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Helping Professionals


Games, Expected Return, House Edge and Payback Percentage, How the House Edge Works, Playing Multiple Hands, Tickets or Bets

Games

Most gambling games fall into one of four categories based on the role probability plays in the outcome.

Games of Pure Chance

In games of pure chance, the underlying events on which gamblers bet are both random and independent, and the player has no real opportunity to make a profit in the long term. The outcome of the game depends entirely on the results of a random number generator such as dice, bingo balls or a slot machine. Examples of games of pure chance include:

  • lotteries,
  • keno,
  • bingo,
  • slot machines,
  • roulette,
  • craps, and
  • baccarat.

Games of Both Skill and Luck

Although a random number generator plays an important role in these games (e.g., a shuffled deck of cards), the player’s success also depends on his/her knowledge, strategies and decisions while playing. A good player can minimize losses with poor hands and maximize wins with good hands. Examples of games of both luck and skill include:

  • blackjack,
  • poker, and
  • dominoes.

Games of Subjective Probability

These games are not actually random, and the outcome of one event (a game this week) is not independent of a later outcome (a game next week with the same players). Teams or horses differ in their actual ability to win. The complex nature of the games, however, and the inclusion of uncertain events (e.g., pitching, batting, catching and running) add some degree of random chance to the outcomes of the games. In these games a bettor is pitted, not against the actual outcomes of the games, but against the subjective guesses of the gambling industry about the outcomes (e.g., the odds, point spread, or money line set by a bookie) or against the mass habits of other bettors (e.g., parimutuel odds or the fluctuating prices of the stock market). Examples of this form of gambling include:

  • sports betting,
  • horse race betting, and
  • stock market investing.

Games of True Skill

Sometimes people place bets on skilled games that they are engaged in. These are private bets between individuals and will only be dealt with briefly in this chapter. Unlike games of both skill and luck, such as cards and dominoes, which include a random number generator, in games of true skill, no random number generator is used. A dart or bowling ball will not always go where the player aims, but as their level of skill increases, uncertainty is reduced. Examples of common games of skill that people bet on include:

  • golf,
  • chess,
  • hoops and one-on-one (forms of basketball),
  • pool, and
  • darts.

Expected Return, House Edge and Payback Percentage

Sometimes people use the expression “the odds are against you” or “those are good odds” when evaluating a particular game. In reality the odds (or more accurately, probability) of a win is not all that important in determining whether a game is a “good” or “bad” bet. If a player placed bets on all 38 slots on a roulette wheel, they would have a 100% chance of winning, but it would result in a net loss of $2. What is important for people who gamble to know is the relationship between the chance of winning and the amount of money they get paid for a win. In all commercial gambling, the player does not get paid enough for a win to make up for their losses when they do not win. This difference is called the house edge. In the case of American roulette, the house edge is $2 for every $38 bet or 5.3%

There is often confusion about terms used to describe the house edge. “House edge,” “house advantage,” “payback percentage” and “expected return” are different words for describing the same basic idea: the games are set up in such a way that, in the long run, the house will make money from the game. The math is simple, but the terminology can be confusing.

The house edge is the percentage of money that the player can expect to lose on each bet, averaged over the long term. The payback is the percentage of money the player can expect to get back from each bet, averaged over the long term. If the payback rate is less then 100%, as it is in all commercial gambling, it means that on average the player will lose money. The house edge and the payback add up to 100%. In American roulette the payback percentage is 94.7% and the house edge is 5.3% (94.7 + 5.3 = 100%).

To make matters a little more confusing, sometimes the phrases “player’s disadvantage” and “expected return” are also used. The expected return is simply the house edge with a negative sign in front of it and is most often used in mathematical discussion of probability. Casinos, however, are more likely to describe their games in terms of payback percentage. Their choice of terms is no doubt a deliberate ploy to emphasize what the player gets back (e.g., 94.7%) rather than what the player does not get back (e.g., 5.3%). To avoid confusion, we will use the term payback percentage

What does a payback of 90% mean? Suppose a man started with $120, played for 2¼ hours on a 25-cent slot machine and now has $20 left. “Where’s my 90% payback?” he asks himself. A 90% payback does not mean the player wins 90% of the time. It does not mean the player wins back 90% of what he/she has lost. It does not mean that the player is ever due to win. It does not mean that the player gets back 90% of what he/she started with. It means that, for each bet, the player loses 10% (house edge).

