Simple Math: "Odds Job" (Special Series, Part I)

“Baccarat King” Nico Zographos was the leader of a notorious group of sharps known as “The Greek Syndicate.” Back in the 1920s, Nico, two other Greek men, an Armenian and a Frenchman roamed the casinos of Cannes and Monte Carlo and made a killing playing baccarat. By 1927, their capital reserve was estimated at over 100 million francs – which nowadays would add up to more than $1 billion. When asked about his success, Nico Zographos would always repeat the same mantra: “There is no such thing as luck. It is all mathematics.”

No doubt, we’d all like to know how Nico did it. And yet, somehow, learning math in order to become better gamblers just doesn’t make sense. You’re probably thinking: “If gambling is just a bunch of mathematics, how come my math teacher used to drive a Volks to school?” The truth is that becoming a mathematician won’t necessarily transform you into a good gambler, but you’ll never become even halfway decent at the felt if you don’t understand and utilize some basic mathematical principals.

Nico Zographos was a mathematician and his father was an economics professor. So don’t beat yourself up if, despite reading the first installment of our gambling-math, you don’t make 100 million euros within the next few years. No matter how much math you know, you’ll never be able to control the outcome of a bet. And what you should instead aim for is using math to gain control over which bets you make; understanding the numbers will only all you to make intelligent betting decisions and place your chips on the right felt. Beyond that, it’s on you.

Finally, one thing we should note before getting into all the gory details is that math actually makes gambling a lot more fun. Betting on anything without understanding your odds is like watching a basketball game without knowing any of the players or understanding the teams’ dynamics: Yes, you might know the rules and understand the game, but if you don’t know that the star player on one team is playing despite an injury or that the opposing coach must win to keep his job, you’re missing half the action. Likewise, in a casino, wheels spin, dice fall and cards are dealt, but a gambler who doesn’t follow the odds and probabilities is missing out on the finer points that can turn a plain, old game into a passion.

Odds v. Probability

You’ll be happy to know that probability and odds are two ways to express the same (or very similar) concepts. Since it sounds more formal, we’ll hit probability first, then swing back and pick up odds. According to Webster’s Collegiate Dictionary, “probability” is “the relative possibility that an event will occur.” What does this mean? Well, let’s put it this way: The outcome of a coin toss is completely random and can only either be heads or tails. So the probability of a coin landing on heads is one-half, or 50 percent, and the probability of hitting tails is exactly the same – one-half, or 50 percent.

This stands to reason because in a coin toss there are only two possible outcomes. And if we look at every coin toss that ever has been and ever will be, a “fair coin” – that is, a coin that is evenly balanced – will land on both heads and tails about fifty times for every 100 tosses. Mind you, this isn’t to say that there will be exactly 50 heads and 50 tails if you flipped a coin 100 times because each flip is independent and completely unrelated to the previous flip. Coins don’t know and don’t care what happened before, and if they do, they’ve certainly never spoken up. Rather, over a nearly infinite number of flips the number of times each outcome shows up will be roughly equal to its given probability.

So, now, moving on to our next principle, an impossible event has the probability of 0 percent. And an event that will definitely happen has a whole one probability, or 100 percent chance, of occurring. The probability of a coin falling on either heads or tails, on the other hand, is certain (it must fall on one or the other), and therefore, the probability that a flipped coin will land on something is one (or 100 percent). Similarly, because a flipped coin must land on either heads or tails (or even on its edge, which is its own little probability problem, and one we won’t get into) the probability that the coin won’t land on something is zero. Sure, this might seem like a pretty stupid thing to point out. But, as we’ll see, it underpins a very important gambling concept: the house edge.

Notice how, in the chart above, the probability of “heads or tails” (100 percent) equals the sum of the “probability of heads” (50 percent) plus the “probability of tails” (50 percent). If we were assigning a money value to these probabilities, then, we could say that heads is $50 and tails is $50, and together they add up to $100. Now, let’s assume you’re flipping in a casino, and every time the coin is flipped, you have to pay the casino one-fifth of your total bet regardless of whether you win or loose. In this case, heads would be $40 and tails would be $40, and together they would not add up to the true probability of the coin landing on either. This is the first major difference between theoretical statistics and gambling. In statistics, the sum of the probabilities of all possible outcomes for a single event will always add up to one (or 100 percent). In gambling, however, the sum never adds up to one – which is what gives us the “house an edge.”

With this in mind, let’s apply our simple, well-known coin probabilities to the world of gambling. Say someone offers you a bet on a coin toss: You know the probability for each outcome is one-half, so now the only information you’re missing in order to decide if the bet you’re being offered is a good one or not is the payout, which is the amount you can expect to get back on your investment if you win. In other words, you should ask yourself how much a winning bet will pay you for each dollar you put in.

In a perfect world, of course, the correct payout for a 50/50 bet is 1-to-1, which means that you stand to win one dollar for every dollar you put in. This is called an “even-money” bet, since you double your investment every time you win. The chance of winning a coin toss is one-half, so theoretically if you bet enough times, you’d win half your bets. If every time you win you double your investments, in the long run you’ll break even.

