• In most gambling games it is customary to express the idea of probability in terms of odds against winning. This is simply the ratio of the unfavourable possibilities to the favourable ones. Because the probability of throwing a seven is 1 / 6, on average one throw in six would be favourable and five would not; the odds against throwing a seven are therefore 5 to 1.
• Perhaps what makes probability theory most valuable is that it can be used to determine the expected outcome in any situation—from the chances that a plane will crash to the probability that a person will win the lottery. History of probability theory Probability theory was originally inspired by gambling.
• In a typical Lottery game, each player chooses six distinct numbers from a particular range. If all the six numbers on a ticket match with that of the winning lottery ticket, the ticket holder is a Jackpot winner- regardless of the order of the numbers. The probability of this happening is 1 out of 10 lakh.
• Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continuedto influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a.

Most of what I learned about odds and probability is what I picked up through independent study. Did you know that odds and probability affect gambling more than most? And there’s a bit of luck involved, of course.

The math underlying odds and gambling can help determine whether a wager is worth pursuing. The first thing to understand is that there are three distinct types of odds: fractional, decimal,.

Today, I’ll be going over some little-known facts that might just give you a better understanding of odds. Here are seven things I know about odds and probability that you probably don’t.

1 – Probability Deals With Random Chance

Math is a broad subject, and like most broad subjects, it’s subdivided into smaller subjects. “Geometry,” for example, is the branch of math that deals with distances, sizes, and shapes. “Trigonometry” is even more specific. It’s the branch of math that deals with angles and triangles.

Probability, along with statistics, is the branch of mathematics that deals with random events and measuring how likely they are to occur.

Understanding probability is especially important in gambling and investing, but it can change your life in all kinds of ways. To get a better idea of how you can apply probability-based thinking to your life, check out a book by David Sklansky called DUCY? Exploits, Advice, and Ideas of the Renowned Strategist.

2 – Probability Measures How Likely Something Is to Happen

If you want to know how far one point is from another, you use “distance” to measure that. In the United States, distances can be measured in inches, feet, yards, and miles.

Another example of a word used to describe a measurement is “volume.” You can buy milk by the quart or by the gallon, for example.

How Is Probability Used In Gambling Rules

“Probability” isn’t just the mathematical study of likelihoods. It’s also the word we use to describe and measure how likely something is to happen.

Probability, by its nature, is measured differently from other kinds of measurements.

3 – It’s Always a Number Between 0 and 1

Most things we measure using arbitrary units. In the previous example, we use inches to measure distance and ounces to measure volume.

But in probability, we use a measurement that’s based on fractions. And the probability of an event happening is always just a fraction that’s less than or equal to 1.

If something has a probability of 1, it’s a sure thing. It will happen every time.

Here’s an example: If you have a jar with 20 marbles in it, and all those marbles are white, and you pick a marble from the jar without looking, the probability of picking a white marble is 1.

The probability of picking a black marble is 0. There aren’t any black marbles in the jar.

It gets more interesting when you put different colored marbles in the jar. If you put 10 white marbles and 10 blackjack marbles, you have a probability of 1/2 for getting a white marble at random. You also have a 1/2 probability of getting a black marble at random.

The formula for probability is simple, too. The probability of an event is the number of ways that event can happen compared to the total number of possible events.

In the marble example, you have 20 possible events (20 possible marbles you could pick at random). 10 of those are white. The probability of getting a white marble at random, then, is 10/20, which reduces to 1/2.

You can express that probability in multiple ways, too, not just as fractions.

• You can express that probability as a decimal, 0.5.
• You can express that probability as a percentage, 50%.
• Or you can express that probability as odds, 1 to 1 or “even odds.”

That last way of expressing a probability, as odds, is especially useful in gambling.

4 – Odds Are A Way of Describing Probability That Are Especially Useful

The odds of something happening are just a comparison of the number of ways it can happen versus the number of ways it can’t happen. In the marble example, you have 10 white marbles versus 10 black marbles, so the odds are 10 to 10 of getting a white marble.

You can reduce that just like you would a fraction to get even odds – 1 to 1.

Let’s change the example, though. Now, let’s suppose you have a jar with 5 white marbles and 15 black marbles in it. Your odds of drawing a white marble are 15 to 5, which reduces to 3 to 1.

For every possible white result, you have three possible black results. Obviously, you’re likelier to get a black marble in this situation than you are to get a white marble.

