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(Updated September 2020)

The following rules form the basis of participation in the Penn National Gaming, Inc. (“Penn”) mychoice patron loyalty program (“mychoice”). A patron’s participation in mychoice will be governed by these rules. It is the patron’s responsibility to read these rules so that they understand mychoice’s rules and benefits, as well as the patron’s responsibilities. A patron’s enrollment as a member of mychoice and/or their use of a mychoice card constitutes their acceptance of these rules. mychoice tier levels are set each year based on the mychoice member’s patronage from the previous earning period. mychoice members may continually upgrade to the next highest tier throughout the year as they attain the Tier Points (as defined below) required to advance.

RULES

1. mychoice members will receive card tier status based on the following accrued Tier Points and/or by applicable casino property management approval:
a. The Owners Club* - 200,000+ Tier Points
b. Elite – 50,000 – 199,999 Tier Points
c. Preferred – 18,000 – 49,999 Tier Points
d. Advantage – 1,000 – 17,999 Tier Points
e. Choice – 0-999 Tier Points

Each player then scores what was won as follows: 1 point for each sweep, ace, and little casino, 2 points for big casino, 1 point for taking the most spades, and 3 points for taking the most cards (unless tied). Game is 11 or 21 points. Three- and four-handed casino games follow the same rules, with four playing in two partnerships. Tier Points are based on the following (each a “Tier Point”): a. Slots and video lottery terminals: $5 coin–in = 1 Tier Point b. Video poker: $10 coin-in = 1 Tier Point c. Table games: Based on game played, average bet and time played d. Electronic table games = $40 coin-in = 1 Tier Point e. Live poker games: 1 hour played = 20 Tier.

*All mychoice members who otherwise earn enough Tier Points to reach The Owners Club status must first be approved by casino management, in its sole discretion.

2. Tier Points are based on the following (each a “Tier Point”):
a. Slots and video lottery terminals: $5 coin–in = 1 Tier Point
b. Video poker: $10 coin-in = 1 Tier Point
c. Table games: Based on game played, average bet and time played
d. Electronic table games = $40 coin-in = 1 Tier Point
e. Live poker games: 1 hour played = 20 Tier Points
f. Non-gaming on-property purchases: $5 spent = 1 Tier Point
g. Horse racing: $10 bet = 1 Tier Point
h. Social gaming (mychoicecasino.com and/or mobile applications): $0.50 = 1 Tier Point
i. Sports Bets
i. Straight Bets, Futures: Every $10 bet = 1 Tier Point
ii. Parlays: Every $5 bet = 1 Tier Point
iii. All other sports bets: Every $10 bet = 1 Tier Point
iv. Any tier points earned from a voided bet or an event that does not take place are subject to forfeiture

Any and all methods of Tier Point calculation are subject to change in Penn’s sole discretion, with or without notice. Please note that the ways in which Tier Points may be earned and the methods of Tier Point calculation may vary by participating casino property and/or regulation.

3. In order to receive Tier Points a mychoice member must have his or her mychoice card properly inserted into an electronic gaming machine.

4. A mychoice member can earn Tier Points on a table game by presenting their mychoice card at a participating table, so that table games staff may record its use.

5. mychoice members’ Tier Points roll-over from one year to the next, but only Tier Points that are over and above a mychoice members’ then-current tier status, capped as follows, shall roll-over:
a. The Owners Club – Tier points over and above 200,000, capped at 30,000 total Tier Points
b. Elite – Tier points over and above 50,000, capped at 15,000 Tier Points
c. Preferred – Tier points over and above 18,000, capped at 5,000
d. Advantage – Tier points over and above 1,000, capped at 500
e. Choice - Choice members may not roll-over any Tier Points.

6. Please note that individual rewards, benefits, comps and/or other similar items earned based, in part or wholly, on tier ranking may vary by participating casino property.

7. It is the mychoice member’s responsibility to ensure that play is registering properly on their mychoice card.

8. Tier Points are awarded at any mychoice participating property.

9. Tier Points have no cash value.

10. The award of any card tier is within the sole discretion of Penn’s Chief Marketing Officer or her designee (“mychoice management”) regardless of a mychoice member’s Tier Points and may be rejected at any time for any reason. Owners Club and Elite members who accrue more than 100,000 Tier Points in a calendar year may be eligible to receive ONE (1) companion card. The companion selected by the mychoice member must also be a mychoice member. The companion card may be cancelled by casino property and or mychoice management at any time. Specific benefits of the companion card will vary based on the benefits available at the participating property from which it is issued. Companion cards are non-transferable.

