Transcript m162-9
Betting By 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 Outline Expected value Pari-mutuel betting Mark Six -- Background 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) 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 Number Name 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/w atch?v=DFYSUX-L9jU Ticket of Mark Six Prize Qualification Name of Prize Condition Payout 1st 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? Example. (winning probability of 6th prize[3 Drawn Numbers + Extra Number]) Number of possible way to buy a selection: C 649 Number of possible way to win 6th prize: C36 C11 C242 Winning probability of 6th prize: C36 C242 C649 17,220 1.2314103 13,983,816 Remarks: nCr n! (n r )!r! Winning Probability Name of Prize 1st Prize Probability 1 C 649 2nd Prize C56 C649 3rd Prize 1 7.1511 108 13,983,816 1 4.2907 10 7 2,330,636 C56 C142 C649 4th Prize C46 C142 C649 5th Prize C46 C242 C649 6th Prize C36 C242 C649 7th Prize 252 1.8021105 13,983,816 630 4.5052105 13,983,816 12915 9.235710 4 13,983,816 17,220 1.2314103 13,983,816 C36 C342 229,600 0.01642 C649 13,983,816 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)=>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 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 ] xi pi i 1 where x i is the amount of payout of the (i)th prize and pi is the probability of getting the (i)th prize Example (Expected Value of Mark Six) Tips: E[ X ] xi pi What is the answer? i 1 Ans: Name of Prize Prize = (1st Prize) x (Probability of 1st Prize) + 1st Prize $23,115,220 (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 $ 320 7th Prize $ 40 E(X) (6th Prize) x (Probability of 6th Prize) + (7th Prize) x (Probability of 7th Prize) 8 7 23115220 7.151110 1165120 4.290710 73100 1.8020105 9600 4.5052 105 640 9.2357 10 4 320 0.001231 40 0.01642 = $5.5446 < $10 which is the wager!!! Pari-mutuel 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 Assumptions You 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 payout? Calculate: Total Pool (TP) The amount wagered on the winning horse (W) Then, the payout (P) (per $1 wagered) TP P W Assumptions Made The operator does not make any profit i.e. TP is distributed to all winners Reality 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 Standard Horse Racing Bets 12% Dividends to winners ≥82.5% HKJC’s Commission ≤5.5% Total 100% Payout distributed By the HKJC Calculate: Total Pool (TP) The amount wagered on the winning horse (W) Commission (C) and Duty (D) Then, the payout (P) (per $1 wagered) TP C D P 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. The Inventor 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 Places He Owned: Places He Owned: Additional Information The website of the HKJC: http://www.hkjc.com/english/betting/betti ng_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) 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 End