One of the reasons that this is so useful is because odds are also used to describe how much a bet pays off. A lot of bets are even money bets. You bet $100, and if you win, you get $100. If you lose, you’re out $100.

But in some bets, you might win more money than you’re risking. For example, you might place a bet where you could win $200 and only risk $100.

You can compare the odds of winning with the payout odds to see whether you or the other party to the bet has the advantage.

This is what makes it possible for poker players to play professionally and win in the long run. They put their money in the pot when they have better payout odds than odds of winning. This is also what makes casinos profitable. They pay out bets at odds less than your odds of winning.

5 – The Casino Has a Mathematical Advantage for Every Game

When you play casino games, the casino always has a mathematical advantage. It’s easier or harder to measure depending on the game you’re playing and the rules.

The easiest example might be roulette. The math behind the house edge is relatively easy to calculate.

Let’s look at an even money bet, a bet that the ball will land on a red number.

You have 18 red numbers on a roulette wheel, 18 black numbers, and 2 green numbers. You have a total of 20 ways to lose and 18 ways to win.

The odds, therefore, of winning are 10 to 9. But the payout is 1 to 1.

Let’s say you place this bet 19 times in a row. You’ll win 9 times, and you’ll lose once on average, in the long run.

If you bet $100 every time, after completing those 19 bets, you’ll have won $900 and lost $1000. This results in a net loss of $100 over 19 bets. Your average loss per bet is $100/19, or $5.26.

Since $5.26 is 5.26% of $100, we say that the house edge for that roulette bet is 5.26%.

The calculations for the house on other bets in other games might be different and even more complicated, but you can count on this – the casino always has the mathematical edge over the player.

6 – The House Edge for Each Casino Game Is a Known Quantity

For every casino game, the house knows what its mathematical edge is. The player can do a little research and find the house edge for every table game, too. You can use this information to inform your gambling decisions.

An easy way to do this would be to only play the games with the lowest house edge.

A simple comparison of the house edge in blackjack, 0.5% in some games, with the house edge in roulette—5.26%—tells you that blackjack is the better game.

But for the gambler, one game in the casino has an unknown house edge. That’s the slot machine.

To calculate the house’s advantage, you must know the probability of each event as well as the payouts for those events.

Modern slot machines use a random number generator (RNG) to determine outcomes. Not every symbol has the same probability of showing up. Some symbols might show up two or three times more often than some other symbols, for example.

The casino, though, has the details for the slot machine games’ probabilities. They know which slots have a house edge of 5% and which slots have a house edge of 15%.

You can’t know, though, unless you clock a large number of spins and extrapolate the data. Even then, you could be way off.

7 – The Theory Behind Probability Has Real-World Implications

You can use the ideas behind probability to inform your decisions in life. Using probability, you can measure the expected value of the various decisions in your life. You just compare the potential loss with the potential win and make your decisions accordingly.

Here’s a silly example: You’re at a bar with a friend, and he decides he’s going to make a pass at every woman in the bar.

What does he risk? He risks potential rejection, but his self-esteem is strong, and rejection is meaningless to him. He’s basically placing a free bet because if he loses, he isn’t out anything.

If your friend is reasonably charming and good-looking, he might have a close to 100% probability of going home with one of these women. On the other hand, if he’s a poor communicator and only average-looking, he might only have a 1 in 20 probability of getting a date.

It’s still a positive expectation bet because he’s risking nothing with a potential gain. You, on the other hand, might suffer terrible anxiety and hate rejection. If that’s the case, you might only make passes at women in the bar who pay attention to you first. The cost of making a pass at every woman might just be too high for you.

In the book by Sklansky I mentioned earlier, he suggests attaching dollar amounts to such decisions. How much money would you pay to avoid rejection? How much money would you pay to get a date? What is the probability of rejection versus the probability of getting a date?

That’s a silly example, but you can use the same probability-based thought process to make career decisions and other life decisions.

Conclusion

Probability might be the most important branch of mathematics. It’s certainly the most practical, especially if you gamble as a hobby or for a living.

I recommend taking a class in probability or at least studying a probability textbook. Definitely explore the ideas in the Sklansky book I recommended in the introduction.

Events and sample spaces in gambling

The technical processes of a game stand for experiments that generate aleatory events. Such events occur wherever the games are played, at home, in a casino or online casino.