*Please note participating casino properties may, at their sole discretion, make this benefit available to other tiered members.

11. Certain expenditures utilizing a mychoice card may result in mychoice members earning mycash rewards. mycash rewards are determined based in the sole discretion of the participating casino property, but may be based in part on a member’s tier, play and/or purchases.

12. mycash can be redeemed in multiple ways according to a mychoice member’s election, as listed below:
$1 in mycash = $2 Food (excluding 3rd party outlets); OR
= $2 Hotel; OR
= $2 on-property retail outlets; OR
= $1 on social gaming (mychoicecasino.com and/or mobile applications); OR
= $1 Bonus Rewards (slot play); OR
= $1 for table play (where available)
= $.50 for cash (available exclusively to mycash Mastercard® credit card members)
= 2 Credits on mychoice Mall

Any and all methods of mycash calculation and/or redemption are subject to change in Penn’s sole discretion, with or without notice. Please note that methods of mycash calculation and/or redemption may vary by participating casino property and/or regulation. mycash earned through social gaming (mychoicecasino.com and/or mobile applications) is not redeemable for cash or free slot play.

At a minimum, a player will earn at least $1 of mycash for every 400 Tier Points earned for Advantage, Preferred, Elite, and Owners Club.

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13. mychoice members playing live poker games may receive mycash per hour played, as determined by each participating property.

14. mycash and/or Tier Points may take up to SIXTY (60) days to appear in a member’s mychoice account.

15. mycash balances may be removed after SIX (6) months of mycash inactivity (i.e., if a mychoice member does not earn any mycash over a period of SIX (6) months, any balance may be removed). Except for guests in the Elite and Owners Club tier, mycash will not expire as long as they retain that tier status. Please note that earning mycash through promotional means shall still be considered “inactivity” for purposes of this Section.

16. mychoice members earn Owners Credits upon reaching the Owners Club tier status. The number of Owners Credits earned upon reaching Owner’s Club tier status, as well as a mychoice member’s continued earning of Owners Credits thereafter, may vary by participating casino property. Any and all methods of Owners Credits calculation and/or redemption are subject to change in Penn’s sole discretion, with or without notice. Please note that items mychoice members may exchange their Owners Credits for may vary by participating casino property.

17. Access to the VIP lounge for Owners Club, Elite and Preferred mychoice members is subject to certain conditions and restrictions as stated below:
a. Preferred tier mychoice members must have accrued a minimum number of Tier Points, as determined by each participating property.
b. All mychoice members must be checked-in and have their mychoice card and a valid government issued photo ID.
c. Any guests of a member must enter with the mychoice member cardholder and may remain in the VIP lounge only as long as the member is present.
d. Guests may be placed on a waitlist if the lounge is approaching its capacity.
e. Any individual whom enters a VIP Lounge, who is not qualified under the mychoice program rules (including guests of qualified members) must have casino marketing host approval or a signed VIP Lounge pass for entry.
f. mychoice members and their guests must be dressed appropriately, as determined by each participating property.
g. Food and beverage may not be removed from any VIP lounge.
h. Casino property management reserves the right to refuse access to a lounge at any time;
i. Casino property management reserves the right to revoke access to the lounge from any guest at any time.
j. Please note that access to any VIP lounge, including the mychoice tier levels allowed in any lounge, may vary.

18. You must be TWENTY-ONE (21) years of age or older to participate in the mychoice program and be able to present valid government issued photo ID.

19. An individual’s mychoice card, prizes, Tier Points, mycash, Owners Credits, entries and any other mychoice benefits or awards are non-transferable and the use of a mychoice card by any person other than the individual named on the card is prohibited. In the event of a divorce of a mychoice member, the named mychoice member retains all mycash unless otherwise stated by a valid and final court order submitted to the applicable casino property. In the event of a death of a mychoice member, all mycash remains non-transferable and shall expire as set forth herein.

20. Falsification of any information as it applies to these rules, a mychoice card, the mychoice program at-large or an individual’s affiliation with Penn and/or its subsidiaries may result in ineligibility.

21. The applicable casino property may adjust mycash balances resulting from any malfunction, fraud or misuse as determined appropriate in its sole discretion.