Sadly, however, we don’t live in a perfect world. Casinos don’t offer payouts that reflect the true probabilities of the bets they host, because if they did they would all go out of business. Probably the closest thing to a coin toss in a casino is betting on black in roulette (or on red, odds, even, etc.), and it perfectly demonstrates how the house gets an edge. Though these bets do pay even money (1-to-1), the true probability that any one of them will occur is slightly less than one-half because there is also a “green” zero pocket on every roulette wheel. This pocket is neither red nor black, neither odd nor even. If a roulette wheel only consisted of an even number of red and black pockets, the probability that the ball would land on red would be one-half, while the probability of the ball landing on black would also be one-half. The probability of either outcome would be a certainty – with a total probability of 100 percent. Yet the green zero slot makes it so these bets do not add up to 100 percent, and outcome of dividing the number of non-even-money bets by the total number of bets makes up the casino’s percentage edge. For this reason, even though the payout for an even-money bet is 1-to-1, your actual chances of winning are not one-half but 47.37 percent.

How to Read the Odds

Odds, like probability, express the likelihood of a given event occurring. However, unlike probability, odds express this concept as the total number of chosen outcomes versus the total number of unchosen outcomes…. OK, OK. So maybe we have to rewind a moment and explain exactly what we mean here. Looking back at probabilities, you’ll notice that they’re expressed as fractions. In our coin-toss example, for instance, the probability of a coin landing on heads is one-half. Really, then, we could say that the fractional probability is the number of chosen outcome (heads) over the total number of possible outcomes (both heads and tails). As we’ve already said, odds are the chosen over the unchosen, so if we want to convert our coin-toss probability to odds, we’d have one outcome we want to happen (heads) over one that we don’t want to happen (tails). Turned into a fraction, that’s one-over-one, or simply, one.

Otherwise, odds and probabilities pretty much explain the same thing, and the only thing you have to figure out now is how to tell the two apart. Casinos and bookmakers generally go by odds because they look more winnable to the uneducated. And since you only need to know how to pick out one – odds or probability – and use the process of elimination to determine the other, you might as well learn the type of expressions the house works with. There are several different notations that operators use to quote their odds, which more often than not, are listed as payouts. This may seem confusing at first, but if you keep in mind that these are all different representations of the same basic idea you’ll pick it up in no time.

Fractional Odds

Most bookmakers in the U.K. and Ireland use fractional odds – which is probably the reason this type of odds is also known as “British odds,” “UK odds” or “traditional odds.” Fractional odds show how many units are paid out to bettors if they bet given proposition and win, relative to the original stake. So the total winnings can be calculated as “stake multiplied by odds,” plus the original stake, which is returned in case of a win. If this sounds confusing, there’s nothing like an example to clarify things:

EXAMPLE

• Say a bet pays out “5:1,” or 5-to-1. If you place a $100 bet and win, you’ll receive $100 x 5 = $500 (that is, the stake is $100, and the odds are 5-to-1, or 5). That, plus the original $100 stake, means you’ll walk away with $600.
• If a bet pays out “1:5,” or 1-to-5, then betting a $100 stake would result in winnings of $100 x 1/5 = $20. Plus the original bet, you’ll walk away with $120 if you win.
• Odds of “1:1” are known as “even money.” That is, you stand to make the same amount you stand to lose, plus your original wager (so a $100 stake that wins on 1-to-1 would pay out $200).

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  Decimal Odds
  

  Commonly used in Europe and Canada, the fundamental difference between decimal odds and fractional odds is that decimal odds include the original stake in the winning quote. In other words, bettors gives up their original stakes as soon as they make their bets, and the odds quoted include all the money the book will return, not just their winnings. Using a similar example, the decimal representation looks like this:

  EXAMPLE

   

• If a bet pays out “6.0,” it is in fact equal to “5:1” fractional odds. You stand to take away a total of 6 times your original stake if you win. A $100 investment will yield $600.
• A “1.2” quote would be the equivalent of a “1:5” bet. You stand to make 1.2 times your original stake, which is $120 for a $100 stake.
• “Even money odds” are represented in decimal form as “2.0,” since you are doubling your money if you win.

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  American Odds
  

  The good people of the United States of America always find ways to use different measurements than the rest of the world, and sports betting is no exception. The good news is that American odds, unlike inches, feet and yards, actually make sense, and are becoming more popular in other countries as well. American odds, also known as “money-line odds,” are divided into two separate categories: positive and negative odds:

   

• Positive odds indicate the amount won for a $100 wager.
• Negative odds are quoting the amount that must be wagered in order to win $100.

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  EXAMPLE

   

• If a bet is quoted as paying out “500,” you stand to win $500 for your initial $100 bet, not including the original stake (which is also returned). This is equal to “5:1” fractional odds and “6.0” decimal odds. Again, a $100 investment will yield a total of $600.
• If a bet pays “-500,” then you must risk $500 to win $100. A $100 investment will yield winnings of $20, which is a total of $120 with the original bet. It is equal to “1:5” fractional odds or “1.20” decimal odds.
• “Even money odds” can be quoted as either 100 or -100. Either way, you have to bet $100 to make $100

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  - Miles Hofex
  miles.hofex@gmail.com

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