Throwing the dice in craps is an experiment that generates events such as occurrences of certain numbers on the dice, obtaining a certain sum of the shown numbers, obtaining numbers with certain properties (less than a specific number, higher that a specific number, even, uneven, and so on). The sample space of such an experiment is {1, 2, 3, 4, 5, 6} for rolling one die or {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), (6, 2), ..., (6, 6)} for rolling two dice. The latter is a set of ordered pairs and counts 6 x 6 = 36 elements.

The events can be identified with sets, namely parts of the sample space. For example, the event occurrence of an even number is represented by the following set in the experiment of rolling one die: {2, 4, 6}.

Spinning the roulette wheel is an experiment whose generated events could be the occurrence of a certain number, of a certain color or a certain property of the numbers (low, high, even, uneven, from a certain row or column, and so on). The sample space of the experiment involving spinning the roulette wheel is the set of numbers the roulette holds: {1, 2, 3, ..., 36, 0, 00} for the American roulette, or {1, 2, 3, ..., 36, 0} for the European. The event occurrence of a red number is represented by the set {1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}. These are the numbers inscribed in red on the roulette wheel and table.

Dealing cards in blackjack is an experiment that generates events such as the occurrence of a certain card or value as the first card dealt, obtaining a certain total of points from the first two cards dealt, exceeding 21 points from the first three cards dealt, and so on.

In card games we encounter many types of experiments and categories of events. Each type of experiment has its own sample space. For example, the experiment of dealing the first card to the first player has as its sample space the set of all 52 cards (or 104, if played with two decks). The experiment of dealing the second card to the first player has as its sample space the set of all 52 cards (or 104), less the first card dealt. The experiment of dealing the first two cards to the first player has as its sample space a set of ordered pairs, namely all the 2-size arrangements of cards from the 52 (or 104).

In a game with one player, the event the player is dealt a card of 10 points as the first dealt card is represented by the set of cards {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦, K♠, K♣, K♥, K♦}.

The event the player is dealt a total of five points from the first two dealt cards is represented by the set of 2-size combinations of card values {(A, 4), (2, 3)}, which in fact counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and symbol).

In 6/49 lottery, the experiment of drawing six numbers from the 49 generate events such as drawing six specific numbers, drawing five numbers from six specific numbers, drawing four numbers from six specific numbers, drawing at least one number from a certain group of numbers, etc. The sample space here is the set of all 6-size combinations of numbers from the 49.

In classical poker, the experiment of dealing the initial five card hands generates events such as dealing at least one certain card to a specific player, dealing a pair to at least two players, dealing four identical symbols to at least one player, and so on. The sample space in this case is the set of all 5-card combinations from the 52 (or the deck used). Dealing two cards to a player who has discarded two cards is another experiment whose sample space is now the set of all 2-card combinations from the 52, less the cards seen by the observer who solves the probability problem.

For example, if you are in play in the above situation and want to figure out some odds regarding your hand, the sample space you should consider is the set of all 2-card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2-size combinations from 47.

All these isolated examples are not the most representative from the respective games. They are presented as an introduction to what mathematics in games of chance means, namely particular probability models, in which probability theory can be applied to obtain the probabilities of the events we are interested in.

Probability models in gambling

A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the field of events. The event is the main unit probability theory works on. In gambling/online gambling, there are many categories of events, all of which can be textually predefined. In the previous examples of gambling experiments we saw some of the events that experiments generate. They are a minute part of all possible events, which in fact is the set of all parts of the sample space. For a specific game, the various types of events can be:
– Events related to your own play or to opponents’ play;
– Events related to one person’s play or to several persons’ play;
– Immediate events or long-shot events.
Each category can be further divided into several other subcategories, depending on the game referred to. From a mathematical point of view, the events are nothing more than subsets and the field of events is a Boole algebra.

The complete mathematical model is given by the probability field attached to the experiment, which is the triple sample space—field of events—probability function. For any basic application in a game of chance, the probability model is of the simplest type—the sample space is finite, the field of events is the set of parts of the sample space, implicitly finite, too, and the probability function is given by the definition of probability on a finite field of events. From this definition and the axioms of a Boole algebra flow all the properties of probability that can be applied in the practical calculus in gambling. Any predictable event in gambling, no matter how complex, can be decomposed into elementary events with respect to the union of sets.

