Lets see what we can figure out Start with one die.

What are the possible outcomes from one roll?

What is the most you would bet to win $100 on a 3 coming up in a single roll?

On a 6?

Rolling a die -1 What could a set of 6 rolls look like?

Set 1: [ 1, 2, 3, 4, 5, 6 ] ?

Set 2: [ 1, 1, 1, 1, 1, 1 ] ?

Set 3: [ 2, 4, 6, 1, 3, 5 ] ?

Set 4: [ 2, 2, 2, 3, 3, 3 ] ?

Is Set 3 the same as Set 1?

Rolling a die - 2

How to characterize the results of a set?

Lets agree that the order does not matter.

Lets do 12 rolls.

Heres a possible result: [ 6, 3, 5, 5, 1, 1, 4, 6, 6, 4, 5, 3 ]

Rolling a die - 3

How to characterize the results of a set?

Consider this result: [ 6, 3, 5, 5, 1, 1, 4, 6, 6, 4, 5, 3 ]

What is the mean of the set?

What would you expect the mean to be (if the die is fair)?

The idea of binning - 1

How to characterize the results of a set?

Consider this result: [ 6, 3, 5, 5, 1, 1, 4, 6, 6, 4, 5, 3 ]

Each roll outcome is between 1 and 6 (integer). Count the number of times each face occurs in the set; save count in the bin labeled 1,2, etc

The idea of binning - 2

Bin this result: [ 6, 3, 5, 5, 1, 1, 4, 6, 6, 4, 5, 3 ]

Bins: [ (1s), (2s), (3s), (4s), (5s), (6s) ]

Counts: [ 2, 0, 2, 2, 3, 3 ]

Do counts total to 12?

The idea of binning - 3

Bin this result: [ 6, 3, 5, 5, 1, 1, 4, 6, 6, 4, 5, 3 ]

Bin counts: [ 2, 0, 2, 2, 3, 3 ]

What is the mean based on bin counts?

Is any information lost in going from the roll set to the bin counts?

Test Case 1: one die, one set

1 set of rolls: nrolls = 12

Outcome: rollSet (1 x nrolls)

How to study the results?

o Mean of rolls: setMean

o Bin counts: bincount (1 x 6)

o Bin percentages: binpct (1 x 6)

Test Case 2: one die, three sets

3 sets of rolls:

nrolls1=12, nrolls2=120, nrolls3=1200

Outcomes:

rollSet1 (1 x nrolls1), rollSet2, rollSet3

How to study the results?

o Mean of rolls: setMean

o Bin counts: bincount (1 x 6)

o Bin percentages: binpct (1 x 6)

Test Case 2: one die, three sets

3 sets of rolls: nrolls1, rollSet1,

How to study the results?

o Array of means: setMeanarr (1x3)

o Bin counts: bincountarr (3x6)

o Bin percentages: binpctarr (3x6)

Test Case 3. Heaven ( pair o dice )

Uh-oh: lets think about this.

One roll of two dice: what are the possible outcomes?

[1,1], [1,2], [2,1], [1,3], [3,1],

Order doesnt matter, so sort by total.

Outcomes by totals: 2, 3, 4, , 11,12

Test Case 3. Heaven outcomes by totals 14

total =

2: 1+1

3: 1+2, 2+1

4: 1+3, 3+1, 2+2

5: 1+4, 4+1, 2+3, 3+2

6: 1+5, 5+1, 2+4, 4+2, 3+3

7: 1+6, 6+1, 2+5, 5+2, 3+4, 4+3

8: ?

9: ?

10: ?

11: 5+6, 6+6 12: 6+6

Test Case 3: two dice, one set of rolls

1 set of rolls: nrolls = 1100 (?)

Outcome: rollSet (1 x nrolls),

where a rollSet entry is the total of 2 dice.

How to study the results?

o Mean of rolls: setMean

o Bin counts: bincount (1 x 11)

o Bin percentages: binpct (1 x 11)

Test Case 4: Coin flip runs

New process: flip a (fair) coin.

Outcome: head or tail

One set of flips: nflips

Outcome: flipSet (1 x nflips)

A flipSet entry is 1 (head) or 2 (tail).

Example: flipSet= [ 1,2,2,2,1, 2,1,1,1,1]

Test Case 4: Coin flip runs

Example: flipSet= [ 1,2,2,2,1, 2,1,1,1,1]

Characterize results:

setMean: (expected value?)

setRuns: [ 1, 3, 1, 1, 4 ]

binRuns: [ 3, 0, 1, 1 ]

--------

function [setMean1,bincount1,binpct1]= dice_1_fcn(nrolls1,rollSet1)

% id: sec00?, your folder name

% INPUT definitions:

% nrolls1, number of rolls

% rollSet1, roll results (1 x nrolls)

% OUTPUT definitions:

% bincount1, nbr of results in each bin

% binpct1, percent of results in each bin

% initialize outputs

bincount1= zeros(1,6);

binpct1= zeros(1,6);

(Your outstanding solution goes in here)

end

-------------------------------------------------------------------------------

Test case 2 (3 pts). Three sets of rolls of one die.

