d graph? . . . . I think so
If so, how?

I may be totally wrong with my approach.
If I am, someone point it out . . . please!


Let the three numbers be \(\displaystyle x,\,y,\,z.\)
. . Their sum can range from \(\displaystyle 0\) to \(\displaystyle 6.\)
We have the plane: \(\displaystyle x\,+\,y\,+\,z\:\leq\:6\) in the first octant.

                |
              6 *
               *| *
              * |   *
             *  |     *
            *   |       *
           *    |         * 6
          *     * - - - - - * - - 
         *    /         *
        *   /       *
       *  /     *
      * /   *
     */ *
  6 *
  /

The volume of a pyramid is: \(\displaystyle \,V\:=\:\frac{1}{3}Bh\)
. . where \(\displaystyle B\) is the area of the base and \(\displaystyle h\) is the height.

This pyramid has a volume of: \(\displaystyle 36\) units³.


For the sum to be between 1 and 2, we want the volume between the planes:
. . \(\displaystyle x\,+\,y\,+\,z\:=\:1\) and \(\displaystyle x\,+\,y\,+\,z\:=\:2\)
(You should be able to find this.)


Then the probability is the ratio of the volumes.