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Transcript of Betting
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
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.
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
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.