--------

function [setMean2arr,bincount2arr,binpct2arr]= dice_2_fcn( nrolls21,rollSet21,nrolls22,rollSet22,nrolls23,rollSet23]

% id: sec00?, your folder name

% INPUT definitions:

% nrolls21, number of rolls for set 1

% rollSet21, roll results for set 1 (1 x nrolls21)

% OUTPUT definitions:

% setMean2arr, mean of each roll set (1 x 3)

% bincount2arr, bincount array for each set (3 x 6)

% binpct2arr, binpct array for each set (3 x 6)

%

% initialize outputs

setMean2arr= zeros(1 x 3);

bincount2arr= zeros(3 x 6);

binpct2arr= zeros(3 x 6)

Test case 3 (2 pts). One set of rolls of two dice.

--------

function [setMean3,bincount3,binpct3]= dice_3_fcn(nrolls3,rollSet3)

% id: sec00?, your folder name

% INPUT definitions:

% nrolls3, number of rolls for set 1

% rollSet3, roll results (total of faces) for set 1 (1 x nrolls3)

% OUTPUT definitions:

% setMean3, mean of roll set

% bincount3, bincount array for roll set (1 x 11)

% binpct3, binpct array for roll set (1 x 11)

%

% initialize outputs

bincount3= zeros(1 x 11);

binpct3= zeros(1 x 11)

(Your outstanding solution goes in here)

end

--------------------------------------------------------------------------------

Test case 4 (2 pts). Two set of rolls of pair of fair dice.

Find differences between bin percentages and expected bin percentages for two sets of rolls of a pair of fair dice.

--------

function [binpct41,binpctdiff41,binpct42,binpctdiff42]= ... dice_4_fcn(nrolls41,rollSet41,nrolls42,rollSet42); %

id: sec00?, your folder name

% INPUT definitions:

% nrolls41, number of rolls in set 41

% rollSet41, roll results (1 x nrolls41)

% nrolls42, number of rolls in set 42

% rollSet42, roll results (1 x nrolls42)

% OUTPUT definitions:

% binpct41, bin percentages (1 x 11)

% binpctdiff41, (binpct41 - bin expected value) (1 x 11)

% binpct42, bin percentages (1 x 11)

% binpctdiff42, (binpct42 - bin expected value) (1 x 11)

% initialize outputs

binpct41= zeros(1,11);

binpctdiff41= zeros(1,11);

binpct42= zeros(1,11);

binpctdiff42= zeros(1,11);

% -------------------------------------------------------------------------

% Test case 1: One set of rolls of one die.

% Test case 2: Three sets of rolls of one die.

% Test case 3: One set of rolls of two dice.

% Test case 4: One set of flips of coin. Study of runs.

% -------------------------------------------------------------------------

clc clear

disp(' ');

mfilename= 'dice_test_1_demo.m'

% -------------------------------------------------------------------------

% Solution setup and solve ------------------------------------------------

% TC1 -- One set of rolls of one die. ------------------------------------

% IN: nrolls1, number of rolls

% IN: rollSet1, roll results (1 x nrolls)

% OUT: setMean1, mean of roll set

% OUT: bincount1, nbr of results in each bin

% OUT: binpct1, percent of results in each bin

nrolls1= 12;

rollSet1= zeros(1,nrolls1);

for nrx=1:1:nrolls1

rollSet1(1,nrx)= randi(6); % generate rollSet1

end

[setMean1,bincount1,binpct1]= dice_1_fcn_ans(nrolls1,rollSet1);

disp(' ');

disp('Test TC1:');

fprintf('nrolls1= %4.0f ',nrolls1);

fmt11= ['setRolls1(1,1:12)= [',repmat('%3.0f',1,12),'] '];

fprintf(fmt11,rollSet1(1,1:12));

fprintf('setMean1= %5.2f ',setMean1);

fmt12= ['bincount1= [',repmat('%4.0f',1,6),'] '];

fprintf(fmt12,bincount1(1,1:6));

fmt13= ['binpct1= [',repmat('%6.2f',1,6),'] '];

fprintf(fmt13,binpct1(1,1:6));

ax11= subplot(2,1,1);

y11= bincount1(1,1:6);

bar(ax11,y11)

title('bin counts for 1 set of 12 rolls');

ax12= subplot(2,1,2);

y12= binpct1(1,1:6);

bar(ax12,y12)

title('bin pcts for 1 set of 12 rolls');

%pause

% eof -