22. By participating in the mychoice program a patron agrees to all official rules, as amended from time to time, and decisions of the applicable casino property and mychoice management whose decisions in all aspects of the mychoice program shall be final and binding.

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23. All mychoice members are responsible for any international, United States federal, state or local taxes and/or government fees that may be imposed in connection with the mychoice program.

24. Each mychoice member shall indemnify and hold harmless Penn, including its subsidiaries and each’s employees, officers, directors, agents and assigns (collectively, the “Penn Indemnitees”) from and against any damage, loss, claim or liability arising from his or her participation in the mychoice program. Each mychoice member agrees that the Penn Indemnitees assume no liability of any kind for any property loss or damage or personal injury occurring in connection with any prizes.

25. mychoice management reserves the right to cancel, change or modify this program or the official rules at any time in accordance with the applicable gaming regulations and/or law.

26. Redemption of mychoice benefits is subject to the terms and conditions of these rules, as may be amended from time to time without notice in Penn’s sole discretion.

27. The applicable casino property and/or mychoice management reserve the right, in their sole discretion, to void any mychoice benefits of a mychoice member who they believe has tampered with or impaired the administration security, fairness or proper use of any mychoice benefit.

28. Any dispute or situation not covered by these rules will be resolved by the applicable casino property, mychoice management and/or Penn, whose decision shall be final and binding on all participants.

29. The mychoice program is subject to all applicable federal, state and/or local rules and regulations, including those applicable to the gaming industry, and all aspects of the program are subject to approval of the appropriate regulatory authorities and are void where prohibited.

30. By participating in the mychoice program, participants hereby grant Penn, including its subsidiaries/affiliates, permission to use his/her name, photograph, and likeness in connection with the advertising and promotion of its products, in any manner and in any medium, without additional compensation, as Penn in its sole discretion, shall deem appropriate or desirable. Participants irrevocably consent to the unrestricted use, in perpetuity, for purposes including, but not limited to, display, advertising, sale and trade, and including any alterations or modifications whatsoever of said photograph, including the negatives and prints made there from and by Penn, its employees, agents, customers, successors, and assigns forever. Participants waive any right, which he/she may otherwise have, to inspect or approve the photographs or prints made from the negative thereof, with respect to: 1) any alterations or modifications, any material or commentary, 2) any publication using the name of the undersigned, no name, or fictitious name, 3) and use for the purpose of publicity, illustration, commercial art and 4) any advertising of products or services.

31. Persons who are on a Disassociated Patrons, Voluntary/State Exclusion or Self Eviction list in jurisdictions in which Penn operates or who have been otherwise excluded from a Penn property are not eligible to participate in the mychoice program.

32. In the event a mychoice member is excluded from any Penn property (either voluntarily or via some other method), or in the sole discretion of Penn has violated these rules are engaged in conduct Penn deems unacceptable or prohibited, Penn reserves the right to require that member to immediately forfeit any and all mycash, Tier Points, Owners Credits or benefits associated with the mychoice program.

33. In the event that Penn and/or any one of its subsidiaries decides to transfer pre-existing points and/or membership status, from an outside membership participating program similar to the mychoice patron loyalty program, discretion with regard to transfer status, point-to-point ratio, the membership level at which an individual is transferred and/or any other similar decision related thereto, shall belong solely to Penn and/or its subsidiaries and affiliates, in their discretion.

34. Benefits may be awarded for hospitality purchases at participating outlets at participating Penn properties. These purchases may include but are not limited to: hotel rooms, food and dining purchases, as well as retail merchandise at Penn operated outlets. The amount earned will not include sales tax, tips or comps. Visit Player Services at a mychoice property for a complete list of participating outlets. Tier Points or mycash earned with hospitality purchases may not count towards eligibility for any gaming promotions. It is the responsibility of the member to present his/her mychoice card for tracking before paying. If the member pays before presenting their mychoice card, points or comps earned will be forfeited. Penn is not responsible for lost ratings, Tier Points or mycash due to failure to present one’s mychoice card. Local regulations regarding earnings on hospitality purchases including alcohol may apply, restrict or otherwise alter this Section. Please contact your local Penn casino property with questions.

35. Owners Club member travel reimbursement is valid once annually and can only be used for airfare or mileage reimbursement for travel to any participating Penn casino property or applicable cruise. The Owners Club member is responsible for booking travel and submitting receipts to their participating casino’s host within THIRTY (30) days of trip completion for reimbursement. Any such trip is valid only when visiting a Penn casino property other than that Owners Club member’s primary (or dominant) Penn casino. Trip must be booked while the member’s Owners Club status is active. Blackout dates may apply.

