Most Frequently Rolled Number

Date: 10/27/1999 at 21:54:54
From: Zoe Nesbitt
Subject: Determining odds

If I had two dice each with the following numbers on them: -3, -2, 
-1, 0, 1, and 2, and rolled them 100 times, what sum would be rolled 
most often and why? Can you help?

Date: 10/28/1999 at 03:29:20
From: Doctor Ian
Subject: Re: Determining odds

Hi Zoe, 

Let's look at the case for regular dice, numbered 1-6.  Here are the
possible sums:

                   second
                   die

           +    1  2  3  4  5  6
              ------------------
           1 |  2  3  4  5  6  7 
           2 |  3  4  5  6  7  8
  first    3 |  4  5  6  7  8  9
  die      4 |  5  6  7  8  9 10
           5 |  6  7  8  9 10 11 
           6 |  7  8  9 10 11 12

How often does each sum appear in the table? 

  Sum  Appears
  ---  -------
   2   1 time
   3   2 times
   4   3 times
   5   4 times
   6   5 times
   7   6 times
   8   5 times
   9   4 times
  10   3 times
  11   2 times
  12   1 time

Now, suppose you had 36 balls, and you marked them this way:

  Mark  Number of balls
  ----  ---------------
   2    1 
   3    2 
   4    3 
   5    4 
   6    5 
   7    6 
   8    5 
   9    4 
  10    3 
  11    2 
  12    1 

Put the balls in a hat, and draw one out.  Toss it back in, and draw one
out again.  If you keep doing this, which mark would you expect to draw
most often?

Do you see why the problem with the dice and the problem with the balls
are really just two versions of the same problem?  Can you see how this
relates to _your_ problem? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/