By Billy Lai and Charlie Law

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
Probability and
Percentage



Expected value
Pari-mutuel betting



Lottery 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)


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 Number
Name of Last One Drawn:
Extra Number


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/w atch?v=DFYSUX-L9jU
Ticket of Mark Six

Prize Qualification
Name of
Prize
Condition
1st Prize Pick all the 6 Drawn
Numbers
Payout
Total number of winning unit investments in the First Division Prize.
Minimum First Division Prize Fund:
$ 8 millions
2nd
Prize
3rd
Prize
Pick 5 Drawn Numbers +
Extra Number
Pick 5 Drawn Numbers
4th Prize Pick 4 Drawn Numbers +
Extra Number
5th Prize Pick 4 Drawn Numbers
Total number of winning unit investments in the Second Division Prize
Total number of winning unit investments in the Third Division Prize
$ 9600
$ 640
6th Prize Pick 3 Drawn Numbers +
Extra Number
7th Prize Pick 3 Drawn Numbers
$ 320
$ 40


Example. (winning probability of 6th prize[
3 Drawn
Numbers + Extra Number]
)

Number of possible way to buy a selection:
C
6
49


Number of possible way to win 6th prize:
C
3
6 
C
1
1

C
2
42
Winning probability of 6th prize:
C
3
6 
C
C
6
49
42
2 
17 , 220
13 , 983 , 816

1 .
2314

10

3
Remarks: nCr

( n
 n !
r )!
r !

Name of Prize Probability
1st Prize
1
C
49
6

1
13 , 983 , 816

7 .
1511

10

8
2nd Prize
C
6
5
C
49
6

1
2 , 330 , 636

4 .
2907

10

7
3rd Prize
C
5
6 
C
1
C
6
49
42

252
13 , 983 , 816

1 .
8021

10

5
4th Prize
C
4
6 
C
1
C
6
49
42

630
13 , 983 , 816

4 .
5052

10

5
5th Prize
6th Prize
7th Prize
C
4
6 
C
C
6
49
42
2 
12915
13 , 983 , 816

9 .
2357

10

4
C
3
6 
C
C
6
49
42
2 
C
3
6 
C
3
C
6
49
42

17 , 220
13 , 983 , 816
229 ,

1 .
2314

10

3
600
13 , 983 , 816

0 .
01642

The Prize Qualification on 17/2/2011 is listed below, What is the expected value of the winnings of one selection?
A:E(X)=>10
B: 7.5<E(X)<10
C: 5<E(X)<=7.5
D: 2.5<E(X)<=5
E: 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 Value ( denoted by E[X] ) is the average value an experiment is expected to produce if it is repeated a large number of times.

E [ X ]
 i


1 x i p i x p i prize and is the probability of getting the (i)th prize

Ans:
Example (Expected Value of Mark Six)
What is the answer? Tips:
E [ X ]
 i



1 x i p i
Name of Prize Prize
E(X)
1st Prize $23,115,220
= (1st Prize) x (Probability of 1st Prize) +
(2nd Prize) x (Probability of 2nd Prize) +
2nd Prize $1,165,120
(3rd Prize) x (Probability of 3rd Prize) + 3rd Prize $73,100
(4th Prize) x (Probability of 4th Prize) + 4th Prize $ 9600
(5th Prize) x (Probability of 5th Prize) +
5th Prize $ 640
(6th Prize) x (Probability of 6th Prize) +
(7th Prize) x (Probability of 7th Prize)
6th Prize $ 320
7th Prize $ 40

23115220

7 .
1511

10

8 
1165120

4 .
2907

10

7

73100

1 .
8020

10

5 
9600

4 .
5052

10

5

640

9 .
2357

10

4 
320

0 .
001231

40

0 .
01642
= $5.5446
< $10 which is the wager
!!!

Game – Horse Racing
Instructions:



Two people (or one person if necessary) in a group
Each 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



You guys are basically gambling addicts so please wager every penny you have on one horse 
One group can only wager on one horse


Calculate:



Total Pool (TP)
The amount wagered on the winning horse (W)
Then, the payout (P) (per $1 wagered)
P 
TP
W

The operator does not make any profit i.e. TP is distributed to all winners



In 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 from
Horse Racing
Type of Bet
Duty
Dividends to winners
HKJC’s Commission
Total
Standard Horse Racing
Bets
12%
≥82.5%
≤5.5%
100%



Calculate:



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 website concerning place 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.

Joseph Oller





Invented pari-mutuel betting in 1867
Sentenced to prison in 1874
Pari-mutuel betting system legalized by the French authorities in 1891
Focused on the entertainment industry from 1876



The website of the HKJC: http://www.hkjc.com/english/betting/betti ng_rule.htm


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


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
Horse 1
Horse 2
Horse 3
Horse 4
Wager
$100
$1200
$1900
$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.