In the process of losing $100 on a 25-cent machine over the course of 2¼ hours (assuming 75 cents per spin, 10 spins per minute), the player will actually make about $1,000 in bets. A 90% payback means the player loses 10% of what he/she actually bets. Ten percent of $1,000 is $100. A loss of $100 is a 90% payback! So the 90% payback is 90% of the $1,000 the player bet, not 90% of the $120 that the player started with. The reason that people often lose most of their money, even when the payback is 90%, is that they reinvest their winnings in the game. The temporary wins in the cycle of bet-win-bet-lose that gradually eats away at a player’s money is called the “churn.” Players who keep reinvesting their winnings will eventually lose it all. A player can test this by using a player’s card. Many casinos offer point cards that give the player 1 point per $10 bet; the points can be redeemed for a rebate. If you played until you lost $100 on the same slot machine, you would find that you had earned close to 100 points (good for a rebate of about $5 in some casinos), indicating that you had in fact placed $1,000 in bets. (Note that actual results will vary.)

A payback of 90% means losing 10%, on average, of what is actually bet. However, if gambling simply meant losing 10 cents on each and every $1 spin of a slot machine, it is doubtful that anyone would play. But gambling involves fundamental uncertainty. Sometimes the players win, sometimes they lose. On average, players lose, but often they do in fact win, at least in the short term. The occasional win makes it difficult for the player to determine the actual house edge while playing.

The volatility of a game is the bet-to-bet variation in actual outcome. For example, on any one bet a player might lose, win back what was bet, win double the bet, or win 10 times the bet. Slots are very volatile; a player could win 1,000 times his/her bet. In any playing session, the erratic nature of the wins and losses and the volatility of the game make it impossible for the player to determine the size of the house advantage (see Turner & Horbay, 2003, for examples).

How the House Edge Works

In gambling, the house achieves its edge by paying back less than the true odds of winning. In many games, the house edge is hidden so that the player cannot exactly determine the edge from merely playing the game. The house edge is accomplished in different ways for different games, but is particularly easy to illustrate in the case of roulette.

Roulette

The roulette wheel contains 36 numbered slots that are coloured black or red (1–36) and two slots that are coloured green (0 and 00), for a total of 38 slots. If a player bet on one number, the probability of winning is 1/38, but the payout for a win is only 36 chips. If the player bets a one-dollar chip on 17 and the ball comes to rest in slot 17, the player is paid $36. (Note that in actual roulette play, the payback comes in the form of $35 placed beside the player’s bet, and the player gets to keep the original $1 bet). The 2 green slots conveniently represent the house’s profit per bet (2 chips for every 38 chips bet). It is not the green slots per se that determine the house edge, but the fact that the player is paid only 36 chips for a win, as if the green slots did not exist. Thus the payback for playing the roulette wheel is 36/38 (94.7%). The difference between the payout odds and the true odds—2 chips—goes to the house.

Because of the random and unpredictable results of each spin of the roulette wheel, a player’s wins and losses vary greatly. A player could win several times in a row or lose for hours and hours. The longer a gambler stays at the table, however, the more his/her betting is likely to approach the situation of a gambler who places a bet on every one of the 38 numbers. (Even if the gambler keeps betting the same number over and over, the random variations of the wheel still make this case.) As a result, the player will lose, on average, 2 chips for every 38 chips bet.

Slots, Lotteries and Keno

The same approach to payouts is used in games with multiple possible wins, such as slots, lotteries and keno. In each case, the total payouts reflect less than the true odds of winning. Figuring out the house edge in these games, however, is more complex. The odds of winning the top prize on a slot machine might be 1 in 200,000 for a payout of 8,000 times the original bet. This may seem like a very poor return—substantially less than the odds of winning—but when all the possible paybacks on every bet are added up, the total payback percentage is usually between 90% and 97%. Table 1 illustrates how to compute the total payback using the payout table of a fictitious slot machine. Notice how most of the payback is returned to the player in the small prizes, not the large prizes. A payout table for a real slot machine would have many more prizes, some smaller, some larger.

 

Table 1. Hypothetical payout table for a slot machine.

Match

Probability

Credits paid

Return

(probability times prize)

3 Double diamonds

0.000004

$8,000

0.032

3 Sevens

0.0000364

$1,000

0.0364

3 Cherries

0.000122

$300.00

0.036607

3 Bars

0.07

$5.00

0.35

1 Cherry

0.1488

$3.00

0.4464

Total probability of a win or hit rate

0.2181 (21.8%)

Total return

(Payback)

0.901

(90.1%)

Note this is just an illustration; real slots payout tables usually have a lot more winning combinations. Note how the small prizes make up most of the payback.

