BettingBettingBy Billy Lai and Charlie Law

Probability and Percentage

Gambling is the wager of something of material value on an event with an uncertain outcome with the primary intent of winning additional material goods

Presentation OutlinePresentation OutlineExpected valuePari-mutuel betting

Mark Six -- BackgroundLottery Betting

Organized by the Hong Kong Jockey Club (HKJC) Since 1975

3 Times a Week (Tuesday, Thursday and Saturday or Sunday)

Hong Kong Jockey Club( 香港賽馬會 )

Non-profit organization founded in1884

Provides betting entertainment in HK such as horse racing and Mark Six

Largest taxpayer (~$1billion each year)

What is Mark Six?Lottery Style Game with 49

colored balls numbered from 1 to 49

In each game, 7 balls are drawn from these 49 balls

Name of First Six Numbers Drawn: Drawn NumberName of Last One Drawn:

Extra Number

Rules Of Mark Six Rule 1:

Choose 6 out of 49 numbers

Rule2:Wager $10 on each Selection

For example, in the ticket,

Selection 1:Selection 2:

Total wager :$10 x = $ 0

http://www.youtube.com/watch?v=DFYSUX-L9jU

Ticket of Mark Six

Prize QualificationName of Prize Condition Payout1st Prize

Pick all the 6 Drawn Numbers

Total number of winning unit investments in the First Division Prize.

Minimum First Division Prize Fund:

$ 8 millions

2nd Prize

Pick 5 Drawn Numbers + Extra Number

Total number of winning unit investments in the Second Division Prize

3rd Prize

Pick 5 Drawn Numbers

Total number of winning unit investments in the Third Division Prize

4th Prize

Pick 4 Drawn Numbers + Extra Number

$ 9600

5th Prize

Pick 4 Drawn Numbers

$ 640

6th Prize

Pick 3 Drawn Numbers + Extra Number

$ 320

7th Prize

Pick 3 Drawn Numbers

$ 40

What is the winning probability?

496C

422

11

63 CCC

3496

422

63 102314.1

816,983,13

220,17 C

CC

!)!(

!

rrn

nnCr

Example. (winning probability of 6th prize[3

Drawn Numbers + Extra Number])

Number of possible way to buy a selection:

Number of possible way to win 6th prize:

Winning probability of 6th prize:

Remarks:

Winning ProbabilityName of Prize

Probability

1st Prize

2nd Prize

3rd Prize

4th Prize

5th Prize

6th Prize

7th Prize

8496

101511.7816,983,13

11 C

4496

422

64 102357.9

816,983,13

12915 C

CC

5496

421

64 105052.4

816,983,13

630 C

CC

5496

421

65 108021.1

816,983,13

252 C

CC

7496

65 102907.4

636,330,2

1 C

C

01642.0816,983,13

600,229496

423

63 C

CC

3496

422

63 102314.1

816,983,13

220,17 C

CC

Question The Prize Qualification on

17/2/2011 is listed below, What is the expected value of the winnings of one selection?

A: E(X)=>10B: 7.5<E(X)<10 C: 5<E(X)<=7.5D: 2.5<E(X)<=5E: 0<E(X)<=2.5

Name of Prize Prize

1st Prize $23,115,220

2nd Prize

$1,165,120

3rd Prize $73,100

4th Prize $ 9600

5th Prize $ 640

6th Prize $ 320

7th Prize $ 40

Expected ValueExpected Value ( denoted by E[X] )

is the average value an experiment is expected to produce if it is repeated a large number of times.

where is the amount of payout of the (i)th prize andis the probability of getting the (i)th prize

1

][i

ii pxXE

ip

ix

Example (Expected Value of Mark Six)What is the answer? Tips:

Name of Prize Prize

1st Prize $23,115,220

2nd Prize $1,165,120

3rd Prize $73,100

4th Prize $ 9600

5th Prize $ 640

6th Prize $ 320

7th Prize $ 40

Ans:

E(X)

= (1st Prize) x (Probability of 1st Prize) +

(2nd Prize) x (Probability of 2nd Prize) +

(3rd Prize) x (Probability of 3rd Prize) +

(4th Prize) x (Probability of 4th Prize) +

(5th Prize) x (Probability of 5th Prize) +

(6th Prize) x (Probability of 6th Prize) + (7th Prize) x (Probability of 7th Prize)

01642.040001231.0320102357.9640

105052.49600108020.173100

102907.41165120101511.723115220

4

55

78

= $5.5446

< $10 which is the wager!!!

1

][i

ii pxXE

Pari-mutuelPari-mutuelGame – Horse Racing Instructions:Two people (or one person if

necessary) in a groupEach group receives a paper on

which there is an amount you can wager on 1 out of the 4 horses provided

Place your paper into the collection box

AssumptionsAssumptionsYou guys are basically gambling

addicts so please wager every penny you have on one horse

One group can only wager on one horse

How to calculate the How to calculate the payout?payout?Calculate:Total Pool (TP)The amount wagered on the

winning horse (W)Then, the payout (P) (per $1

wagered)

P TP

W

Assumptions MadeAssumptions Made

The operator does not make any profit

i.e. TP is distributed to all winners

RealityRealityIn reality, the operator usually

takes away a certain percentage of the TP as the commission

In Hong Kong, under the Betting Duty Ordinance (BDO), the duty accounts for a certain percentage of the proceeds from horse racing

The distribution of Proceeds The distribution of Proceeds from Horse Racingfrom Horse Racing

Type of Bet Standard Horse Racing Bets

Duty 12%

Dividends to winners ≥82.5%

HKJC’s Commission ≤5.5%

Total 100%

Payout distributed By the Payout distributed By the HKJCHKJCCalculate:Total Pool (TP)The amount wagered on the

winning horse (W)Commission (C) and Duty (D)Then, the payout (P) (per $1

wagered)

P TP C D

W

An extract from the HKJC’s An extract from the HKJC’s website concerning place website concerning place betting:betting:The Net Pool will be divided into

two or three parts, according to whether two Place Betting or three Place Betting is being conducted, then each such divided part will be divided by the number of Unit Bets on the horse to which the divided part relates.

The InventorThe Inventor

Joseph Oller

Invented pari-mutuel betting in 1867

Sentenced to prison in 1874Pari-mutuel betting system

legalized by the French authorities in 1891

Focused on the entertainment industry from 1876

Places He Owned:Places He Owned:

Places He Owned:Places He Owned:

Additional InformationAdditional InformationThe website of the HKJC:http://www.hkjc.com/english/

betting/betting_rule.htm

Homework(Question1)Based on the Prize Qualification of

Mark Six on 17/02/2011, Calculate the expected income (expected value) of the Jockey Club if the total turnover is $ 46,613,470.

Name of Prize

Prize

1st Prize

$23,115,220

2nd Prize

$1,165,120

3rd Prize

$73,100

4th Prize

$ 9600

5th Prize

$ 640

6th Prize

$ 320

7th Prize

$ 40

Homework(Question2)Homework(Question2)Using the figures in the game

(which will be provided in the uploaded ppt), calculate the rate of payout if the game was organized by the HKJC, given the commission is 5.5% of TP

Horses Wager

Horse 1 $100

Horse 2 $1200

Horse 3 $1900

Horse 4 $1200

Homework (Extra Credit Problem)Given that the total turnovers in a

Mark Six game and a horse racing game are the same and the prize qualification of both games and the BDO(in P.18) are the same as shown in questions 1 and 2. Then, from which game, Mark Six or horse racing, does the Hong Kong Jockey Club receive more commission? State your reasons.

The EndThe End