Elsewhere, we have seen that fair six-sided dice will produce about equal numbers of 'ones,' 'twos,' 'threes,' 'fours,' 'fives,' and 'sixes,' in the long run. Those are the elementary outcomes for each die, and each occurs with equal probability.
But what about other events? What about the sum of two dice? What do we expect to get?
Well, we know that there are many possible outcomes. The two dice could add up to any number from 2(if both dice came up 'one') to 12 (if both dice came up 'six'). Clearly 2 and 12 are fairly rare outcomes, but they are possible. It is not possible to get 1or 13, however.
And this idea that 2 and 12 are pretty rare should put you on the track to figuring out what we do actually expect. We expect the sum which has the most ways of occurring. Notice that 2 and 12 each have only one way of occurring. But 3 can be the sum if either the first die comes up 'one' and the second comes up 'two' or if the first die comes up 'two' and the second die comes up 'one'. So there are two ways for a sum of 3 to occur. Similarly, there are two ways for a sum of 11 to occur.
Can you figure out that there are three ways for a sum of 4 to occur? How many ways can a sum of 7 occur? It turns out that a sum of 7 has the most ways of occurring, so we would expect it to occur the most often in the long run.
The Flash simulation below gives you a way to test this reasoning against the real world. The simulation rolls the dice five times and shows you the sums of those rolls. The red histogram produced by the simulation simply shows the frequency of each face of the die. The blue histogram is a histogram of the sum of the dice.
