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.

Investigations

Be sure you can explain the difference between the shape of the red histogram
and the shape of the blue histogram.

Notice how much (or little) the average and standard deviation of the blue
histogram change.

Compare the predicted mean and standard deviation with the values you get
from this simulation.