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The word “dice”
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Historically, dice is the plural of die.
In modern standard English, dice is used as both the
singular and the plural.
Example of 19th Century bone dice
randi function
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Generate uniformly distributed pseudorandom integers
randi(imax) returns a scalar value between 1 and imax.
randi(imax,m,n) and randi(imax,[m,n])return an m-by-n matrix containing pseudorandom integer values drawn from the discrete uniform distribution on the interval [1,imax].
randi(imax) is the same as randi(imax,1).
randi([imin,imax],...) returns an array containing integer values drawn from the discrete uniform distribution on the interval [imin,imax].
We have already seen the rand and randn functions.
randi function: Example
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close all; clear all; N = 1e3; % Number of trials (number of times that the coin is tossed) s = (rand(1,N) < 0.5); % Generate a sequence of N Coin Tosses. % The results are saved in a row vector s. NH = cumsum(s); % Count the number of heads plot(NH./(1:N)) % Plot the relative frequencies
LLN_cointoss.m
Same as
randi([0,1],1,N);
Dice Simulator
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http://www.dicesimulator.com/
Support up to 6 dice and also has some background
information on dice and random numbers.
Two Dice
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Two-Dice Statistics
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Classical Probability
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Assumptions The number of possible outcomes is finite. Equipossibility: The outcomes have equal probability of
occurrence. Equi-possible; equi-probable; euqally likely; fair
The bases for identifying equipossibility were often
physical symmetry (e.g. a well-balanced die, made of homogeneous material in a cubical shape)
a balance of information or knowledge concerning the various possible outcomes.
Formula: A probability is a fraction in which the bottomrepresents the number of possible outcomes, while the number on top represents the number of outcomes in which the event of interest occurs.
Two Dice
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A pair of dice
Double six
Two dice: Simulation
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[ http://www2.whidbey.net/ohmsmath/webwork/javascript/dice2rol.htm ]
Two dice
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Assume that the two dice are fair and independent.
P[sum of the two dice = 5] = 4/36
Two dice
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Assume that the two dice are fair and independent.
Two-Dice Statistics
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Leibniz’s Error (1768)
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Though one of the finest minds of his age, Leibniz was not immune to blunders: he thought it just as easy to throw 12 with a pair of dice as to throw 11.
The truth is...
Leibniz is a German philosopher, mathematician,and statesman who developed differential and integral calculus independently of Isaac Newton.
2[sum of the two dice = 11]
36
1[sum of the two dice = 12]
36
P
P
[Gorroochurn, 2012]
0 100 200 300 400 500 600 700 800 900 10000
0.05
0.1
0.15
0.2
0.25
Two dice: MATLAB Simulation
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n = 1e3;
Number_Dice = 2;
D = randi([1,6],Number_Dice,n);
S = sum(D); % sum along each column
RF11 = cumsum(S==11)./(1:n);
plot(1:n,RF11)
hold on
RF12 = cumsum(S==12)./(1:n);
plot(1:n,RF12,'r') 0 100 200 300 400 500 600 700 800 900 1000
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
[SumofTwoDice.m]
Concatenation
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Concatenation is the process of joining arrays to make larger ones.
The pair of square brackets [] is the concatenation operator.
Horizontal concatenation: Concatenate arrays horizontally using commas.
Each array must have the same number of rows.
Vertical concatenation: Concatenate arrays vertically using semicolons.
The arrays must have the same number of columns.
More on plot: Line specification
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Various line types (styles), plot symbols (markers) and colors
may be obtained with plot(X,Y,S) where S is a character
string made from one element from any or all the following 3
columns:
y yellow . point - solid
m magenta o circle : dotted
c cyan x x-mark -. dashdot
r red + plus -- dashed
g green * star
b blue s square
w white d diamond
k black v triangle (down)
^ triangle (up)
< triangle (left)
> triangle (right)
p pentagram
h hexagram
Character Strings
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A character string is a sequence of any number of characters enclosed in single quotes.
If the text includes a single quote, use two single quotes within the definition.
These sequences are arrays, like all MATLAB variables.
Their class or data type is char, which is short for character.
You can concatenate strings with square brackets, just as you concatenate numeric arrays.
To convert numeric values
to strings, use functions,
such as num2str or
int2str.
