Rolling a Die 600 and 5000 Times

Date: 06/03/2001 at 11:47:40
From: Tory Swanson
Subject: Rolling a die 600 times and 5000 times and frequency 
occurring

1. Toss a die 600 times. How many times does each number come up? 
Approximately how many would you expect?

2. Toss 2 die 5000 times. How many times does each number come up? 
Approximately how many would you expect?

How can you use Microsoft Excel to solve these problems? I understand 
that this involves the law of large numbers and you would expect 1/6 
for a number coming up when rolling one die out of 6^600 outcomes, and 
you would expect (1/36 for 2; 2/36 for 3; etc., etc.) when rolling 2 
dice out of 11^5000 outcomes.

I just need a little insight to get this problem started.  Thanks for 
your time and consideration.

Date: 06/07/2001 at 13:20:03
From: Doctor Twe
Subject: Re: Rolling a die 600 times and 5000 times and frequency 
occurring

Hi Tory - thanks for writing to Dr. Math.

First, I would treat each of the problems as two separate problems. 
The first part of the problem is to use a spreadsheet program (in this 
case, Excel) to simulate rolling dice n times and summing the results. 
The second part of the problem is to use probability to determine the 
expected number of each result.

For the first part, you can use Excel's random number generating 
function to simulate the roll of a die. Use the formula:

     =INT(6*RAND())+1

to simulate the roll of one die. For question 1, you will need to copy 
this function into 600 different cells to simulate 600 different rolls 
of the die. I would put them in a column, for example, in cells A1 
through A600.

For question 2, you'll need to copy it to 10,000 different cells and 
add the pairs. I would copy the formula into two columns of 5,000 each 
(for example, A1:A5000 and B1:B5000) then use a third column to total 
each pair. For example, in cell C1 use the formula:

     =A1+B1

then copy that formula to cells C1 through C5000. (Note that Excel 
will automatically adjust the cell row references in the formula when 
copying, as long as you don't put a '$' in front of the number part of 
the cell address. That is, use =A1+B1, not =$A$1+$B$1.)

Next, you'll have to count the number of times each result occurred. 
You can use Excel's COUNTIF() function to do this. To count the number 
of times a 1 occurred in cells A1 through A600, for example, you'd 
use:

     =COUNTIF($A$1:$A$600,1)

where the first parameter is the cell range and the second parameter 
is the result number you're counting up. You can use similar functions 
to count the number of 2's, 3's, etc. (Note: for question 1 you'll 
need to count 1's, 2's, ..., 6's; while for question 2 you'll need to 
count 2's, 3's, ..., 12's.)

That covers the simulation of rolling the dice. Note that every time 
you "recalculate" the spreadsheet - including any time you re-open it 
after saving it - you'll get a brand new set of rolls, and the results 
will be different.

For the second part of the questions, you'll have to use probability 
formulas. You have the right idea when you say that the probability of 
rolling a 1 is 1/6, but the number of rolls is 600, not 6^600. So the 
expected number of 1's would be:

     E(1) = n * P(1)
          = 600 * (1/6)
          = 100

You can compute the expected number for each of the other outcomes. 
Remember that the sum of all expected number of outcomes (for each 
question separately) should equal the total number of rolls made.

I hope this helps. If you have any more questions, write back.

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

Date: 06/07/2001 at 17:48:53
From: Tory Swanson
Subject: Re: Rolling a die 600 times and 5000 times and frequency 
occurring

Thank you - I didn't even think about a random number generator.

Tory