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Probability — the likelihood of something happening — can be a challenging concept for kids to grasp. However, it is an important idea to understand, both as a foundation for studying statistics, and just for comprehending the random events that happen throughout life.

Probability theory is the branch of mathematics that studies probability, and pretty complex it is too. However, it’s easy to give your kid a simple, fun taste of probability in action, using nothing more than two dice and a simple spreadsheet! Along the way they’ll get to practice their addition skills, learn how to create spreadsheets, find out how to use formulas and charts in spreadsheets, and get to grips with the concept of simulation.

This project is fairly straightforward to do, but it does involve a fair amount of maths, reading and computer skills. It’ll work best for ages 8 and up.

Read on to find out how you can do this fun, easy project with your child!

The Count of Monte Carlo

The project you’re going to do uses the Monte Carlo method, named after the famous Monte Carlo Casino in Monaco. The basic idea is to show how random events — in this case, rolling dice — can, if done enough times, produce a statistical pattern.

Here are the experiments you’re going to do:

Experiment 1: One die.
Roll one die several times, and view the results in a spreadsheet chart.
Experiment 2: Two dice.
Repeat the experiment with two dice.
Experiment 3: Simulated dice.
Repeat the two-dice experiment, replacing real rolls with simulated rolls. This will let you easily “roll” the dice thousands of times!
Experiment 4: More dice.
Try the experiment with six dice instead of two, and look at how the chart pattern changes.

What you’ll need

All you need to do these experiments are:

Two dice
An hour or so of time, and
A spreadsheet app. I’ve used Apple’s Numbers app on the Mac, but you can just as easily use Numbers on the iPad, Microsoft Excel on a PC, or OpenOffice Calc.

To save time, you can download all the spreadsheets used in this article. I’ve included them in Numbers, Excel and OpenOffice formats.

Experiment 1: One die

Let’s start a probability experiment with just one die. Follow these steps:

Step 1: Create a new blank spreadsheet and call it Monte Carlo (One Die).
Step 2: Create a column called Die Roll.
Step 3: Roll one die 10 times, and type each result into a new row in your Die Roll column, like this:

Creating a Die Roll column and recording die rolls.

Step 4: Create two more columns called Roll and Frequency.
Step 5: In the Roll column, add the numbers 1 to 6, one number per row.
Step 6: In the Frequency column, create a new cell next to the “1” column. In this new cell, you want to create a formula that uses the COUNTIF() function to count all the occurrences of the number 1 in your Die Roll column. In Numbers this is easily done, like this:

Using the COUNTIF() function in Numbers. COUNTIF() counts all occurrences of a value in a range of cells.

In Excel, you can use =COUNTIF(A:A,'=1') (assuming your Die Roll column is column A). In OpenOffice Calc, you need to do something along the lines of =COUNTIF($A$2:$A$9999;'=1').

Step 7: Click this new cell in the Frequency column, and choose Edit > Copy.
Step 8: Select the remaining 5 rows in the Frequency column, and choose Edit > Paste.
Step 9: Go through the formula in each new cell, changing the '=1' to the new number: '=2', '=3' and so on.

You should now have a neat summary that shows how often each number cropped up during your die rolls:

Our spreadsheet data. The raw die roll data is on the left, and the summary data is on the right.

Step 10: Let’s plot these figures on a chart so we can see the data more easily. Click the Roll column’s header button (probably called A, B or C), then hold down Shift and click the Frequency column’s header button to select both columns.
Step 11: Now choose Insert > Chart, and choose a simple 2-D column chart. Make sure you select the “First column as label” option, if shown. Your chart should look something like this:

This chart shows the frequency distribution after 10 rolls of the die.

Step 12: Keep rolling the die and adding the results to the Die Roll column. As you add more and more rolls, you’ll see the chart start to even out. Here is my chart after 50 rolls:

And after 150 rolls:

The distribution after 150 rolls of a single die.

As you’d expect, the more results you add to the table, the more the chart evens out, since each number on the die has a roughly equal chance of coming out on top. Eventually, after thousands of rolls, each column in the chart would be more or less the same height. (For the record, this type of pattern is known as a discrete uniform distribution.)

From this your kid can see that, even though each throw of the die is unpredictable, we can predict the overall way that the die will behave. This is probability theory in action!

Experiment 2: Two dice

Let’s try the same experiment with two dice.

