Here are the three greatest casino games to play if you would like decent probability of winning cash: Blackjack’s Odds of Winning: 49%. Blackjack is a really simple card video game using a part of ability with it particular. The probability of winning are not too bad. The probability of winning or losing a casino game is said to be its backbone. Probability is nothing but the chances of winning or losing. The probability value may lie between 0-1 and can be given as decimal or fraction. Zero probability means the chances are quite zero to happen.
The math behind gambling is endlessly fascinating. In fact, without the branch of mathematics called “probability”, we wouldn’t even have gambling—or at least we wouldn’t be able to talk about it intelligently.
Few bets are fair bets. One side almost always has an edge over the other. Being able to determine that edge is a critical part of being an educated gambler. This post starts with an overview of what probability is and how it’s calculated, then it continues with 7 examples of how it’s used in practical applications.
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Probability concerns itself with measuring how likely it is that certain things will happen. For purposes of this post, I’ll call those things “events”. You probably use probability to talk about possible events without even knowing it.
Probably the most common expression of probability happens with percentages, especially when you’re watching the nightly news. When the meteorologist says that there’s a 50% chance of thunderstorms tomorrow, she’s telling you what the probability is that there will be rain. And most people understand that 50% means that half the time it’s going to rain, and half the time it’s not.
A probability is just a number that describes how likely an event is. And that number is always a number between 0 and 1. Something with a probability of 0 won’t ever happen. Something with a probability of 1 (which is also 100%) will always happen.
You can express probabilities as percentages, but that’s not the only way to express a probability. You can also express it as a fraction. 50% is the same thing as ½.
You can also express a probability as a decimal. 50% is the same thing as 0.5.
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Probability can also be expressed in odds format. In this case, 50% is the same thing as 1 to 1, or even odds.
Each of those ways of expressing probability is useful in different situations. Stating a probability as odds is especially useful when comparing the payoff of a bet with the odds of winning that bet.
Calculating probability is actually pretty simple, too. For a single event, you look at the number of ways that event can happen versus how many ways things might turn out total. You put the single event on top of the fraction, and you put the total number of potential events as the bottom of the fraction. Of course, if you have any math experience at all, you know that you can use division to turn a fraction into a decimal or a percentage.
If you want to calculate the probability of multiple events, you either multiply or add depending on whether you want to know if multiple events will happen or if you want to know the odds of a certain number of events happening.
The key words to look for in such a problem are “and” and “or”.
If you want to know the probability that event A will happen AND event B will happen, you multiply the probability of each.
If you want to know the probability that event A will happen OR event B will happen, you add the probability of each.
The following examples will show how these probability calculations happen time and again in the gambling world.
1. Roulette Math
Roulette is a simple game, and it’s a great example of probability in action. An American roulette wheel has 38 possible events, numbered 0, 00, and 1-36. The 0 and the 00 are green. Half of the other numbers are black, and half of them are red.
With this information, you can calculate the probability of just about any outcome or combination of outcomes. You can compare those probabilities with the payoffs for the bet to see if one side has an edge, and if so, how much that edge is.
Let’s start by thinking about some of the more common bets in roulette—the outside bets. These bets are on odd/even, high/low, or red/black. They all pay out at even odds. You bet $1 on one of these outcomes, you win $1 if you win.
At first glance, that sounds like a fair enough bet, but when you look at these bets a little more closely, the house has a distinct advantage.
Here’s why:
Suppose you bet on black. There are 18 numbers on the wheel that are black, but there are 20 numbers on the wheel that are not. (18 of the numbers are red, and 2 more numbers are green.) So out of 38 possible outcomes, only 18 of them win your bet.
That makes the probability 18/38. It’s probably easiest to understand this bet by converting it into a percentage, 47.37%.
Casino Probability Games
So 52.63% of the time, the casino will win this bet, and the rest of the time, you will. It’s clear to see how if you play this game long enough, eventually the casino will win all your money.
You can even calculate the amount of each bet the casino will win over the long run—this number is called the house edge.