More on plot : Examples
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PLOT(X,Y,'c+:')plo
ts a cyan dotted line with a
plus at each data point
0 1 2 3 4 5 6 7 8 9 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
PLOT(X,Y,'bd')plots
blue diamond at each data point
but does not draw any line
More on plot : Axes and Title
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0 1 2 3 4 5 6 7 8 9 10-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1sin and cos functions
x
y
y = sin(x)
y = cos(x)
x = linspace(0,10,100);
y1 = sin(x);
y2 = cos(x);
plot(x,y1,'mo--',x,y2,'bs--')
title('sin and cos functions')
xlabel('x')
ylabel('y')
legend('y = sin(x)','y = cos(x)')
grid on
Two dice: MATLAB Simulation
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[SumofTwoDice.m]
0 100 200 300 400 500 600 700 800 900 10000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Number of Rolls
Rela
tive F
requency
Sum = 11
Sum = 12
n = 1e3;
Number_Dice = 2;
D = randi([1,6],Number_Dice,n);
S = sum(D); % sum along each column
RF11 = cumsum(S==11)./(1:n);
plot(1:n,RF11)
hold on
RF12 = cumsum(S==12)./(1:n);
plot(1:n,RF12,'r')
xlabel('Number of Rolls')
ylabel('Relative Frequency')
legend('Sum = 11', 'Sum = 12')
grid on
figure
S_Support = (1*Number_Dice):(6*Number_Dice);
hist(S,S_Support)
N_S_Sim = hist(S,S_Support);
2 3 4 5 6 7 8 9 10 11 120
20
40
60
80
100
120
140
160
180
Loops
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Loops are MATLAB constructs that permit us to execute a sequence of statements more than once.
There are two basic forms of loop constructs:
while loops and
for loops.
The major difference between these two types of loops is in how the repetition is controlled.
The code in a while loop is repeated an indefinite number of times until some user-specified condition is satisfied.
By contrast, the code in a for loop is repeated a specified number of times, and the number of repetitions is known before the loops starts.
for loop: a preview
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Execute a block of statements a specified number of times.
k is the loop variable (also known as the loop index or iterator variable).
expr is the loop control expression, whose result usually takes the form of a vector.
The elements in the vector are stored one at a time in the variable k, and then the loop body is executed, so that the loop is executed once for each element in the array produced by expr.
The statements between the for statement and the endstatement are known as the body of the loop. They are executed repeatedly during each pass of the for loop.
for = expr k
body end
for k = 1:5 x = k end
for_ex1.m
Two dice: Probability Calculation
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Generate all the 36 possibilities.
Find the corresponding sum.
Count how many, among the
36, have a particular value.
Nested loops
Number_Dice = 2;
Dice_Support = 1:6;
S_Support = (1*Number_Dice):(6*Number_Dice);
S = [];
for k1 = Dice_Support
for k2 = Dice_Support
S = [S k1+k2];
end
end
Size_SampleSpace = length(S);
Number_11 = sum(S==11)
Number_12 = sum(S==12)
% Count all possible cases at once
N_S = hist(S,S_Support)
P = sym(N_S)/Size_SampleSpace
Concatenation
Vectorization and Preallocation
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LLN_cointoss.m
LLN_cointoss_for_preallocated.m
close all; clear all;
N = 1e3; % Number of trials
s = zeros(1,N); % Preallocation
s(1) = randi([0,1]);
for k = 2:N
s(k) = randi([0,1]);
end
NH = cumsum(s); % Count the number of heads
plot(NH./(1:N)) % Plot relative frequencies
The script will still run without this line.
Revisiting an old script:
close all; clear all; N = 1e3; % Number of trials (number of times that the coin is tossed) s = randi([0,1],1,N); % Generate a sequence of N Coin Tosses. % The results are saved in a row vector s. NH = cumsum(s); % Count the number of heads plot(NH./(1:N)) % Plot the relative frequencies
Preallocating Vectors (or Arrays)
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Used when the content of a vector is computed/added/known while the loop is executed.
One method is to start with an empty vector and extend the vector by adding each number to it as the numbers are computed. Inefficient.
Every time a vector is extended a new “chunk” of memory must be foundthat is large enough for the new vector, and all of the values must be copied from the original location in memory to the new one. This can take a long time.
A better method is to preallocate the vector to the correct size and then change the value of each element to store the desired value. This method involves referring to each index in the output vector, and placing
each number into the next element in the output vector.
This method is far superior, if it is known ahead of time how many elements the vector will have.
One common method is to use the zeros function to preallocate the vector to the correct length.
zeros, ones, eye
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zeros
Create array of all zeros.
zeros returns the scalar 0. zeros(n) returns an n-by-n matrix of zeros. zeros(n,m) returns an n-by-m matrix of
zeros.
ones
Create array of all ones.
eye
Create identity matrix.
eye returns the scalar, 1. eye(n) returns an n-by-n identity matrix with
ones on the main diagonal and zeros elsewhere.
eye(n,m) returns an n-by-m matrix with ones on the main diagonal and zeros elsewhere.
Exercise
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The following script was used to calculate the probabilities related to the sum resulting from a roll of two dice.
Modify the script here so that the vector S is preallocated.
Modify the script to also create a matrix SS. Its size is 6×6. The value of its (k1,k2) element should be k1+k2.