For example, if we consider the event player 1 is dealt a pair in a Texas Hold’em game before the flop, this event is the union of all combinations of (xx) type, x being a value from 2 to A. Each such combination (xx) is in turn a union of the elementary events (x♣ x♠), (x♣, x♥), (x♣, x♦), (x♠, x♥), (x♠, x♦) and (x♥, x♦), all of which are equally possible. The entire union counts 13C(4, 2) = 78 elementary events (2-size combinations of cards as value and symbol).

There are also applications in gambling involving events related to the long-run play, whose suitable probability models are chosen from classical probability distributions such as Bernoullian, Poisson, Polynomial, or Hypergeometric distribution.

Probability calculus in gambling

Probability calculus actually means to use of all the properties of the probability in order to obtain explicit formulas of the probabilities of the measured events and apply these formulas in the given circumstances for obtaining a final numerical result.

The basic principle that makes the probability calculus performable in gambling is that any compound event can be decomposed into equally possible elementary events, then the probability properties and formulas can be applied to it to find its numerical probability. Besides the basic properties of probability, the formulas from the classical distributions are of great help for some complex gaming events.

In most probability computations in gambling, the application of the formulas reverts to combinatorial calculus, which is an essential tool for this type of applications.

The hardest task of the gaming mathematician performing probability calculus is to provide explicit formulas in algebraic form, which express the sought probabilities.

Expected value

The mathematical model of a game of chance involves not only probability, but also other statistical parameters and indicators, of which the expected value is the most important.

In gambling, probabilities are associated with stakes in order to predict an average future gain or loss. This predicted future gain or loss is called expectation or expected value (EV) and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times.

For example, an American roulette wheel has 38 equally possible outcomes. Assume a bet placed on a single number pays 35–to–1 (this means that you are paid 35 times your bet and your bet is returned, so you get back 36 times your bet). So, the expected value of the profit resulting from a one dollar repeated bet on a single number is , which is about –$0.05. Therefore one expects, on average, to lose over five cents for every dollar bet.

The mathematical expectation (expected value) is defined as follows:
Definition: If X is a discrete random variable with values and corresponding probabilities , , the sum or sum of series (if convergent) is called mathematical expectation, expected value or mean of variable X.

How Is Probability Used In Gambling

So, the mathematical expectation is a weighted mean, in the sense of the definition given above. It terms of gambling, this value means the amount (positive or negative) a player should expect, if performing same type of experiment (game or gaming situation) in identical conditions and by making the same bet, via mathematical probability. The expected value being negative is the sign for that bet being profitable for the house, by ensuring its house edge. In practice, expected value is a statistical parameter assigned to every bet that has a computable probability and a payout, even though one cannot run that bet infinitely many times. Together with probability, expected value stands as criterion for decisions in games and betting where the bets have specific payouts.

The role of probability in gambling strategies

How Is Probability Used In Gambling Losses

A strategy only makes sense if related to both game and player. That is because it is the player who builds and applies a strategy, according to his/her own goals. Among all the criteria used within a strategy, there are subjective personal criteria related to player's profile, but also objective criteria, of which probability is the most important. Acting in a certain way in a particular gaming situation basing on the evaluation and comparison of odds/probabilities means to make a decision based on probability as the most objective measure of possibility we have. These might be decisions related to gaming situations during the game, but also of choosing a certain game or another, quitting a game for another or even not playing at all. Even objective, the criteria standing at the base of such decisions still can have a subjective component, which is the threshold of afforded risk. This parameter is an average probability to which each player refers when making decisions on his/her next action and is the level of the probability of failure with which the player is comfortable staying in the game. A general criterion using the threshold of afforded risk in a gaming strategy would be: “If the odds for my opponents (or house) to beat (gain an advantage over) me at moment t are higher than p, then I will quit (stand, fold, etc.).” The threshold of afforded risk is p and the value of p is different for each player and even can change several times during a game, for the same player.

A probability-based strategy consists (and is defined as) only of decisions resulting from the evaluation and comparison of probability results. Mathematics proved that a probability-based strategy is theoretically optimal among other types of strategies, in the respect of gaining advantage during the game, at its end, and on the long-run play.

There exist strategies for any game of chance, either played against with opponents or the house, and all of them can be probability-based. While in games like poker strategy includes the interaction with the opponents and applies to each round (and probability is essential in expressing the strength of a hand), in simple games like lottery or slots the only strategy is the strategy of choosing (choosing what numbers to play and how frequent to play in lottery, choosing which game to play, how many paylines to enable, and choosing not to play a particular game in slots, etc.) and this strategy can also be based on probability.