36. All benefits relating to hotel stays are only valid at Penn owned properties.

37. Owners Club members are guaranteed a comped room when staying in a Penn hotel in accordance with the following restrictions: a) FORTY EIGHT (48) hour notice must be given, b) a maximum of 6 room nights per month for each Owners Club member and c) a maximum of THREE (3) room nights in a row. This Section is contingent upon there being an available room at the Penn hotel in question.

38. Gambling problem? Penn cares about the welfare of its guests and encourages players to play responsibly and only within their means. If gambling is causing you to experience financial problems or problems with your personal, family or professional life, help is just a phone call away at the numbers below:

39. Gambling problem? Please use the following:
• Missouri: 1-888-BETSOFF;
• Indiana: 1-800-9WITHIT;
• Massachusetts: 1-800-426-1234;
• Ohio: 1-800-589-9966;
• Pennsylvania/Illinois/New Jersey/West Virginia: 1-800-GAMBLER;
• New Mexico: 1-800-572-1142;
• Mississippi: 1-888-777-9696;
• Michigan: 1-800-270-7117;
• All Else: 1-800-522-4700.

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Introduction

At its core the business of casino gaming is pretty simple. Casinos make money on their games because of the mathematics behind the games. As Nico Zographos, dealer-extraordinaire for the 'Greek Syndicate' in Deauville, Cannes, and Monte Carlo in the 1920s observed about casino gaming: 'There is no such thing as luck. It is all mathematics.'

With a few notable exceptions, the house always wins - in the long run - because of the mathematical advantage the casino enjoys over the player. That is what Mario Puzo was referring to in his famous novel Fools Die when his fictional casino boss character, Gronevelt, commented: 'Percentages never lie. We built all these hotels on percentages. We stay rich on the percentage. You can lose faith in everything, religion and God, women and love, good and evil, war and peace. You name it. But the percentage will always stand fast.'

Puzo is, of course, right on the money about casino gaming. Without the 'edge,' casinos would not exist. With this edge, and because of a famous mathematical result called the law of large numbers, a casino is guaranteed to win in the long run.

Why is Mathematics Important?

Critics of the gaming industry have long accused it of creating the name 'gaming' and using this as more politically correct than calling itself the 'gambling industry.' The term 'gaming,' however, has been around for centuries and more accurately describes the operators' view of the industry because most often casino operators are not gambling. Instead, they rely on mathematical principles to assure that their establishment generates positive gross gaming revenues. The operator, however, must assure the gaming revenues are sufficient to cover deductions like bad debts, expenses, employees, taxes and interest.

Despite the obvious, many casino professionals limit their advancements by failing to understand the basic mathematics of the games and their relationships to casino profitability. One casino owner would often test his pit bosses by asking how a casino could make money on blackjack if the outcome is determined simply by whether the player or the dealer came closest to 21. The answer, typically, was because the casino maintained 'a house advantage.' This was fair enough, but many could not identify the amount of that advantage or what aspect of the game created the advantage. Given that products offered by casinos are games, managers must understand why the games provide the expected revenues. In the gaming industry, nothing plays a more important role than mathematics.

Mathematics should also overcome the dangers of superstitions. An owner of a major Las Vegas strip casino once experienced a streak of losing substantial amounts of money to a few 'high rollers.' He did not attribute this losing streak to normal volatility in the games, but to bad luck. His solution was simple. He spent the evening spreading salt throughout the casino to ward off the bad spirits. Before attributing this example to the idiosyncrasies of one owner, his are atypical only in their extreme. Superstition has long been a part of gambling - from both sides of the table. Superstitions can lead to irrational decisions that may hurt casino profits. For example, believing that a particular dealer is unlucky against a particular (winning) player may lead to a decision to change dealers. As many, if not most, players are superstitious. At best, he may resent that the casino is trying to change his luck. At worst, the player may feel the new dealer is skilled in methods to 'cool' the game. Perhaps he is even familiar with stories of old where casinos employed dealers to cheat 'lucky' players.