Here’s how you do it:
Assume that you make 100 bets and that you see the mathematically expected results. (That never happens in real life, but if you play long enough, the actual results will start to resemble the expected results.)
In this case, you will win $47.37, but you’ll lose $52.63. That’s a net loss of $52.63 – $47.37, or $5.26.
Since you bet $100 on those 100 wagers, you lost an average of 5.26% of each bet.
And that’s the house edge.
As it turns out, that’s the house edge for all the bets at the roulette table (except for one).
In a sense, the green 0 and the green 00 are where the house gets its edge. The payouts for all the bets on the table would offer neither side an edge if those numbers weren’t on the wheel.
But they ARE on the wheel. And that makes all the difference.
2. The Math Behind a Coin Toss
An even simpler example of probability in action is a coin toss. Most people don’t actually place wagers on the outcomes of a coin toss, but they could. And depending on the payout structure, one side might or might not have an edge over the other side.
Here’s the simplest version of this calculation. You want to know the probability that you’ll get heads on a coin toss. Since there are 2 potential events, and since only 1 of them is heads, your probability is ½, or 50%.
In cases where you want both sides to have an even shot at winning something, you’ll flip a coin. This is how they determine who gets to kick off during a football game, for example.
I should point out that there’s no advantage to being the one to call heads or tails. The probability is the same, and I don’t believe in psychic phenomena. I’ve never seen any evidence that anyone has any kind of precognitive ability that would improve their chances of predicting the outcome of a coin toss.
But let’s try a more interesting calculation. Let’s say we want to know the probability of getting heads twice in a row. That means you want to know the probability of getting heads on the first flip AND the probability of getting heads on the second flip.
Remember I said earlier that if we’re using the word “and” in the problem, we multiply. In this case, we’re multiplying ½ by ½, which is ¼. Or we could call it 0.5 X 0.5 and get 0.25. Either of those ways can be expressed as 25%.
Another way to look at this is to look at the total number of outcomes when you toss a coin twice in a row:
You could get heads on the first toss and heads on the second toss.
You could get tails on the first toss and tails on the second toss.
You could get heads on the first toss and tails on the second toss.
You could get tails on the first and heads on the second toss.
Those are literally the only 4 outcomes, but only 1 of them is the outcome you were solving for. That’s ¼, or 25%, which is what we’d determined earlier.
Suppose you wanted to create a simple gambling game based on the outcome a coin toss. Let’s say you’re running a back room casino in a bar or something.
You might have a game where you toss a coin, and so does the dealer. If you get heads and the dealer gets tails, you win. If the dealer gets tails, and you get heads, then the dealer wins.
But if you both get heads or both get tails, you have to put up another coin in order to get to toss the coins again.
The catch is that the dealer does NOT have to put up another coin. If you win this second toss, you win a coin, but if you lose it, you lose both coins that you put up.
It’s pretty clear in this example how the casino has an edge, right?
3. Poker Math
I could spend the rest of this post talking about poker math. But I’ll try to limit it to just this bullet point.
Anyone who knows anything about poker knows that you have just as good a chance of getting a better hand as I do. We’re both getting cards from the same 52 card deck, after all.
It’s what you do with those cards after that make a difference.
Let’s suppose that you’re playing 5 card draw and you’re dealt a hand with 4 cards to a flush in it. You’re going to discard a card and hope to draw to that flush.
What is the probability that you will succeed?
There are 47 cards left in the deck. 9 of them are of the suit you need. (There are 13 cards in each suit, and 4 of them are already in your hand.) So your probability of getting the card you need is 9/47, or 19.1%. That’s almost 1 in 5, or 20%.
If you assume that you have to make this hand in order to win the pot, you can calculate how much money needs to be in the pot in order for you to profitably calla bet.
Let’s suppose that there is $10 in the pot, and it costs you $1 to stay in and draw that extra card. If you win, you’ll win 10 to 1 on a 4 to 1 draw. You’ll lose almost 80% of the time, but you’ll win so much when you do win that it will make up for it and give you a tidy profit.
