The word “dice”  Historically, dice is the plural of die.  In modern standard English, dice is used as both the

singular and the plural.

52

Example of 19th Century bone dice

“Advanced” dice

53

[ http://gmdice.com/ ]

We have already seen the rand and randn functions.

randi function  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].

54

randi function: Example LLN_cointoss.m 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

Same as randi([0,1],1,N);

55

Dice Simulator  http://www.dicesimulator.com/  Support up to 6 dice and also has some background

information on dice and random numbers.

56

Two Dice

57

Two-Dice Statistics

58

Classical Probability  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 bottom

represents the number of possible outcomes, while the number on top represents the number of outcomes in which the event of interest occurs.

59

Two Dice  A pair of dice

60

Double six

Two dice: Simulation

[ http://www2.whidbey.net/ohmsmath/webwork/javascript/dice2rol.htm ]

61

Two dice  Assume that the two dice are fair and independent.  P[sum of the two dice = 5] = 4/36

62

Two dice  Assume that the two dice are fair and independent.

63

Two-Dice Statistics

64

Leibniz’s Error (1768)  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... 0.25

2 36 1 P[sum of the two dice = 12]  36

P[sum of the two dice = 11] 

0.2

0.15

0.1

0.05

0

0

100

200

300

400

500

600

700

800

900

1000

 Leibniz is a German philosopher, mathematician,

and statesman who developed differential and integral calculus independently of Isaac Newton.

65

[Gorroochurn, 2012]

Two dice: MATLAB Simulation 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

0.08 0.07 0.06 0.05 0.04 0.03 0.02

RF12 = cumsum(S==12)./(1:n); plot(1:n,RF12,'r')

[SumofTwoDice.m]

66

0.01 0

0

100

200

300

400

500

600

700

800

900

1000

Concatenation  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.

67

More on plot: Line specification 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 m c r g b w k

68

yellow magenta cyan red green blue white black

. o x + * s d v ^ < > p h

point circle x-mark plus star square diamond triangle (down) triangle (up) triangle (left) triangle (right) pentagram hexagram

: -. --

solid dotted dashdot dashed

Character Strings  A character string is a



 



69

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  PLOT(X,Y,'c+:')plo

ts a cyan dotted line with a plus at each data point

blue diamond at each data point but does not draw any line

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

70

 PLOT(X,Y,'bd')plots

0

1

2

3

4

5

6

7

8

9

10

-1

0

1

2

3

4

5

6

7

8

9

10

More on plot : Axes and Title sin and cos functions 1 y = sin(x) y = cos(x)

0.8 0.6 0.4 0.2

y

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

0 -0.2 -0.4 -0.6 -0.8 -1

71

0

1

2

3

4

5 x

6

7

8

9

10

Two dice: MATLAB Simulation 0.5

RF11 = cumsum(S==11)./(1:n); plot(1:n,RF11) hold on RF12 = cumsum(S==12)./(1:n); plot(1:n,RF12,'r')

0.4 0.35

Relative Frequency

n = 1e3; Number_Dice = 2; D = randi([1,6],Number_Dice,n); S = sum(D); % sum along each column

0.3 0.25 0.2 0.15 0.1 0.05 0

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)

Sum = 11 Sum = 12

0.45

0

100

200

300

400 500 600 Number of Rolls

700

180 160 140 120 100

N_S_Sim = hist(S,S_Support);

80 60

[SumofTwoDice.m]

40 20

72

0

2

3

4

5

6

7

8

9

10

11

12

800

900

1000

Loops  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. 73

for loop: a preview  Execute a block of statements a specified number of times. for_ex1.m for k = expr body end

for k = 1:5 x = k end

 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 end 74

statement are known as the body of the loop. They are executed repeatedly during each pass of the for loop.

Two dice: Probability Calculation Generate all the 36 possibilities.

75

Number_Dice = 2; Dice_Support = 1:6; S_Support = (1*Number_Dice):(6*Number_Dice); S = []; for k1 = Dice_Support for k2 = Dice_Support Nested loops S = [S k1+k2]; end Concatenation end Size_SampleSpace = length(S);

Find the corresponding sum.

Number_11 = sum(S==11) Number_12 = sum(S==12)

Count how many, among the 36, have a particular value.

% Count all possible cases at once N_S = hist(S,S_Support) P = sym(N_S)/Size_SampleSpace

Vectorization and Preallocation 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

LLN_cointoss.m close all; clear all; N = 1e3; % Number of trials s = zeros(1,N); % Preallocation

The script will still run without this line.

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

LLN_cointoss_for_preallocated.m 76

Preallocating Vectors (or Arrays)  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 found that 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.

77

zeros, ones, eye  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. 78

Exercise  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. 79

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 [Gorroochurn, 2012] (1620)  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:

80

Exercise

0.5

0.4

0.3

0.2

0.1

0

0

100

200

300

400 500 600 Number of Rolls

700

800

900

1000

0.14 probability relative frequency

0.12

0.1

0.08

0.06

0.04

0.02

0

81

Sum = 11 Sum = 12

0.6

Relative Frequency

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.

0.7

2

4

6

8 10 12 sum of three dice

14

16

18

Three-Dimensional Arrays  Arrays in MATLAB are not limited to two dimensions.

82

Three-Dimensional Arrays  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.

83

Empty Array  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 84

removed from a matrix.

Scandal of Arithmetic 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)? P ( A) 

4

6 5 5  1     .518 4 6 6 4

4

36 24  3524  35  P( B)   1     .491 24 36  36  24

85

[http://www.youtube.com/watch?v=MrVD4q1m1Vo]

“Origin” of Probability Theory  Probability theory was originally inspired by gambling  

 

86

problems. 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  General form:

if expression statements elseif expression statements else statements end

The ELSE and ELSEIF parts are optional.

 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 ~=. 87

Example: Assigning Grades Pass vs. Fail if_ex_1.m % Generate a random number score = randi(100, 1) if score >= 50 grade = 'pass' else grade = 'fail' end

true grade = ‘pass’ 88

score ≥ 50

false grade = ‘fail’

Example: Assigning Letter Grades

true grade = ‘A’

score ≥ 80

true grade = ‘B’

false

score ≥ 70

true grade = ‘C’

89

% 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

false

score ≥ 60

true grade = ‘D’

false

score ≥ 50

false grade = ‘F’

Exercise  Write a MATLAB script to evaluate the relative frequencies

involved in the scandal of arithmetic. 0.8 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

0.7

relative frequency

0.6 0.5 0.4 0.3 0.2 0.1 0

90

0

100

200

300

400 500 600 number of rolls

700

800

900

1000

Exercise  Write a MATLAB script to evaluate the relative frequencies

involved in the Monty Hall game. 1 Not Switch Swicth

0.9

Relative Frequency of Winning

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

91

0

100

200

300

400 500 600 Number of Trials

700

800

900

1000