Suppose two people agreed to bet on the toss of a coin. If it comes up heads, Jane wins. If it comes up tails, John wins. Which one of them do you think would be the clear winner at the end of 50 tosses?
If you said there’s no reliable way to predict the outcome, you’re way ahead of most amateur gamblers. However, you can calculate the probability, or most likely outcome, of those coin tosses.
Since a coin has two sides, there’s a one in two chance that it will land heads up, meaning that Jane wins, or tails up, meaning that John wins. This is expressed in probability as a ½ chance, or 50 percent. So no matter how long Jane and John toss their coin, the way their wagering game is set up – one coin, two sides, one outcome per play – means the odds of winning will always be 50-50.
Now imagine that you were offered $10 to choose a queen from a standard deck of 52 playing cards having one queen in each suit – diamonds, hearts, clubs and spades. The deck has been shuffled well to mix up the cards. It costs you $1 each time you draw a card from the deck. What would your odds of winning be this time? Here’s how to figure them out:
You know that a standard playing card deck has 52 cards with four queens. This means that your probability of drawing a queen is 4/52, or 1 in 13 tries. Gambling always expresses odds (probability) as the chance against your winning. So the odds here are 48 to 4 (52 cards minus 4 queens), or 48/4. Since in mathematics you always want to reduce fractions to the lowest expression, your odds of winning would be 12/1 or 12:1 against.
Now let’s add another wrinkle. The winnings you’re being offered are $10 against a $1 wager, or playing fee. In other words, the $10 win you’ve been offered, minus the $1 you’re paying to play, equals a $9 profit, or odds of 9:1. At this point, it becomes clear that paying you $10 to pull a queen out the deck at a paying rate of $1 per play isn’t a fair game, because you’re not playing for the true odds of 12 to 1. Like the coin toss, if playing for the true odds, both you and the “house” would eventually wind up even IF you were being paid $12 each time you won wagering $1. You’d be paid $12 in winnings and get to keep your $1 bet.
In the end, at 9:1 odds, the “house” is winning this “pick a queen” game. That’s because you’d have to play 13 times, at $1 per bet, in order to win $10. The “house” would earn $3, or a 23 percent profit on your wagering. So just how long do you think you’d be able to keep playing at this rate?
Each time you’re offered an arrangement similar to this in a casino game, the difference between the odds of the reduced payoff and the actual odds is known as “the house edge.” This percentage going to the house changes from game to game because of the probabilities involved. In some games, such as craps (throwing dice to make combinations of numbers), the percentage to the house changes with play because of the large number of probabilities on a pair of six-sided numbered dice.
In some cases, the probabilities are so complex that the house charges a playing fee to ensure that it earns its edge. For instance, in baccarat, the house may charge a commission on some bets, while in poker, there’s frequently an entry fee for playing.
Looking at gambling from the perspective of you, the player, the house edge can also be called “the player’s expectation.” This is the true amount of money you can expect to win or lose from each dollar you bet against the house. For example, in the “pick a queen” game, your player expectation would be -23 percent, or minus 23 percent. That means you can expect to lose 23 cents out of each dollar you bet.
Understanding and practicing this rule of gambling can improve your wagering experience, because you’ll be able to calculate for yourself the probability – the odds – of winning. The first rule to keep in mind is this: Over time, players will always lose at a game with a negative rate of player expectation. This is sometimes expressed by the saying, “The house always wins.” However, once you master the ability to calculate the odds for yourself, you vastly increase your chances of success.
Always keep in mind that your return on any bet – your payoff – equals 100 percent minus the house edge. The house edge equals a percentage of your bet that the house keeps. It’s what you “pay to play.” In order for the house to make sure it keeps its percentage, or edge, winnings are reduced from the true odds.
The house’s expectations
In the following table, examples show various games along with what the house expects to earn from your bets, whether you win or lose. This expectation is based on the rules of the game, what the house odds are, and what the true odds are.
In the table, player’s expectations are calculated according to the percentage of each dollar you bet that you can plan on winning or losing in the long run. A positive expectation, shown by a plus sign (+), is the percentage you can expect to win, and a negative expectation, shown by a minus sign (–), is what you can expect to lose, over time playing a particular game consistently.
