# GAMING MANAGEMENT SEMINAR SERIES. Prices of Casino Games. Jason Zhicheng Gao 高 志 成

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1 GAMING MANAGEMENT SEMINAR SERIES Prices of Casino Games Jason Zhicheng Gao 高 志 成

2 Casinos-Games-Players Casinos make money by offering games to the players. Games are casinos major products, and players are the customers.

3 Demonstration of Some Casino Games Keno( 澳 门 白 鸽 票 ) Slot Machines( 老 虎 機 ) Video Poker Baccarat( 百 家 乐 ) Blackjack(21 点 )

4 Prices of Casino Games What price does a player pay when playing a casino game? Keno 1-spot example.

5 Keno 1-Spot Example. The 1-spot game has payback 3 for 1, i.e., the player risks 1 unit to win 2 units. Suppose the player bets \$100 for the 1-spot game. If the marked number appears among the 20 drawn numbers, the player gets paid back \$300 (i.e. wins \$200); otherwise the player loses \$100.

6 House Percentage as the Theoretical Price The probability that the marked number appears among the 20 drawn numbers is 20/80=1/4. Hence the house s losing probability is 1/4, and the house s winning probability is 3/4. The house s expected value is 100x(3/4)-200x(1/4)=25(dollars) The house percentage is defined as the house s expected value divided by the betting amount, which is μ=25/100=0.25=25%.

7 Actual Price v.s. Theoretical Price Suppose the game is repeated n times, and let S(n) denote the house net win. Then S(n)/n is the house s average win per game (actual price of the game). The law of average says that S(n)/n approaches the house percentage μ as n gets large.

8 Fluctuation of The Actual Price Fluctuation of actual price can be measured by the standard deviation σ of the game. In the keno 1-spot example, we have 2 σ = (3/ 4)(1 µ ) + (1/ 4)( 2 µ ) = According to The Central Limit Theorem, the probability that S(n)/n lies in the interval σ S kσ can be approximated by the µ k < n < µ n n + n 2 integral k k e x 2 / 2 dx 2π

9 Since / 2 e x 2π dx = , we can be 99.73% sure that S(n)/n lies in the interval 3σ S n 3σ µ < < µ + n n n n ,000 1,000,000 Actual Price S(n)/n [0, 0.50] [0.211, ] [0.2461, ] House Net Result S(n) [0, 122] [2110, 2897] [246100, ]

10 Slot Machine

11 A Slot Machine Example Pay Combination Payout WW XX XX 2 XX WW XX 2 XX XX WW 2 WW WW ~~ 5 WW ~~ WW 5 ~~ WW WW 5 WB WB WB 5 W1 W1 W1 10 W6 W6 W6 50 W8 W8 W8 200 WW WW WW 400 SYM R1 R2 ~~ B 9 7 6B 4 4 8B 1 1 WW 1 1 XX : any symbol. W1 : WW or 1B. W6 : WW or 6B. W8 : WW or 8B. WB : WW,1B,6B,or 8B. R

12 The Slot Machine Example A B C D House Net Result Probability AxB Bx(A-µ)^ Total µ= Standard Deviation σ=

13 Video Poker

14 Video Poker Example Note that video poker games require skills. Skilled players pay lower prices. For the Jacks-or-better machine with a fixed 9/6 payback schedule, the price which skilled players pay is about 0.5%. If the machine pays progressive jackpot for royal flushes, and if the jackpot reaches the level 1800 for 1. Then a skilled player has about 2.0% advantage over the casino.

