Lesson 7. Does God play the dice?
Objective:
Coin tossing and dice rolling are the familiar means to simulate events
occurring randomly. They are however not a practical means to generate
a long sequence of random events. Hence, a pseudo-random number
generator is used to quickly simulate several thousands of coin tossing
(Prog#7a), dice rolling (Prog#7b – 7d), and random tree branching
(Prog#7e – 7g). For one die roll, the chances for facing up any side are
the same and equal to one out of six, so that they have a uniform
distribution
But, the sums of faced-up pips of two dice favor the median value 7, and
obey a triangle distribution.
Moreover, the distribution of sums of three dice gets rounded in the
middle and falls off at the edges.
As more and more dice are rolled, the distribution of their sums becomes a
bell-shaped curve. This is the Central Limit Theorem. The branch
scaling factor and spreading angle controls the growth of binary tree
introduced in Lesson 1. When we assign only the scaling factor randomly
the binary tree grows into all different sizes. But, when both the scaling
factor and spreading angle are chosen randomly, there appear more
middle-sized binary trees and less of the extreme small and large ones, as
suggested by the Central Limit Theorem.
1
Scary words: Randomizer, pseudo-random number generator, random walk, event,
sample space, probability distribution, uniform distribution, bell-shaped
(normal) distribution, random variable, Central Limit Theorem.
Last update: 8 Jan 2003
2
Lesson
Randomizer
We encounter the laws of chance in our daily life; for example, tossing a coin at
the beginning of a football game, throwing dice for a board game, picking a winning
lottery ticket, and forecasting the probability of rain. Of these, coin tossing is the simplest
with two possible outcomes of head (H) and tail (T). It is the randomizer to generate
either H or T in an unpredictable manner. Given a fair coin, our intuition tells us the
chances for H or T are the same and hence equal to one in two, i.e., 12 . After 10, 50,
and 100 throws of a coin, you record the numbers of H and T in table 1 and then compute
the ratio of H or T to the total number of throws. Are the ratios close to 12 ? Do they
approach
1
2
, as the number of coin throws increases?
Table 1. Coin tossing
Throws
Number of
occurrences
Compute
the ratio
10
50
100
H
T
H
T
H
/10 =
/10 =
/50 =
/50 =
/100 =
T
/100 =
It must be noted that the long-term probability of 12 is one of many tests to guarantee a
random coin tossing. For instance, {H, T, H, T, H, T,…} is obviously not random,
though the probability for H or T turns out exactly 12 . Now, a die throw generates one
out of six numbers {1, 2, 3, 4, 5, 6}. To test the randomness of 60 die throws, you
record in table 2 the number of occurrences of (1, 2, 3, 4, 5, 6) and then compute the
ratio of occurrences to the total number of throws. In theory, for a true die the odds for
any side to face up are one out of six, i.e., 16 . But, are the ratios equal to 16 in table 2?
Table 2. 60 die throws
Pip
Number of
occurrences
Compute
the ratio
1
2
3
4
5
6
/60 =
/60 =
/60 =
/60 =
/60 =
/60 =
Although coin tossing and dice throwing are the familiar randomizers, it is
difficult to visualize how a winning ticket can be chosen randomly out of millions of
3
lottery tickets. Perhaps, randomness is seriously lacking if one imagines a large metal
cage containing millions of lottery tickets being tumbled, prior to pulling out a winning
ticket. Moreover, the uncertainties in weather forecasting are due to the random
variations in geophysical flow computations over several days, for which there are no
simple randomizers to simulate the meteorological fluctuations. For this reason, the realworld randomizers are based on pseudo-random number generator. First of all, a
random number generator is a recipe to come up with a number that is not in an obvious
manner related to all the numbers that it has previously generated. Since it is difficult to
prove that such numbers are truly random, we simply call them pseudo-random numbers.
Random walk by coin tossing
Suppose that you are standing in the middle of a room facing north, as indicated
by the green head with a nose pointing to north. If head, you turn right 90 and take a
step forward, or turn left 90 and take a step forward if tail, as denoted by the orange
heads with H or T, respectively
Figure 1 shows the path of {H, T, T, H, T} and you can extend it for the next 10 steps by
coin tossing. Did you return to the starting position in figure 1? How about after 20,
30 or more steps?
Figure 1. Five steps by {H, T, T, H, T}
4
This is called a random walk to simulate the zigzag path of a drunken person.
When a ray of sun pierces through a window, we sometimes can see the dust particles
floating in the air and their path of movement resembles a random walk. A pseudorandom number generator coded into Prog#7a simulates coin tossing. To facilitate
visualization, Prog#7a displays the first 25% of random walk path in red, the second
25% in blue, the third 25% in green, and the last 25% in red. Also, the final location of
random walk is indicated by a red  . You record the final positions after 1,000, 2,000,
3,000, 4,000, and 5,000 steps of random walk. Are you getting closer to or moving
away from the starting point as the number of random steps increases?
Uniform distribution
For one die roll, the chances for any one of the six sides to face up are the same
and equal to 16  0.167. This is indicated in figure 2 by six bars of the equal height 16 .