In fact, let’s do the same calculation we did above, where we assume that you do this 100 times in a row. You’ll lose $80.90, but you’ll win $190.10, for a profit of $109.20. These are excellent pot odds.
On the other hand, if there were only $3 in the pot, and it cost you $1 to get in, you wouldn’t get a big enough payout to make this a profitable bet. You’d still lose $80.10, but you’d only win $57.30, for a net loss of $22.50.
Of course, in a real poker game, you’d have other probabilities to take into account. For example, you might raise in this situation, hoping to scare your opponents out of the pot. You have to estimate the probability that this tactic will work when you try this. You can add that to your expected value.
This is where reading other players becomes important. Some people think that reading people is all about gauging what they’re going to do 100% of the time.
But the reality is that you make educated guesses about their likelihood of doing something. If you estimate that your opponent will fold to your bluff 50% of the time, then that makes a big difference to your strategy.
4. Video Poker Math
Video poker is a little bit like poker and a little bit like slot machines, but it’s like nothing so much as it’s like itself. Most of the math, though, is similar to the math of traditional poker. The difference is that you have an exact payoff you can expect when you achieve a certain hand. You don’t have to worry about what your opponents have.
For example, if you have a pair of jacks in a poker game, and your opponent also has a pair of jacks, you could wind up in a situation where you tie and split the pot.
But in a Jacks or Better video poker game, you get paid even odds any and every time you get a pair of jacks or higher. And you don’t get a higher payout for a pair of queens or a pair of kings. For purposes of these payouts, all 3 hands are the same, even though there’s a clear hierarchy among those 3 hands in a real poker game.
Video poker is based on draw poker, so every time you get a hand, you get to decide which cards to keep and which ones to throw away. You compare the probability of making certain hands with their payoffs in order to decide which decision has the best expected value.
Here’s an example:
The best possible hand you can get in most video poker games is a royal flush, which pays off at a whopping 800 to 1. (I’m assuming you’re making the max coin bet—if you don’t, you’re only getting a 250 to 1 payoff. But you should never play for less than max coins.)
But you can win even odds with a pair of jacks or higher. That’s clearly a much lower payoff.
But suppose you have to choose between those 2 options? Let’s say you have the ace of hearts, the king of hearts, the queen of hearts, and the jack of hearts. But your 5th card is the king of spades.
You have a pair of kings. You can keep that and have a 100% chance of getting an even money payoff.
Or you can throw away the king of spades and try to get the royal flush. Only 1 card of the 47 remaining cards will make your hand, which is a slightly better than 2% chance of success.
What happens over 100 perfect iterations?
98 times you lose your bet. But twice you get 800 coins. That’s 1600-98, or 1502. Divided by 100 bets, that’s 15.02 per bet that you won.
In the other case, you win 100 times, but you only win 100 coins total. Casino bonus reviews.
Would you rather average $15 in winnings per bet, or $1 in winnings per bet?
Of course, this example ignores the possibility that you could draw to another random winning hand, but that has a more or less equal probability with both decisions. We’ll just assume that it evens out.
On the other hand, if you only had 3 cards to a royal flush, the odds of hitting your hand get much smaller. 2% X 2% is 0.04%. With odds like that, you’ll need a lot more than an 800 to 1 payoff to make that decision worthwhile.
But no matter what hand you are dealt initially, you have one decision which has a higher expected value than any of the others.
That expected value is determined by looking at all the possible moves in that situation and the likelihood that each of them will result in a particular payoff amount.
5. Craps Math
Oldest casino in las vegas. Craps is an interesting exercise in probability because it’s a great example of a bell curve. That’s when some results happen so seldom that the drawing of the curve is low on either end, but the odds of the results in the middle happening are much higher.
Here are the possible outcomes when rolling a pair of dice:
2 – 1 +1 – Only one possible way of getting this total.
3 – 2+1 or 1+2 – Only 2 possible ways of getting this total.
4 – 3+1, 2+2, or 1+3 – Only 3 possible ways of getting this total.