Horse Race Betting

In horse racing, the racetrack takes a cut off the top of the total pool of money bet, and then distributes the rest to the winners. The clever part, however, is how the racetrack uses odds to make the bettor’s chances seem better than they really are. Mathematically, odds are a means of reporting probability that expresses the ratio of chances of losing to chances of winning. A probability of 1/10 or 10% translates into odds of 9 to 1 (9 chances of losing to 1 chance of winning). The odds reported by the racetrack, however, are not an estimate of probability but a statement of the payout for each horse. If a horse’s posted odds were translated into a percentage, it would be higher than the true probability of the horse winning.

A racetrack might quote a horse at 2 to 1 (a 33% chance of winning), but its true odds might be 3 to 1 (a 25% chance of winning). If the player places a $2 bet on the horse and wins, he/she gets back $6 ($4 win + $2 bet). If the player were paid according to the true odds, he/she would get back $8 ($6 win + $2 bet).

If the posted odds for every horse in a race were converted into percentages and then added up, the total would be over 130%. Of course the total of all the horses’ actual chances of winning must add up to 100% because only one horse can win.

Suppose 5 horses are running in a race. The favourite is quoted at 3 to 2 (a 40% of winning), another horse is quoted at 2 to 1 (33%), another is quoted at 3 to 1 (25%), one is quoted at 4 to 1 (20%), and last horse is quoted at 5 to 1 (16.6%). Adding up the percentages (40 + 33 + 25 + 20 + 16.6) gives a total of 134%. By overestimating each horse’s chances, the racetrack pays out less than the true risk of each bet, thus ensuring itself a profit. The use of odds, however, to describe the horses’ chances of winning hides this fact.

Other Betting Games

In other games, the house edge is achieved through a commission (e.g., baccarat, craps and the stock market), a vigorish, which is a fee charged by a bookie to accept a bet (e.g., sports betting), or a rake percentage of the win (e.g., poker). In blackjack, the house edge comes from the fact that when the dealer and the player both bust (exceed 21), the house still wins. In games such as slots or bingo, the odds are unknown to the player, but the mechanics of how the house makes its money are the same.

Table 2 gives a summary of the payback percentages of a variety of games. Note that in many games the payback can depend on how well the game is played, so Table 2 is more an approximation than an authoritative guide. A comprehensive source of information on casino games and how the bets work can be found at the internet site The Wizard of Odds (www.wizardofodds.com).

 

Table 2. Approximate payback percentage of different types of gambling.

Game

Specific Bets

Approximate Payback

Notes

Craps

Pass, don’t-pass, come,
don’t-come

  1. 6%

Higher than 99% if the player adds an additional free-odds bet.

Craps

Other bets

84%-98.3%

 

Roulette

American (0, 00)

  1. 7%

 

 

European (0)

  1. 4%
  2. 7%

Most bets

Even-money bets with surrender

Blackjack

Any

With card counting

94%-99.5%

99%-105%

Payback depends on skill and the specific rules used at the casino.

Baccarat

Banker bets

Player bets

Tie bets

  1. 8%
  2. 6%
  3. 0%

 

Slots

Line games

85%-98%

Higher denomination (e.g., $5) slots have a higher payback.

Bingo

Any

85%-100%

Depends on size of prize and the number of bingo cards in play.

Horse racing

Ordinary bets

Exotic bets (daily double)

83%

75%-83%

Depends on the skill of the player and of other players.

 

Sports bets

Any

  1. 4%

Based on 9% commission charged by the bookie, but is also influenced by skill.

Poker

Casino games such as
hold ’em or 7-card stud.

95% to 98%

Depends on the size of the rake (e.g., 5%). The rake percentage varies and is usually capped at some maximum.

Games with larger bets usually have a smaller rake or a flat rate table fee.

Stock Market

Short term

90%-99%

 

Based on the size of the broker’s commission. For larger investments, some brokers charge a lower commission.

Note that many of the games have variations and different bets that make a specific payback percentage difficult to pin down. In this table we have tried to indicate the approximate payback only. For stock market investments, the average return is positive for long-term investments (as high as +10% per year), but with frequent short-term investments (e.g., day trading) the commission decreases the profitability of investments.

Playing Multiple Hands, Tickets or Bets

Understanding the house edge helps explain why playing multiple hands, numbers or bingo cards provides no real advantage. The actual loss is based on the total amount of money that is gambled. If a player bought two tickets instead of one, the probability of winning doubles, but the expected loss also doubles. As discussed above, playing all 38 numbers of an American roulette wheel would guarantee a loss of $2 per spin of the wheel. The payback percentage (2/38) when a player covers the entire wheel is exactly the same as when a player covers only one number. The same is true for multiple hands, tickets, cards or bets. It does not matter how many tickets a player buys, the payback percentage remains the same. Making multiple bets, however, usually means betting more money. When the bet is doubled, the expected loss is also doubled. This is an important point, if the payback is less than 100%, the more that is bet the more that is lost in the long run.

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