Three dice roll: Let's now roll three dice of white and blue and red. The smallest sum ‘3’ is given by (, , ) and the largest sum ‘18’ occurs when ( , , ). Hence, the sums of three faced-up pips have the sample space of {3, 4, 5, …, 17, 18}. Similar to table 3, we can list the events of three dice roll. For instance, ( , , ) and ( , , ) and ( , , ) ) give sum ‘4’. However, since it gets tedious to list all events, we summarize in table 4 the number of events of three dice rolls, the total of which is 216. Then, the probabilities for sum ‘3’ and ‘18’ are 1/216, sum ‘4’ and ‘17’ are 3/216, and finally sum ‘10’ and ‘11’ are 27/216 =1/8, as they are shown in figure 4 by vertical bars over the sample space of {3, 4, 5, ..., 17, 18}. We notice that the distribution of figure 4 is more rounded in the middle and falls off more quickly at the low and high ends than the triangle distribution of figure 3. It begins to look like what is commonly called the bell-shaped distribution curve. You can also experiment with Prog#7d to see how closely the probability distribution of figure 4 is observed as the number of dice throws becomes large.

Table 4. Three dice roll

Pip sum

3 or 18

4 or 17

5 or 16

6 or 15

7 or 14

8 or 13

9 or 12

10 or 11

Number of events

1

3

6

10

15

21

25

27

 

Figure 4. Three dice roll

To sum up, the probability distributions of figures 2, 3, and 4 have the following message. Let us begin by defining a random variable, which varies from one instance to another in an unpredictable way, like the outcome of a coin tossing and die rolling. Other examples are the measurement errors in experiment, wind speed and temperature variations in weather forecasting, daily stock price fluctuations, etc. Although a single random variable obeys a constant distribution (figure 2), the sum of two random variables follows a quite different distribution of triangle shape (figure 3). Now, for the sum of three random variables we begin to see a bell-shaped distribution (figure 4). According to the Central Limit Theorem, the distribution is truly bell-shaped or normal as the number of random variables becomes very large.

Note:
Prog#7d is available for download on the index page.
If you use a Macintosh, view the Flash versions.

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