5 – 4+1, 3+2, 2+3, or 1+4 – Only 4 possible ways of getting this total.
6- 5+1, 4+2, 3+3, 2+4, 1+5 – Only 5 possible ways of getting this total.
7 – 6+1, 5+2, 4+3, 3+4, 2+5, 1+6 – Only 6 possible ways of getting this total.
8 – 6+2, 5+3, 4+4. 3+5, 2+6 – Only 5 possible ways of getting this total.
9 – 6+3, 5+4, 4+5, or 3+6 – Only 4 possible ways of getting this total.
10 – 6+4, 5+5, or 4+6 – Only 3 possible ways of getting this total.
11 – 6+5 or 5 +6 – Only 2 possible ways of getting this total.
12 – 6+6 – Only one possible way of getting this total.
You only have 11 possible totals, but you have a total of 36 different outcomes.
Knowing this, you can divide the number of ways of achieving each total by 36 in order to determine the probability of getting that total.
So getting a total of 2 or 12 has a probability of 1/36.
3 or 11 has a probability of 2/36, or 1/18.
4 or 10 has a probability of 3/36, or 1/12.
5 or 9 has a probability of 4/36 or 1/9.
6 or 8 has a probability of 5/36.
7 has a probability of 6/36, or 1/6.
So your most likely outcome is a total of 7, but that still only happens 1 time out of 6.
Probability Casino Games
But you can bet on any of these totals at various times in the game. You can compare the payoffs on these bets with the odds of winning to determine the house edge on each of those bets.
For example, you can make a place bet on any 8 or any 6 and get a payoff of 7 to 6 if you win. But the odds of winning that bet are 5/36. That can be converted into a percentage, and we can calculate the house edge for that bet. The odds of winning this bet are 13.89%.
Place this bet 100 times, and you will win 13.88 bets with winnings of $1.17 each time (7 to 6). That’s $16.24 in winnings. Worst odds in a casino. But you lose 86.12 times, losing $1 each time, for losses of $86.12. You’ve lost way much more than you’ve won over these 100 bets–$69.88. That makes the house edge 6.99% on this bet, which is almost 7%. That’s worse than roulette with its 5.26% edge.
Luckily, many of the bets on the craps table have a much lower house edge.
6. Blackjack Math
My favorite kind of gambling math relates to blackjack. It’s such an elegant game, and it’s also one of the only casino games where a skilled player can get an edge. What’s so interesting about the game is that it has a memory.
Here’s what I mean:
When you play roulette, the odds are the same on every spin of the wheel. The outcome of one spin has no effect on the odds of the outcome of the next spin. There are 38 possibilities every time you spin the wheel, and each of them is equally as likely as the others.
But if you eliminated a slot on the wheel once it got hit, you’d wind up with odds that changed on every spin.
Here’s an example:
You bet on black. The probability of winning that bet is 18/38.
You win. The croupier (the roulette dealer) leaves the ball in that slot, so it can’t be landed on again.
You bet on black again. This time the probability of winning is only 17/38, because one of the options has been removed.
This is exactly what happens every time a card is dealt in blackjack. One of the 52 options is no longer available to be dealt in subsequent rounds.
This continues until the dealer reshuffles the pack of cards.
Of course, in a game with a continuous shuffling machine, the odds stay the same no matter what.
But most games are still dealt without the benefit of such a machine. In these games, you can keep rough track of which cards have been dealt and raise your bets when you have a better chance of winning more money.
Here’s how that works:
A “natural”, or a “blackjack”, pays off at 3 to 2. That’s a 2 card hand that totals 21. There are only 2 values of cards which can result in such a hand—the aces, which count as 11, and the 10, J, Q, and K, each of which counts as 10.
If all of the aces in a deck are gone, it’s impossible to get a blackjack. You just can’t do it.
Every time a 10 gets dealt, your chances of getting a blackjack decrease, too.
But at the same time, every time a lower ranked card gets deal, like a 2, 3, 4, 5, or 6, the odds improve a little bit in the player’s favor.
