Sharing Out the Energy - Molecular Modelling - page 99

7.1.1 Two Dice

Table 7.4 and Figure 7.3 show what happens when we repeat the experiment with two
dice. In this case the possible scores are 2, 3, ..., 12 and this is certainly not the same as
having a single 12-sided die. To model the experiment, use your random number gener-
ator for each die and add the result; obviously your values will not be exactly the same
as mine.

Table 7.4 Throwing two dice n times

n = 100 n = 1000 n = 1 000 000

< S > 7.010 6.980 7.003

σ 2.500 2.357 2.415

Throwing two dice n times

0.2

0.15

0.1

0.05

n = 100

0

123456789 0 1

Score-1

Figure 7.3 Throwing two dice

I will tell you that if we toss p dice then th e theoretical mean score is p times the mean for
a single die, and the standard deviation is √ p times. The distribution has obviously settled
down by the time we get to a million throws, but what has it settled down to?

7.2 Enumeration

It should be clear that we can deduce the theoretical probabilities by asking how many
ways a given event can occur from all allowed possibilities. The probability is given by the
ratio of the number of times an event can happen to the total number of ways. Table 7.5
shows the number of ways the two dice can fall in order to make up a given score. The
theoretical probabili ti es are 1/36, 2/36, ..., 1/36 whilst the mean and standard deviation

work out as 7 and 2 √ 2. These probabilities agree completely with the 'experimental'values
shown.