15 Final Hand Probability Distribution Final Hand Payout Royal Flush 800 for 1 Straight Flush 50 for 1 Four of a Kind 25 for 1 Full House 9 for 1 Flush 6 for 1 Straight 4 for 1 Three of a Kind 3 for 1 Two Pair 2 for 1 High Pair (JJ-AA) 1 for 1 No Pay 0 for 1 House Percentage μ Standard Deviation σ Final Hand Probability % 0.011% 0.236% 1.151% 1.10% 1.123% 7.45% 12.93% 21.46% 54.54% 0.5% 4.387

16 Baccarat( 百 家 乐 )

17 House Percentages Banker Player Tie Winning Probability 45.8% 44.6% 9.6% Payoff 0.95 to 1 1 to 1 8 to 1 House Percentage 1.1% 1.2% 13.6%

18 The Price of The Pair Bet In Macao, players may bet that the player s (and/or banker s) initial two cards form a pair. The payoff for the pair bet is 11 to 1. 1-deck 6-deck House Percentage 29.41% 11.25% Standard Deviation

19 Blackjack(21 点 )

20 Prices for Playing Blackjack If the player mimics the dealer s strategy, i.e., always hits when the hand is below 17, and stands when the hand is 17 or higher. Then the house percentage is about 5.5% (7.8% - 2.3%). A skilled player can use the basic strategy to reduce the house percentage to about 0.7%. An expert player can use card-counting techniques to get an advantage over the house.

21 Use Basic Strategy to Reduce House Percentage Option Mimic the dealer Proper standing Proper doubling-down Proper pair-splitting Proper hitting soft 17,18 Total μ Standard Deviation σ House Percentage 5.5% -2.78% -1.34% -0.4% -0.3% 0.68% 1.12

22 The Price of Insurance Bet When the dealer s up-card is an Ace, the player may put up a side bet (insurance bet), which is equal to half of the original bet, that the dealer s hand is a blackjack. The payoff of the insurance bet is 2 to 1. The house percentage of the insurance bet depends on the exposed cards and the number of decks being used.

23 The Price of Insurance Bet Single Deck Examples Player s cards (10,10) (8,9) House Percentage 14.28% 2.04%

24 Thank You Q & A The next talk will be near the end of November.

25 The Standard Normal Distribution y = e x 2 / 2 2π

26 Banker s Third Card Rule- Continued If Player's hand draws a third card and Banker hand has point equal to 3,4,5, or 6. Then Banker s hand Banker s hand draws when Player s third card is 0,1,2,3,4,5,6,7,9 2,3,4,5,6,7 4,5,6,7 6,7 Banker s hand does not draw when Player s third card is 8 0,1,8,9 0,1,2,3,8,9 0,1,2,3,4,5,8,9

27 Example: The 10-spot Keno Ticket Catch Payoff 5 2 for for for for for for 1 House Percentage Standard Deviation Probability % % % % % % 27.65%

28 Red/Black Bet The house percentage of the bet is 1 (20/ 38) 1 (18/ 38) = 2 / 38 = 5.26% The variance and standard deviation of the bet are ( ) 2 (20/ 38) + ( ) (18/ 38) Hence at confidence level 90%, the volatility index is V. I. = = 2 =

29 美 式 轮 盘 庄 家 百 分 率 Straight Up Bet: The house percentage of the bet is 1 37 / / 38 = 2 / 38 = 5.26% The standard deviation of the bet is ( ) 2 (37 / 38) + ( ) (1/ 38) = Hence at confidence level 90%, the volatility index is V. I. = =

30 Volatility Comparisons Straight-Up Bet V.I Confidence Limits ± 9.479/ n n =10, ± n =1,000, ± Red/Black Bet ± / n ± ±

31 A Slot Machine Example Pay Combo #Per Reel #Hits Remark JW XX XX 1, 31, x31-12x10 XX JW XX 31,1, x31-14x10 XX XX JW 31,31, x31-14x12 JW JW XX 1,1,31 21 XX=~~ JW XX JW 1,31,31 19 XX=~~ XX JW JW 31,1,31 17 XX=~~ AB AB AB 15,13, x13x J 1J 1J 10,8, x8x7-1 5J 5J 5J 5,5,4 99 5x5x4-1 7J 7J 7J 2,2,2 7 2x2x2-1 JW JW JW 1,1,1 1 1x1x1

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