A range of probabilities for the player’s expectations means that the odds of winning or losing vary with the amount of the bet, the player’s skill level, and the game’s playing conditions or rules. Since there is considerable variance with most games, only a few examples are shown on the following chart.
Certain games on the chart are the only ones in which a player with good skills can bet with a positive expectation at times. These games and their characteristics include:
- Blackjack, under favorable conditions;
- Bingo, if playing in Las Vegas;
- Sports wagering, when you’ve learned how to “read the sheet” and assess opportunities;
- Video poker, when playing a machine that you trust to work properly.
Poker can be another game with positive expectations. However, it takes many years and a lot of wagering to develop the experience and skill to create a consistent positive expectation as a poker player, and even longtime poker champions can run “hot” and “cold.” As the saying goes, that’s why they call it “gambling.”
Table #1: Your Odds of Winning or Losing at Certain Games
Win (+) or Loss ((–)
|Expected Earnings per Game
from a $100 bet
|Bingo||-15 % to +0 %||$85 to $100 or more|
|Blackjack (single deck)||-6 % to +1.5 %||$94 to $101.50|
|Caribbean Stud||-5.2 % to -2.6 %||$95 to $97.40|
|Craps||-13% to -.6%||$87 to $99.40|
|Let It Ride||-3.5%||$97|
|Slot machines||-20% to -2.7%||$80 to $97.30|
|-6.3% to +1%|
In the following articles, the advantages and disadvantages to each game will be explained in greater detail. In general, remember that when you’re playing with a positive expectation, the advantage goes to the player (you). With a negative expectation, the advantage goes to the house. When you have a positive expectation – when the odds are in your favor – the longer you play, the more likely it is that you’ll win. Whenever you face a negative expectation – when the odds are with the house – it’s more likely that you’ll lose the longer you play.
In other words, in games with a negative expectation for the player, it’s the same situation we noted earlier – “the house always wins.” When you determine that you’re in the latter situation, it’s time to drop out of the game, or as experienced gamblers say, “cut your losses.”
The influence of the Bell Curve
A bell curve is a graphic way of showing statistical probability. Mathematically, the bell curve is called “a symmetrical distribution of numbers around an average, or mean.” Statisticians use the bell curve as a standard frame of reference when they calculate the outcome of events.
You can draw your own bell curves once you understand the chances of winning or losing any particular game, such as in the previous chart. For instance, say you’re playing a game with a -5 percent player expectation, and you play the game 100 times. Your chart could be drawn with the vertical axis representing the probability and the horizontal access representing the actual results. Then you would note the number of times you win versus the odds, and draw the bell curve to connect the points of intersection.
It would quickly become apparent by this chart that in playing a game with a -5 percent negative expectation, the more hands you play the more your chances of winning line up with the average of -5 percent. The ultimate expected outcome of such a game means you’re losing 5 percent of each dollar you bet. Clearly, you’re not making a profit, or reward, on the risk you’re taking.
When you calculate the odds accurately, you begin to say that the old saying about coming out ahead in the long run isn’t always true when it comes to gambling. If you’re playing a game with a negative player expectation, you can be assured that the more you play, the more you will lose. However, if you calculate that the game you’re playing has a positive player expectation, then the more you play, the more likely it is that you will end up winning.
So what do you do when faced with a negative-expectation game? You have two options:
- Limit your participation by playing less, making fewer bets. Then quit while you’re still ahead, because you will undoubtedly lose if you keep going.
- Play games with a higher expectation that involve a certain level of skill, rather than chance. For example, a game with a -1 percent chance of losing will give you a larger return than a game with a -10 percent chance of losing.
Playing Odds Outside the Norm
There’s another tool that statisticians use when calculating probabilities. It’s called the Standard Deviation (SD). It helps you figure out your likelihood of winning or losing based on our old friend, the Bell Curve. It works like this:
Say you bet $1 on each of 100 plays of a game with a -5 percent player expectation. The odds say that you’ll lose $5. In this particular game, the Standard Deviation from this -5 percent is plus or minus 1, or about 68 percent of the time you play. Thus your chances of losing line up somewhere between losing $15 and winning $5. You have only about a 16 percent chance of winning more than $5 or losing more than $15.
So now we go back to the basic rules we mentioned earlier for all games with a negative expectation: don’t play too long, because the odds favor the house over the long run, and play games with a higher player expectation.