Number_Dice = 2;
Dice_Support = 1:6;
S_Support = (1*Number_Dice):(6*Number_Dice);
S = [];
for k1 = Dice_Support
for k2 = Dice_Support
S = [S k1+k2];
end
end
Size_SampleSpace = length(S);
Number_11 = sum(S==11)
Number_12 = sum(S==12)
% Count all possible cases at once
N_S = hist(S,S_Support)
P = sym(N_S)/Size_SampleSpace
Galileo and the Duke of Tuscany
(1620)
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When you toss three dice, the chance of the sum being 10 is
greater than the chance of the sum being 9.
The Grand Duke of Tuscany “ordered” Galileo to explain a paradox
arising in the experiment of tossing three dice:
“Why, although there were an equal number of 6 partitions of the
numbers 9 and 10, did experience state that the chance of throwing a
total 9 with three fair dice was less than that of throwing a total of
10?”
Partitions of sums 11, 12, 9 and 10 of the game of three fair dice:
[Gorroochurn, 2012]
Exercise
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Write a MATLAB script to
Simulate N = 1000 repeated rolls of three dices.
Calculate the sum of the three dices from the rolls above.
Plot the relative frequency for the event that the sum is 9.
In the same figure, plot (in red) the relative frequency for the event that the sum is 10.
In another figure, create a histogram for the sum after N rolls of three dice.
Calculate the actual probability for each possible value of the sum.
In another figure, compare the probability with the relative frequency obtained after the N simulations.
2 4 6 8 10 12 14 16 180
0.02
0.04
0.06
0.08
0.1
0.12
0.14
sum of three dice
probability
relative frequency
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Number of Rolls
Rela
tive F
requency
Sum = 11
Sum = 12
Three-Dimensional Arrays
82
Arrays in MATLAB are not limited to two dimensions.
Three-Dimensional Arrays
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Three-dimensional arrays can be created directly using functions such as the zeros, ones, rand, and randi functions by specifying three dimensions to begin with.
For example, zeros(4,3,2) will create a 4×3×2 matrix of all 0s.
Empty Array
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An array that stores no value
Created using empty square
brackets: []
Values can then be added by
concatenating.
Can be used to delete elements
from vectors or matrices.
Individual elements cannot be
removed from matrices.
Matrices always have to have the same
number of elements in every row.
Entire rows or columns could be
removed from a matrix.
Scandal of Arithmetic
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Which is more likely, obtaining at least one
six in 4 tosses of a fair dice (event A), or
obtaining at least one double six in 24 tosses
of a pair of dice (event B)?
44 4
4
2424 24
24
6 5 5( ) 1 .518
6 6
36 35 35( ) 1 .491
36 36
P A
P B
[http://www.youtube.com/watch?v=MrVD4q1m1Vo]
“Origin” of Probability Theory
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Probability theory was originally inspired by gamblingproblems.
In 1654, Chevalier de Méré invented a gambling system which bet even money on case B.
When he began losing money, he asked his mathematician friend Blaise Pascal to analyze his gambling system.
Pascal discovered that the Chevalier's system would lose about 51 percent of the time.
Pascal became so interested in probability and together with another famous mathematician, Pierre de Fermat, they laid the foundation of probability theory.
best known for Fermat's Last Theorem
Branching Statements: if statement
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General form:
The statements are executed if the real part of the expression has
all non-zero elements.
Zero or more ELSEIF parts can be used as well as nested if’s.
The expression usually contains ==, <, >, <=, >=, or ~=.
if expression
statements
elseif expression
statements
else
statements
end
The ELSE and ELSEIF parts
are optional.
Example: Assigning Grades
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Pass vs. Fail
% Generate a random number
score = randi(100, 1)
if score >= 50
grade = 'pass'
else
grade = 'fail'
end
score ≥ 50true false
grade = ‘pass’ grade = ‘fail’
if_ex_1.m
Example: Assigning Letter Grades
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score ≥ 80
score ≥ 70
score ≥ 60
score ≥ 50
true false
true false
true false
true false
grade = ‘A’
grade = ‘B’
grade = ‘C’
grade = ‘D’ grade = ‘F’
% Generate a random number
score = randi(100, 1)
if score >= 80
grade = 'A'
elseif score >= 70
grade = 'B'
elseif score >= 60
grade = 'C'
elseif score >= 50
grade = 'D'
else
grade = 'F'
end
Exercise
90
Write a MATLAB script to evaluate the relative frequencies
involved in the scandal of arithmetic.
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
number of rolls
rela
tive f
requency
Event A: At least one six in 4 tosses of a fair dice
Event B: At least one double six in 24 tosses of a pair of dice
Exercise
91
Write a MATLAB script to evaluate the relative frequencies
involved in the Monty Hall game.
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of Trials
Rela
tive F
requency o
f W
innin
g
Not Switch
Swicth
