Random is no different, it just takes more samples. An example:

Say I give you these results:

       *
     *     * 
                *   
  *        *
              *

No real pattern, yea? So sample some more:

       *
     *     *  *
        *       * * *
  *   *  * *
              *

Maybe nothing. So try more:

       *                *
     *     *  *
        *    ** *** *
  * * ** * *     *
        *      *

And more:

    *  * *    *    *
     *     *  *  *
    *   *   *********
  **********    *  *
   *   *  **  *   *

Zomg. You have a discernable behavior. Adding more randomness would just give you the same easily-visible results after adding, say, 2x as many points, at which point you have this (expanded a little):

    *  *  * *  * *    *
     **  *  ** * * * *
    *   * *** * **********
  ***************    *  *
   *   *  ** ** * *   *