Step 1: Make a copy of the “one die” spreadsheet and call it Monte Carlo (Two Dice).
Step 2: Use Edit > Copy and Edit > Paste to extend the Roll and Frequency columns so that the Roll column goes up to 12. Don’t forget to edit the '=x' value inside each row’s COUNTIF() function: '=7', '=8', '=9' and so on.
Step 3: Delete the “1” row (since of course you can’t roll a 1 with two dice!). Your new spreadsheet should look something like this:

Altering the spreadsheet to track two dice instead of one.

Step 4: You might also need to update your chart to include the new rows in the Roll and Frequency columns. For example, in OpenOffice, double-click the chart and choose Format > Data Ranges. Alternatively, just delete the chart, select both the Roll and Frequency columns, and create a new chart.
Step 5: Delete all the data in the Die Roll column, and rename the column to Dice Roll.
Step 6: Roll both dice, and add the total to the first row in the Dice Roll column. As you add the results, the chart starts to fill up. Here’s my chart after 10 rolls:

After 10 rolls with two dice, my chart looked like this.

Step 7: Keep rolling and adding values, and after a while you should see a pattern start to emerge. Here’s my chart after 100 rolls:

Notice that the pattern in this chart looks quite different to the chart for the one-die experiment.

Why do they look different? The answer is that, for one die, each number has an equal chance of coming up. For two dice, however, the total values have different chances of coming up. For example, there is only one way to roll a 2 (1+1), but there are two ways to roll a 3 (2+1 or 1+2). So a 3 is twice as likely to come up as a 2.

This type of pattern, by the way, is known as triangular distribution, and it’s not hard to see why!

Experiment 3: Simulated dice

How many dice rolls would you need to make a perfect triangular chart? Quite a lot!

Rather than tiring ourselves out rolling dice hundreds of times, let’s make the spreadsheet do the hard work! We’ll create a dice-rolling simulation.

Step 1: Make a copy of the “two dice” spreadsheet and call it Monte Carlo (Two Dice, Simulated). Delete all the values in the Dice Roll column.
Step 2: Click inside the Dice Roll column, and add two columns before this column. In Numbers, you can do this by choosing Table > Add Column Before. In OpenOffice, choose Insert > Columns. Call the first column Die 1 Roll, and the second column Die 2 Roll.
Step 3: Now we need to simulate some dice rolls. Handily, most spreadsheets come with a RANDBETWEEN() function that lets us add a random number to a cell. You want to add the following to the topmost cell in the Die 1 Roll column:

=RANDBETWEEN(1,6)

(In OpenOffice, you need to use a semicolon instead of the comma.) This “rolls” a random number between 1 and 6.

Step 4: Copy this cell and paste it into the topmost cell in the Die 2 Roll column. You now have two random numbers:

Creating two random “dice rolls” with the RANDBETWEEN() function.

Step 5: In the topmost cell in the Dice Roll column, create the following formula to add up the values in the two die roll cells (depending on your spreadsheet, you might need to use values other than A2 and B2):

=SUM(A2:B2)

Your spreadsheet should now look something like this:

Using the SUM() function to add together the two dice rolls.

Step 6: Select all three cells in the first row of your table. Now you want to duplicate this row many times. In Numbers, you can choose Edit > Copy, then choose Insert > Copied Rows to copy them once to a new row:

Once you’ve done that, drag the little “corner” button to stretch the table vertically, adding copies of the row as you go:

Dragging the “corner” icon to duplicate rows in Numbers.

In OpenOffice/Excel, you can simply drag the little black dot in the bottom right corner to duplicate the rows:

Duplicating rows in OpenOffice by dragging the little black dot down.

As you add more rows, notice that the chart becomes smoother and smoother. How many rows do you need to add to make a perfect triangle? Chances are you’d never do it, but it does get pretty close after a while. Here’s my chart after 5,000 “dice rolls”:

My two-dice chart after 5,000 simulated dice rolls.

Once again, you can see that if you take a random event like a dice roll and repeat it enough times, a regular pattern often emerges.

Experiment 4: More dice

Try adding four more columns of random numbers before the Die 1 Roll column to make six simulated dice. Update the Roll and Frequency columns to handle results from 6 to 36. You’ll end up with this type of chart:

This is the pattern you get if you “roll” six simulated dice enough times.

Why do you think the chart is this shape? (This pattern is known as a normal distribution, by the way.)

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