So a card counter will use a system to keep rough track of the ratio of high cards to low cards. They count the low cards as +1 and the high cards as -1. If and when the count gets high on the positive side, the counter knows he has a better than average chance of getting that 3 to 2 payout. So he raises his bets accordingly. The higher the count, the more he bets.
He lowers his bet when the count is 0 or negative.
There’s a lot more to counting cards than that, but those are the basics. And they’re rooted in math.
7. Sports Betting Math
Most bookmakers require you to risk $110 in order to win $100, but that’s not all they do. They also handicap teams by giving them points or taking them away. The goal of this handicapping is to make a bet on either side a 50/50 proposition. Since these sports bets don’t pay off at even odds, a 50/50 proposition is profitable for the bookmaker but not the player.
But the bookmakers aren’t always right when they set the lines. And they don’t always leave the lines the way they are. A bookmaker’s goal is to get an equal amount of action on either side of an event. They do this so that they can pay off the winning bets with the losers’ money. That extra $10 that the losers bet is how they prefer to make their profit.
But what if they don’t get an equal amount of bets on each side?
Most bookmakers move the line in order to stimulate action on the other side. Sharp sports bettors—those who know how the business work—know that it’s usually best to bet against the public.
Here’s an example of how this works:
The Washington Redskins are playing the Dallas Cowboys, and they’re favored by 7 points. This means that before paying off a bet on the Redskins, the bookmaker subtracts 7 from their score.
They set this line early in the week, but they don’t get nearly as many bets on the Cowboys as they expect. So they move the line to 7.5, which is meant to encourage more action on the other side. A smart bettor is going to bet against the public in this situation, because the public is usually wrong.
The really interesting effect of the vigorish on a sports bettor is what it does to the required winning percentage just to break even. If you’re right 50% of the time and wrong 50% of the time, you’ll lose money. You’re losing $110 half the time, and you’re only winning $100 the other half the time.
If you can bet on the right side a little over 53% of the time, you can break even and even make a tiny profit. If you can get over 55% and start nearing 60%, you’re on your way to becoming a world class sports bettor. You can make 6 figures a year with a win rate like that, but you need to have enough money in your bankroll to be able to weather any losing streaks you might run into.
Losing streaks in the short term are inevitable, too. That’s just the nature of a game of chance. Also, the handicappers who work for the bookmakers are almost always right. In order to make a profit betting at sports, you have to be adept at finding profitable situations. This means outthinking the handicappers and the bookmakers most of the time.
Finding value in sports betting is an endlessly interesting topic.
Conclusion
As you can see from these 7 examples, it’s unusual that anyone ever gets a fair bet. Someone almost always gets an edge. Figuring out who has the edge and by how much is just a matter of comparing the odds of winning for each side and the payouts for winning those bets.
Casinos always have an edge over the players. I can only think of 2 bets in a Las Vegas casino which offer fair odds—the double up bet in video poker and the odds bet in craps. But you can find occasional bets in Vegas casinos where the player has an edge, but these are the exceptions, not the rules.
When you’re playing games like slots, craps, and roulette, there’s really not much you can do to even out the odds. Some people claim that they can affect the outcome of a roll of the dice, but I’m skeptical.
On the other hand, if you’re a skillful blackjack player or a skillful video poker player, you might be able to get a small edge over the casino. If you’re counting cards as a blackjack player, most casinos will refuse to let you continue to play, though. And they’re pretty good at catching advantage players now.
Skilled poker players and sports bettors can get the odds in their favor, but they still have to be skillful enough to overcome a house edge of sorts. In poker games, the cardroom hosting the games charges a percentage of each pot as rent for the table—this is called the rake. When betting on sports, you have to bet $110 to win $100. That extra $10 you have to risk on every bet is called the vigorish.
But no matter what betting activity you choose, you’ll enjoy it more if you have a clear understanding of the math behind the game and your bets. That’s why I write posts examining the math behind gambling.
It’s worth it to try to get an edge, but it’s impossible to get an edge if you don’t have at least a rudimentary understanding of gambling math. Seeing it in action helped me a lot when I got started.
