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ProbabilityPredicting the Future

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If we roll two dice at once and add up their scores we could get results from up to . However, not all outcomes are equally likely. Some results can only happen one way (to get 12 you have to roll + ) while others can happen in multiple different ways (to get 5 you could roll + or + ).

This table shows all possible outcomes:


The most likely result when rolling two dice is 7. There are outcomes where the sum is 7, and outcomes in total, so the probability of getting a 7 is 636=0.1666.

The least likely outcomes are 2 and 12, each with a probability of 136=0.0277.

It is impossible to forecast the outcome of a single coin toss or die roll. However, using probability we can very accurately predict the outcome of many dice.

If we throw a die 30 times, we know that we would get around 16×30=5 sixes. If we roll it 300 times, there will be around 16×300=50 sixes. These predictions get more and more accurate as we repeat the predictions more and more often.

In this animation you can roll many “virtual” dice at once and see how the results compare to the predicted probabilities:

Rolling Dice

${ probTable(d) }

We roll ${d} dice at once and record the SUM of their scores. The green lines represent the probabilities of every possible outcome predicted by probability theory and the blue bars show how often each outcome happened in this computer generated experiment.

Notice how, as we roll more and more dice, the observed frequencies become closer and closer to the frequencies we predicted using probability theory. This principle applies to all probability experiments and is called the Law of large numbers.

Similarly, as we increase the number of dice rolled at once, you can also see that the probabilities change from a straight line (one die) to a triangle (two dice) and then to a “bell-shaped” curve. This is known as the Central Limit Theorem, and the bell-shaped curve is called the Normal Distribution.