Understanding the mathematics of a game also is important for the casino operator to ensure that the reasonable expectations of the players are met. For most persons, gambling is entertainment. It provides an outlet for adult play. As such, persons have the opportunity for a pleasant diversion from ordinary life and from societal and personal pressures. As an entertainment alternative, however, players may consider the value of the gambling experience. For example, some people may have the option of either spending a hundred dollars during an evening by going to a professional basketball game or at a licensed casino. If the house advantage is too strong and the person loses his money too quickly, he may not value that casino entertainment experience. On the other hand, if a casino can entertain him for an evening, and he enjoys a 'complimentary' meal or drinks, he may want to repeat the experience, even over a professional basketball game. Likewise, new casino games themselves may succeed or fail based on player expectations. In recent years, casinos have debuted a variety of new games that attempt to garner player interest and keep their attention. Regardless of whether a game is fun or interesting to play, most often a player will not want to play games where his money is lost too quickly or where he has a exceptionally remote chance of returning home with winnings.

Mathematics also plays an important part in meeting players' expectations as to the possible consequences of his gambling activities. If gambling involves rational decision-making, it would appear irrational to wager money where your opponent has a better chance of winning than you do. Adam Smith suggested that all gambling, where the operator has an advantage, is irrational. He wrote 'There is not, however, a more certain proposition in mathematics than that the more tickets [in a lottery] you advertise upon, the more likely you are a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty.'

Even where the house has an advantage, however, a gambler may be justified if the amount lost means little to him, but the potential gain would elevate him to a higher standing of living. For example, a person with an annual income of $30,000 may have $5 in disposable weekly income. He could save or gamble this money. By saving it, at the end of a year, he would have $260. Even if he did this for years, the savings would not elevate his economic status to another level. As an alternative, he could use the $5 to gamble for the chance to win $1 million. While the odds of winning are remote, it may provide the only opportunity to move to a higher economic class.

Since the casino industry is heavily regulated and some of the standards set forth by regulatory bodies involve mathematically related issues, casino managers also should understand the mathematical aspects relating to gaming regulation. Gaming regulation is principally dedicated to assuring that the games offered in the casino are fair, honest, and that players get paid if they win. Fairness is often expressed in the regulations as either requiring a minimum payback to the player or, in more extreme cases, as dictating the actual rules of the games offered. Casino executives should understand the impact that rules changes have on the payback to players to assure they meet regulatory standards. Equally important, casino executives should understand how government mandated rules would impact their gaming revenues.

The House Edge

The player's chances of winning in a casino game and the rate at which he wins or loses money depends on the game, the rules in effect for that game, and for some games his level of skill. The amount of money the player can expect to win or lose in the long run - if the bet is made over and over again - is called the player's wager expected value (EV), or expectation. When the player's wager expectation is negative, he will lose money in the long run. For a $5 bet on the color red in roulette, for example, the expectation is -$0.263. On the average the player will lose just over a quarter for each $5 bet on red.

When the wager expectation is viewed from the casino's perspective (i.e., the negative of the player's expectation) and expressed as a percentage, you have the house advantage. For the roulette example, the house advantage is 5.26% ($0.263 divided by $5). The formal calculation is as follows:

EV = (+5)(18/38) + (-5)(20/38) = -0.263
(House Advantage = 0.263/5 = 5.26%)

When this EV calculation is performed for a 1-unit amount, the negative of the resulting value is the house edge. Here are the calculations for bets on a single-number in double-zero and single-zero roulette.

Double-zero roulette (single number bet):
EV = (+35)(1/38) + (-1)(37/38) = -0.053
(House Advantage = 5.3%)

Single-zero roulette (single number bet):
EV = (+35)(1/37) + (-1)(36/37) = -0.027
(House Advantage = 2.7%)

The house advantage represents the long run percentage of the wagered money that will be retained by the casino. It is also called the house edge, the 'odds' (i.e., avoid games with bad odds), or just the 'percentage' (as in Mario Puzo's Fools Die). Although the house edge can be computed easily for some games - for example, roulette and craps - for others it requires more sophisticated mathematical analysis and/or computer simulations. Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.

Because this positive house edge exists for virtually all bets in a casino (ignoring the poker room and sports book where a few professionals can make a living), gamblers are faced with an uphill and, in the long run, losing battle. There are some exceptions. The odds bet in craps has zero house edge (although this bet cannot be made without making another negative expectation wager) and there are a few video poker machines that return greater than 100% if played with perfect strategy. Occasionally the casino will even offer a promotion that gives the astute player a positive expectation. These promotions are usually mistakes - sometimes casinos don't check the math - and are terminated once the casino realizes the player has the edge. But by and large the player will lose money in the long run, and the house edge is a measure of how fast the money will be lost. A player betting in a game with a 4% house advantage will tend to lose his money twice as fast as a player making bets with a 2% house edge. The trick to intelligent casino gambling - at least from the mathematical expectation point of view - is to avoid the games and bets with the large house advantages.