It is called the probability distribution, indicating how the probabilities of each event are
distributed over the sample space of outcomes {1, 2, 3, 4, 5, 6}. It is a uniform
distribution of equal probabilities. You may also indicate in figure 2 the six ratios of
table 2 by vertical bars for {1, 2, 3, 4, 5, 6}. Do they obey a uniform probability
distribution after 60 die throws?
Figure 2. One die roll
A pseudo-random number generator simulates the dice roll in Prog#7b, so that you can
quickly generate thousands of die throws. For instance, beginning from 500 die throws
you can test for the equal probability of 16 by incrementing the number of die rolls by 500
with Prog#7b.
5
Bell-shaped distribution
Two dice roll: We now roll two dice of white and blue, and add up the pips of faced-up
sides. The smallest sum ‘2’ is given by (
(
,
) and the largest sum is 12 when
), so that {2, 3, 4, …, 11, 12} is the sample space.
,
But, sum ‘3’ can result
from two events (
,
) and (
,
), and similarly (
,
) and (
,
)
give sum ‘11.’ In table 3, we list all possible events for the sums of {2, 3, 4, …, 11, 12}
of two dice roll.
Table 3. Sum of the facing-up pips of two dice roll
Pips
2
3
4
5
6
7
8
9
10
11
12
Events
(
,
)
(
,
), (
,
)
(
,
), (
,
), (
,
)
(
,
), (
,
), (
,
), (
,
)
(
,
), (
,
), (
,
), (
,
), (
,
)
(
,
), (
,
), (
,
), (
,
), (
,
), (
(
,
), (
,
), (
,
), (
,
), (
,
)
(
,
), (
,
), (
,
), (
,
)
(
,
), (
,
), (
,
)
(
,
), (
,
)
(
,
)
,
)
Since there are in all 36 events in table 3, the probabilities of sum ‘2’ and ‘12’ are 1/36,
sum ‘3’ and ‘11’ are 2/36, sum ‘4’ and ‘10’ are 3/36, and so on, as they appear in the bar
graph of figure 3 over the sample space {2, 3, 4, …, 11, 12}. Here comes a surprise.
The uniform distribution (figure 2) of one die roll has become a triangle distribution
(figure 3) when two dice are rolled and the faced-up pips are added up. That is, you are
more likely to get sum ‘7’ than any other sums in two dice throw. With Prog#7c you
6
can check how closely a triangle distribution is realized by increasing the number of dice
throws up to 5,000.
Figure 3. Two dice throw
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 or17
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
7
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.
Binary tree
In Lesson 1, the growth of binary tree depends on two parameters, the scaling
factor a and spreading angle  . As shown in figure 5, the main trunk of, say, 1 meter
gives rise to two shorter branches of ( 1a ) meter for the first branching, since a has a value
in the range (1.2, 1.8). Also, the tree height is controlled by angle  ; that is, the smaller  ,
the taller the binary tree. Now, there are four branches of ( 1a )  ( 1a ) meter for the second
branching, eight branches of ( 1a )  ( 1a )  ( 1a ) meter for the third branching, and so on.
Here, we assign the values of a and  randomly in a given range.
8
Figure 5. A binary tree
One random variable: We first fix the spreading angle at   40 , but let a to vary
randomly at each stage of branching. This is coded into Prog#7e by a pseudo-random
number generator, which picks a value for a randomly in the range (1.2, 1.8). After
running Prog#7e a dozen of times or more, you record in table 5 the number of small,
medium, and large binary trees that are observed. Be as objective as possible, though no
quantitative measures are provided here for the classification of binary trees.
Table 5. Binary trees with random scaling factor
Size
Small
Medium
Large
Number of observations
Two random variables: Not only the random variation of a in the range (1.2, 1.8),
Prog#7f also assigns angle  randomly in the range ( 20 , 60 ). Again, after running
Prog#7f a dozen of times or more, you record in table 6 the number of small, medium,
and large binary trees that are observed.
Table 6. Binary trees with random scaling factor and spreading angle
Size
Small
Medium
Large
Number of observations
Comparing tables 5 and 6, you may notice that there are more or less the same number of
small and medium and large binary trees in table 5, whereas more medium binary trees
are observed in table 6 than either the small or large ones. This is because randomizing
9
both a and  decreases the small and large binary trees by the triangle distribution of
figure 3.
Pythagoras tree (Modification #2)
The Pythagoras tree of modification #2 introduced in Lesson 3 has the motif
which involves three parameters, the branching ratio r and two angles  1 and  2 , as
shown in figure 6. Here, we assign the three parameters randomly at each stage of
branching. That is, we choose r in the range (0.3, 0.5) and both  1 and  2 in the range
( 5 , 35 ), but angle  1 is always smaller than  2 . All this has been coded into
Prog#7g. It appears that Prog#7g generates a more naturally looking tree than the
original program Prog#3f, because the extreme small and large variations are being
suppressed by a bell-shaped distribution of figure 4. This is a testament of the Central
Limit theorem, favoring a median over the extremes.
Figure 6. Motif for Pythagoras tree (modification #2)
10