2.1 Random Variables
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A natural inquiry at this point is, “What is the probability distribution of the sum on three fair dice?” It is more difficult to work out the distribution here than it was
for two dice. While we will show another solution later, we give one approach to the problem at this time. Consider, for example, a sum of 10 on three dice. The sum could
have arisen from these combinations of results showing on the individual dice which do not indicate which die showed which face:
. 2; 2; 6; .3; 3; 4; .2; 4; 4;
. 3; 1; 6; .3; 2; 5; .5; 1; 4:
Each of the first three of these combinations could occur in 3 different orders corresponding to the three different dice, while each of the last three could occur in 6
different orders. This gives a total of 27 possibilities, each of which has probability
1 216
. Therefore P. X = 10 =
27 216
. A similar process could be followed for other values of the sum; the complete probability distribution can be found to be
P. X = x = 8
:
1 216
if x = 3 or 18
3 216
if x = 4 or 17
6 216
if x = 5 or 16
10 216
if x = 6 or 15
15 216
if x = 7 or 14
21 216
if x = 8 or 13
25 216
if x = 9 or 12
27 216
if x = 10 or 11 0;
otherwise. A computer algebra system may also be used to find the probability distribution for
X: Many systems will give all the permutations, each of which may be summed and the relative frequencies recorded. This is shown in Appendix 1. There are other methods
that can be used to solve the problem; one of these will be discussed in Chapter 4. A graph of this function is shown in Figure 2.3. It begins to show what we will call
a normal probability distribution shape. As the number of dice increases, the “curve” the eye sees smooths out to resemble a normal probability distribution; the distribution
for six or more dice is remarkably close to the normal distribution. We will discuss the normal distribution in Chapter 3.
Ex ample 2.1.4
We saw in Example 2.1.2 that a single die could be loaded so that the probability of the occurrence of a face is proportional to the face. Can we
load a die so that when the die is thrown twice the probability of a sum is proportional to the sum?
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Chapter 2 Discrete Random Variables and Probability Distributions
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