Some casino games are pure chance - no amount of skill or strategy can alter the odds. These games include roulette, craps, baccarat, keno, the big-six wheel of fortune, and slot machines. Of these, baccarat and craps offer the best odds, with house advantages of 1.2% and less than 1% (assuming only pass/come with full odds), respectively. Roulette and slots cost the player more - house advantages of 5.3% for double-zero roulette and 5% to 10% for slots - while the wheel of fortune feeds the casino near 20% of the wagers, and keno is a veritable casino cash cow with average house advantage close to 30%.

Games where an element of skill can affect the house advantage include blackjack, video poker, and the four popular poker-based table games: Caribbean Stud poker, Let It Ride, Three Card poker, and Pai Gow poker. For the poker games, optimal strategy results in a house edge in the 3% to 5% range (CSP has the largest house edge, PGP the lowest, with LIR and TCP in between). For video poker the statistical advantage varies depending on the particular machine, but generally this game can be very player friendly - house edge less than 3% is not uncommon and some are less than 1% - if played with expert strategy.

Blackjack, the most popular of all table games, offers the skilled player some of the best odds in the casino. The house advantage varies slightly depending on the rules and number of decks, but a player using basic strategy faces little or no disadvantage in a single-deck game and only a 0.5% house edge in the common six-deck game. Despite these numbers, the average player ends up giving the casino a 2% edge due to mistakes and deviations from basic strategy. Complete basic strategy tables can be found in many books and many casino-hotel gift shops sell color-coded credit card size versions. Rule variations favorable to the player include fewer decks, dealer stands on soft seventeen (worth 0.2%), doubling after splitting (0.14%), late surrender (worth 0.06%), and early surrender (uncommon, but worth 0.24%). If the dealer hits soft seventeen it will cost you, as will any restrictions on when you can double down.

Probability versus Odds

Probability represents the long run ratio of (# of times an outcome occurs) to (# of times experiment is conducted). Odds represent the long run ratio of (# of times an outcome does not occur) to (# of times an outcome occurs). If a card is randomly selected from a standard deck of 52 playing cards, the probability it is a spade is 1/4; the odds (against spade) are 3 to 1. The true odds of an event represent the payoff that would make the bet on that event fair. For example, a bet on a single number in double-zero roulette has probability of 1/38, so to break even in the long run a player would have to be paid 37 to 1 (the actual payoff is 35 to 1).

Confusion about Win Rate

There are all kinds of percentages in the world of gaming. Win percentage, theoretical win percentage, hold percentage, and house advantage come to mind. Sometimes casino bosses use these percentages interchangeably, as if they are just different names for the same thing. Admittedly, in some cases this is correct. House advantage is just another name for theoretical win percentage, and for slot machines, hold percentage is (in principle) equivalent to win percentage. But there are fundamental differences among these win rate measurements.

The house advantage - the all-important percentage that explains how casinos make money - is also called the house edge, the theoretical win percentage, and expected win percentage. In double-zero roulette, this figure is 5.3%. In the long run the house will retain 5.3% of the money wagered. In the short term, of course, the actual win percentage will differ from the theoretical win percentage (the magnitude of this deviation can be predicted from statistical theory). The actual win percentage is just the (actual) win divided by the handle. Because of the law of large numbers - or as some prefer to call it, the law of averages - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage.

Because handle can be difficult to measure for table games, performance is often measured by hold percentage (and sometimes erroneously called win percentage). Hold percentage is equal to win divided by drop. In Nevada, this figure is about 24% for roulette. The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues. Suffice it to say that the casino will not in the long term keep 24% of the money bet on the spins of roulette wheel - well, an honest casino won't.

To summarize: House advantage and theoretical win percentage are the same thing, hold percentage is win over drop, win percentage is win over handle, win percentage approaches the house advantage as the number of plays increases, and hold percentage is equivalent to win percentage for slots but not table games.