And if you have any aspirations of gambling professionally, understanding these examples is a must.
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The mathematics of gambling are a collection of probability applications encountered in games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events.
Experiments, events, probability spaces[edit]
The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples:
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, and obtaining numbers with certain properties (less than a specific number, higher than 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 generates 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 draw 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.
The probability model[edit]
A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the space (field) of events. The event is the main unit probability theory works on. In 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 theset 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. These events can be literally defined, but it must be done very carefully when framing a probability problem. From a mathematical point of view, the events are nothing more than subsets and the space of events is a Boolean algebra. Among these events, we find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events.
In the experiment of rolling a die:
Event {3, 5} (whose literal definition is occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};
Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
Events {3, 5} and {4} are incompatible orexclusive because their intersection is empty; that is, they cannot occur simultaneously;
Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;
In the experiment of rolling two dice one after another, the events obtaining 3 on the first die and obtaining 5 on the second die are independent because the occurrence of the second event is not influenced by the occurrence of the first, and vice versa.
In the experiment of dealing the pocket cards in Texas Hold’em Poker:
The event of dealing (3♣, 3♦) to a player is an elementary event;
The event of dealing two 3’s to a player is compound because is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
The events player 1 is dealt a pair of kings and player 2 is dealt a pair of kings are nonexclusive (they can both occur);
The events player 1 is dealt two connectors of hearts higher than J and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur);
The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the occurrence of the first, while the same deck is in use).
These are a few examples of gambling events, whose properties of compoundness, exclusiveness and independency are easily observable. Theseproperties are very important in practical probability calculus.
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 game of chance, the probability model is of the simplest type—the sample space is finite, the space 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 space of events:
Combinations[edit]
Games of chance are also good examples of combinations, permutations and arrangements, which are met at every step: combinations of cards in a player’s hand, on the table or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on, and the like. Combinatorial calculus is an important part of gambling probability applications. In games of chance, most of the gambling probability calculus in which we use the classical definition of probability reverts to counting combinations. The gaming events can be identified with sets, which often are sets of combinations. Thus, we can identify an event with a combination.
For example, in a five draw poker game, the event at least one player holds a four of a kind formation can be identified with the set of all combinations of (xxxxy) type, where x and y are distinct values of cards. This set has 13C(4,4)(52-4)=624 combinations. Possible combinations are (3♠ 3♣ 3♥ 3♦ J♣) or (7♠ 7♣ 7♥ 7♦ 2♣). These can be identified with elementary events that the event to be measured consists of.
Expectation and strategy[edit]
Games of chance are not merely pure applications of probability calculus and gaming situations are not just isolated events whose numerical probability is well established through mathematical methods; they are also games whose progress is influenced by human action. In gambling, the human element has a striking character. The player is not only interested in the mathematical probability of the various gaming events, but he or she has expectations from the games while a major interaction exists. To obtain favorable results from this interaction, gamblers take into account all possible information, including statistics, to build gaming strategies. The oldest and most common betting system is the martingale, or doubling-up, system on even-money bets, in which bets are doubled progressively after each loss until a win occurs. This system probably dates back to the invention of the roulette wheel. Two other well-known systems, also based on even-money bets, are the d’Alembert system (based on theorems of the French mathematician Jean Le Rond d’Alembert), in which the player increases his bets by one unit after each loss but decreases it by one unit after each win, and the Labouchere system (devised by the British politician Henry Du Pré Labouchere, although the basis for it was invented by the 18th-century French philosopher Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet), in which the player increases or decreases his bets according to a certain combination of numbers chosen in advance.[1][2] The predicted average gain or loss is called expectation or expected value 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. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game. The attribute fair refers not to the technical process of the game, but to the chance balance house (bank)–player.
Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table—shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win. It has been mathematically proved that, in ideal conditions of randomness, and with negative expectation, no long-run regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win either in the short term or over the long run.
