GEK 1544Mathematics of Games

Your lecturer: Assoc. Prof. Leung Man Chun

Dept. of Mathematics

S- 17- 06 - 01

Tel. No. 6516-2758

e-mail: matlmc@nus.edu.sg

IVLE

Expect to find lecture PowerPoint files (simply scroll down the front page) and other information on the course.

Lectures will be webcasted. Not under my control. Please be mindful of poor quality and delays on the webcast.

Lectures: Mon & Thur 6 – 8 PM in U-Town Auditorim 3 (Alice & Peter Tan Colleage)

Tutorials: Start on the third week.

Assessment:

•One Open Book Test without Internet or Mobile Communication (about 1 hour; 20%).

Date of the Test: 16 March 2015 (Monday; 6:15pm – 7:15pm in U-Town Auditorim 3 ).

•Final Exam (80%): can bring in one (1) hand-written (on both sides) A-4 size help sheet.

Date of the Exam: 27-April-2014 (Mon evening).

Please double check the IVLE timetable.

Refer to the Library for past

exam. Papers.

Stata Centre in MIT (funded partially by Bill Gate)designed by the famed architect Frank Gehry,an example of “deconstructionist” architecture.

In the movie "21", when Ben is celebrating his birthday, the cake `says’

1, 1, 2, 3, 5, 8, 13, ...

These are known as Fibonacci numbers ,which follow the relation

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

21

).81321..(11

ge

ccc nnn

nc

Classic TV game show Let's Make A Deal:

1

The better strategy is to switch doors.

How do we calculate probability?

Suppose there are countable (finite) number of events, separated from one another, and each one has equal likelihood, then (roughly speaking)

.#

#

eventsoftotal

favouryourineventsofP

Dealer Player

High cards, A, K, Q, J & 10,

Low cards, 2, 3, 4, 5 & 6.

A high concentration of Aces and Tens give a higher chance of Blackjacks (which pay at 3:2 unless the

dealer also has Blackjack).

Low cards are good for the dealer, because the dealer must hit stiff hands (that is, 12 – 16 points), unlike the player who can hit or stand according to the strategy. Tens bust all stiff hands and will increase the chance

the dealer will lose.

Basic Hi - Lo system:

-1 for each dealt of Ten and Ace,

+1 one for any value 2 - 6.

Values like 7, 8 & 9 do not affect the count.

Lord of the Dies.Chevalier de Mere (1607-1684)

To get at least one `six’ in 4 rolls.

In keeping with his Salon methods, Gombaud enlisted two famous `mathematicians’, Pascal and Fermat to solve the problems. In a series of letters they laid the foundation for the modern theory of probability

Let’s start with the renowned mathematicianDescartes. We’ve heard of his famous motto:

"Cogito; ergo sum"

“I think, therefore I am.”

René Descartes (1596-1650)

Legend tells us that Descartes often woke up late and spent the cold mornings in a stove-heated room, contemplating mathematics and philosophy. One day he came out and got the idea of

Why at this time and why at this place.

doubtAccording to Descartes, we should apply our mind to question everything that is

not firmly rooted in “reason”.

Thus, to doubt is to think:

“I doubt, therefore I am.”

This is referred as Descartes’ doubt.

Believe

Idea

First action

Pascal (1623 – 1662)

Fermat (1601-1665)

Applying reason to study chances.

A `six’ in four rolls:-direct method.

A six in the first roll.

.6

1)(

.6

1)(

.6

1)(

.6

1)(

4

3

2

1

DP

DP

DP

DP

4

3

2

1

D

D

D

D

A six in the second roll.

A six in the third roll.

A six in the fourth roll.

At least one `six’ in four rolls ?6

4

6

1

6

1

6

1

6

1

What’s wrong? (Profit is based on 1 : 1.)

A six in four rolls:-indirect method.

Not a six in the first roll.

.6

5)(

.6

5)(

.6

5)(

.6

5)(

4

3

2

1

CP

CP

CP

CP

4

3

2

1

C

C

C

C

Not a six in the second roll.

Not a six in the third roll.

Not a six in the fourth roll.

Not a `six’ in four rolls .4321 CCCC

?)( 4321 CCCCP

?)][( 4321 CCCCNotP

Together.

.BthenABtogetherA

if and only if events A and B are independent.

That is, the results of the first experiment A have no effect on the results of the second experiment B, and vis versa.

).(*)()( BPAPBAP

Pascal became interested in problem that dates to medieval times, if not earlier, the problem of the points. Suppose two players

agree to pay a certain number of games, and whoever be the first to score five wins is the one who wins.

Unfortunately, they are interrupted before they can finish. How should the stake be divided among them if, say, one has won four games and the other has won three? Given that in each game the

two players have equal chance of winning.

Player AWon 4 games.

Player BWon 3 games.

Game 8

A wins A winsB wins B wins

Each case has equal probability for A & B .

Game 9

A wins B winsA wins B wins

Overall

A winsA winsA winsB wins

2

1

Blank page for you to write notes.

Generalizations, binomial expansion and expectation.

Flipping two fair coins simultaneously.

TT

TH

HH

Results: Cases.

Only one case as both coins have to be head.

Two cases:

ab

ba

TH

TH

Only one case as both coins have to be tail.

When you toss two pennies there are three possibilities, Two Heads

Two Tails

One of each =

So does that mean there is a 1/3 chance of getting a double head?

Everyone learning statistics knows that is wrong. In fact, if we label the coins A and B, there are four possibilities,

Two Heads Two Tails A=head, B=tail A=tail, B=head

So the chance of getting a double head is 1/4 not 1/3.

One day Bose was demonstrating the 'ultra-violet catastrophe' to his students, that is, he was showing them that theory predicts a curve far different from experiments.

He made the equivalent error on predicting that two pennies come down double heads one time in three –

Surprisingly, the calculation came out in accordance with experiment.

An embarrassing mistake if you set out to prove that it didn't. An even more embarrassing error when he realized he made a “kindergarten” mistake.

In desperation, he wrote to Einstein, who saw at once what it meant:

Two photons (light particles) are fundamentally indistinguishable.

Therefore the mathematics should make no distinction between photon A and photon B, just treat them as 'two photons'.

Dice.

Two dice totals

Die 1

Die 2

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 11 1210

TotalNumber of

combinationsProbability

2 1 2.78%

3 2 5.56%

4 3 8.33%

5 4 11.11%

6 5 13.89%

7 6 16.67%

8 5 13.89%

9 4 11.11%

10 3 8.33%

11 2 5.56%

12 1 2.78%

Total 36 100%

Cards and probabilities.

* Probability of being dealt a pair in two cards is 3/51.

* The probability of being dealt 5 Spades (called a Spade flush) in 5 cards is

.2000

1

48

9.

49

10.

50

11.

51

12.

52

13

Four French suits: diamonds (♦), spades (♠), hearts (♥) and clubs (♣),

~0.0000154

~0.00024

~0.0014

~0.004Excluding Straight flush

Later in the course, we will see that:

~0.00197