· Hold % = Win/Drop
· Win % (actual) = Win/Handle
· H.A. = Theoretical Win % = Limit(Actual Win %) = Limit(Win/Handle)
· Hold Percentage ¹ House Edge

Furthermore, the house advantage is itself subject to varying interpretations. In Let It Ride, for example, the casino advantage is either 3.51% or 2.86% depending on whether you express the advantage with respect to the base bet or the average bet. Those familiar with the game know that the player begins with three equal base bets, but may withdraw one or two of these initial units. The final amount put at risk, then, can be one (84.6% of the time assuming proper strategy), two (8.5%), or three units (6.9%), making the average bet size 1.224 units. In the long run, the casino will win 3.51% of the hands, which equates to 2.86% of the money wagered. So what's the house edge for Let It Ride? Some prefer to say 3.51% per hand, others 2.86% per unit wagered. No matter. Either way, the bottom line is the same either way: assuming three $1 base bets, the casino can expect to earn 3.5¢ per hand (note that 1.224 x 0.0286 = 0.035).

The question of whether to use the base bet or average bet size also arises in Caribbean Stud Poker (5.22% vs. 2.56%), Three Card Poker (3.37% vs. 2.01%), Casino War (2.88% vs. 2.68%), and Red Dog (2.80% vs. 2.37%).

For still other games, the house edge can be stated including or excluding ties. The prime examples here are the player (1.24% vs. 1.37%) and banker (1.06% vs. 1.17%) bets in baccarat, and the don't pass bet (1.36% vs. 1.40%) in craps. Again, these are different views on the casino edge, but the expected revenue will not change.

That the house advantage can appear in different disguises might be unsettling. When properly computed and interpreted, however, regardless of which representation is chosen, the same truth (read: money) emerges: expected win is the same.

Volatility and Risk

Statistical theory can be used to predict the magnitude of the difference between the actual win percentage and the theoretical win percentage for a given number of wagers. When observing the actual win percentage a player (or casino) may experience, how much variation from theoretical win can be expected? What is a normal fluctuation? The basis for the analysis of such volatility questions is a statistical measure called the standard deviation (essentially the average deviation of all possible outcomes from the expected). Together with the central limit theorem (a form of the law of large numbers), the standard deviation (SD) can be used to determine confidence limits with the following volatility guidelines:

Volatility Analysis Guidelines
· Only 5% of the time will outcomes will be more than 2 SD's from expected outcome
· Almost never (0.3%) will outcomes be more than 3 SD's from expected outcome

Obviously a key to using these guidelines is the value of the SD. Computing the SD value is beyond the scope of this article, but to get an idea behind confidence limits, consider a series of 1,000 pass line wagers in craps. Since each wager has a 1.4% house advantage, on average the player will be behind by 14 units. It can be shown (calculations omitted) that the wager standard deviation is for a single pass line bet is 1.0, and for 1,000 wagers the SD is 31.6. Applying the volatility guidelines, we can say that there is a 95% chance the player's actual win will be between 49 units ahead and 77 units behind, and almost certainly between 81 units ahead and 109 units behind.

A similar analysis for 1,000 single-number wagers on double-zero roulette (on average the player will be behind 53 units, wager SD = 5.8, 1,000 wager SD = 182.2) will yield 95% confidence limits on the player win of 311 units ahead and 417 units behind, with win almost certainly between 494 units ahead and 600 units behind.

Casino Card Game Point System

Note that if the volatility analysis is done in terms of the percentage win (rather than the number of units or amount won), the confidence limits will converge to the house advantage as the number of wagers increases. This is the result of the law of large numbers - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage. Risk in the gaming business depends on the house advantage, standard deviation, bet size, and length of play.

Player Value and Complimentaries

Using the house advantage, bet size, duration of play, and pace of the game, a casino can determine how much it expects to win from a certain player. This player earning potential (also called player value, player worth, or theoretical win) can be calculated by the formula:

Earning Potential = Average Bet ´ Hours Played ´ Decisions per Hour ´ House Advantage

For example, suppose a baccarat player bets $500 per hand for 12 hours at 60 hands per hour. Using a house advantage of 1.2%, this player's worth to the casino is $4,320 (500 ´ 12 ´ 60 ´ .012). A player who bets $500 per spin for 12 hours in double-zero roulette at 60 spins per hour would be worth about $19,000 (500 ´ 12 ´ 60 ´ .053).