House advantage or edge[edit]
Casino games provide a predictable long-term advantage to the casino, or 'house', while offering the player the possibility of a large short-term payout. Some casino games have a skill element, where the player makes decisions; such games are called 'random with a tactical element.' While it is possible through skilful play to minimize the house advantage, it is extremely rare that a player has sufficient skill to completely eliminate his inherent long-term disadvantage (the house edge or house vigorish) in a casino game. Common belief is that such a skill set would involve years of training, an extraordinary memory and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette. For more examples see Advantage gambling.
The player's disadvantage is a result of the casino not paying winning wagers according to the game's 'true odds', which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. However, the casino may only pay 4 times the amount wagered for a winning wager.
The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits.
Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38.
The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%. Therefore, the house edge is 5.26%. After 10 rounds, play $1 per round, the average house profit will be 10 x $1 x 5.26% = $0.53.Of course, it is not possible for the casino to win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round.
Best Probability Casino Games
The house edge of casino games varies greatly with the game. Keno can have house edges up to 25% and slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%.
The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task.
In games which have a skill element, such as Blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting or shuffle tracking), on the first hand of the shoe (the container that holds the cards). The set of the optimal plays for all possible hands is known as 'basic strategy' and is highly dependent on the specific rules, and even the number of decks used. Good Blackjack and Spanish 21 games have house edges below 0.5%.
Online slot games often have a published Return to Player (RTP) percentage that determines the theoretical house edge. Some software developers choose to publish the RTP of their slot games while others do not.[3] Despite the set theoretical RTP, almost any outcome is possible in the short term.[4]
Standard deviation[edit]
The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be simply calculated because of the binomial distribution of successes (assuming a result of 1 unit for a win, and 0 units for a loss). For the binomial distribution, SD is equal to , where is the number of rounds played, is the probability of winning, and is the probability of losing. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. Therefore, SD for Roulette even-money bet is equal to , where is the flat bet per round, is the number of rounds, , and .
After enough large number of rounds the theoretical distribution of the total win converges to the normal distribution, giving a good possibility to forecast the possible win or loss. For example, after 100 rounds at $1 per round, the standard deviation of the win (equally of the loss) will be . After 100 rounds, the expected loss will be .
The 3 sigma range is six times the standard deviation: three above the mean, and three below. Therefore, after 100 rounds betting $1 per round, the result will very probably be somewhere between and , i.e., between -$34 and $24. There is still a ca. 1 to 400 chance that the result will be not in this range, i.e. either the win will exceed $24, or the loss will exceed $34.
The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation.
Unfortunately, the above considerations for small numbers of rounds are incorrect, because the distribution is far from normal. Moreover, the results of more volatile games usually converge to the normal distribution much more slowly, therefore much more huge number of rounds are required for that.
As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is practically impossible for a gambler to win in the long term (if they don't have an edge). It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.
The volatility index (VI) is defined as the standard deviation for one round, betting one unit. Therefore, the VI for the even-money American Roulette bet is .
The variance is defined as the square of the VI. Therefore, the variance of the even-money American Roulette bet is ca. 0.249, which is extremely low for a casino game. The variance for Blackjack is ca. 1.2, which is still low compared to the variances of electronic gaming machines (EGMs).
Additionally, the term of the volatility index based on some confidence intervals are used. Usually, it is based on the 90% confidence interval. The volatility index for the 90% confidence interval is ca. 1.645 times as the 'usual' volatility index that relates to the ca. 68.27% confidence interval.
It is important for a casino to know both the house edge and volatility index for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the volatility index tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field.
See also[edit]
References[edit]
^'Roulette'. britannica.
^'D'Alembert roulette system'.
^'Online slots Return to Player (RTP) explained - GamblersFever'.
^'Return to Player and Hit frequency - What do these mean? - GetGamblingFacts'.
Further reading[edit]
The Mathematics of Gambling, by Edward Thorp, ISBN0-89746-019-7
The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN0-12-240761-X
The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN0-88385-646-8
Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN973-87520-3-5excerpts
Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN1-56881-210-8introduction.
External links[edit]
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