1.5 Close to certainty
It’s not possible to predict the outcome of a single roll of a pair of dice. But what if you keep on rolling the dice? In this case, probability can be used to determine how many of each outcome can be expected in the long run.
For example, imagine rolling a pair of dice 10 000 times. Using the probability information from Table 1, you would expect a total of 6 (or a total of 8) to come up approximately 5/36 x 10 000 = 1389 times, or 13.9% of the time, and you would expect a double six to come up approximately one-fifth of that, or 2.78% of the time.
Similarly, while the outcome of a single toss of a coin cannot be predicted, in the long run you would expect it to come up heads and tails in almost equal measure. In either case, if your expectations fail to be fulfilled by a significant margin, you would be justified in questioning the fairness of the dice or the coin.
Activity 3 Rolling many dice
To observe what happens to the frequencies when two or more dice are rolled together a large number of times, try using an online dice roller (this is much quicker and easier than using real dice and noting results!)
The site linked below lets you choose the number of dice to roll and provides a bar chart which shows the frequency of the dice roll results. Use this site to roll two and then three dice at least 2000 times each, and watch the bar chart evolve. What do you see?
Dice roll simulator [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] (make sure to open the link in a new tab/window)
Discussion
As the number of rolls increases, the shape of the bar chart becomes more symmetrical about the most common rolls (the rolls of greatest frequency). For two dice, this is 7, and for three dice, this is 10 and 11 (as demonstrated in Tables 1 and 2 earlier). As you increase the number of dice rolled – try six – you will see that the shape of the bar chart becomes increasingly bell-shaped. It approaches what is known as a bell curve (more formally known as ‘normal distribution’), an important notion in statistics and one which commonly arises in the study of data, especially in nature.