Many casinos set comp (complimentary) policies by giving the player back a set percentage of their earning potential. Although comp and rebate policies based on theoretical loss are the most popular, rebates on actual losses and dead chip programs are also used in some casinos. Some programs involve a mix of systems. The mathematics associated with these programs will not be addressed in this article.

Casino Pricing Mistakes

In an effort to entice players and increase business, casinos occasionally offer novel wagers, side bets, increased payoffs, or rule variations. These promotions have the effect of lowering the house advantage and the effective price of the game for the player. This is sound reasoning from a marketing standpoint, but can be disastrous for the casino if care is not taken to ensure the math behind the promotion is sound. One casino offered a baccarat commission on winning banker bets of only 2% instead of the usual 5%, resulting in a 0.32% player advantage. This is easy to see (using the well-known probabilities of winning and losing the banker bet):

EV = (+0.98)(.4462) + (-1)(.4586) = 0.0032
(House Advantage = -0.32%)

A casino in Biloxi, Mississippi gave players a 12.5% edge on Sic Bo bets of 4 and 17 when they offered 80 to 1 payoffs instead of the usual 60 to 1. Again, this is an easy calculation. Using the fact that the probability of rolling a total of 4 (same calculation applies for a total of 17) with three dice is 1/72 (1/6 x 1/6 x 1/6 x 3), here are the expected values for both the usual and the promotional payoffs:

Usual 60 to 1 payoff: EV = (+60)(1/72) + (-1)(71/72) = -0.153
(House Advantage = 15.3%)

Casino Card Game Point System

Promotional 80 to 1 payoff: EV = (+80)(1/72) + (-1)(71/72) = +0.125
(House Advantage = -12.5%)

In other promotional gaffes, an Illinois riverboat casino lost a reported $200,000 in one day with their '2 to 1 Tuesdays' that paid players 2 to 1 (the usual payoff is 3 to 2) on blackjack naturals, a scheme that gave players a 2% advantage. Not to be outdone, an Indian casino in California paid 3 to 1 on naturals during their 'happy hour,' offered three times a day, two days a week for over two weeks. This promotion gave the player a whopping 6% edge. A small Las Vegas casino offered a blackjack rule variation called the 'Free Ride' in which players were given a free right-to-surrender token every time they received a natural. Proper use of the token led to a player edge of 1.3%, and the casino lost an estimated $17,000 in eight hours. Another major Las Vegas casino offered a '50/50 Split' blackjack side bet that allowed the player to stand on an initial holding of 12-16, and begin a new hand for equal stakes against the same dealer up card. Although the game marketers claimed the variation was to the advantage of the casino, it turned out that players who exercised the 50/50 Split only against dealer 2-6 had a 2% advantage. According to one pit boss, the casino suffered a $230,000 loss in three and a half days.

In the gaming business, it's all about 'bad math' or 'good math.' Honest games based on good math with positive house advantage minimize the short-term risk and ensure the casino will make money in the long run. Players will get 'lucky' in the short term, but that is all part of the grand design. Fluctuations in both directions will occur. We call these fluctuations good luck or bad luck depending on the direction of the fluctuation. There is no such thing as luck. It is all mathematics.

Gaming Regulation and Mathematics

Casino gaming is one of the most regulated industries in the world. Most gaming regulatory systems share common objectives: keep the games fair and honest and assure that players are paid if they win. Fairness and honesty are different concepts. A casino can be honest but not fair. Honesty refers to whether the casino offers games whose chance elements are random. Fairness refers to the game advantage - how much of each dollar wagered should the casino be able to keep? A slot machine that holds, on average, 90% of every dollar bet is certainly not fair, but could very well be honest (if the outcomes of each play are not predetermined in the casino's favor). Two major regulatory issues relating to fairness and honesty - ensuring random outcomes and controlling the house advantage - are inextricably tied to mathematics and most regulatory bodies require some type of mathematical analysis to demonstrate game advantage and/or confirm that games outcomes are random. Such evidence can range from straightforward probability analyses to computer simulations and complex statistical studies. Requirements vary across jurisdictions, but it is not uncommon to see technical language in gaming regulations concerning specific statistical tests that must be performed, confidence limits that must be met, and other mathematical specifications and standards relating to game outcomes.

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Summary Tables for House Advantage

The two tables below show the house advantages for many of the popular casino games. The first table is a summary of the popular games and the second gives a more detailed breakdown.

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Note: This summary is the intellectual property of the author and the University of Nevada, Las Vegas. Do not use or reproduce without proper citation and permission.