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I know you won't be moving back to your old casino if you've got one after you've had your first real money spin. Here are a bunch of charts and tables for different probabilities in European roulette that might be useful to know. There's also some handy but not necessarily easy information at the bottom about working out roulette probabilities, plus a little bit on the gambler's fallacy. Note: The following probabilities are for European roulette.

Check out the American probabilities page if you're playing on a wheel with the double-zero. How unlikely is it to see the same colour 2 or more times in a row? What's the probability of the results of 5 spins of the roulette wheel being red? The following chart highlights the probabilities of the same colour appearing over a certain number of spins of the roulette wheel.

Example: The probability of the same colour showing up 4 times in a row is 5. As the graph shows, the probability of seeing the same colour on consecutive spins of the roulette wheel more than halves well, the ratio probability doubles from one spin to the next. There are a number of ways to display probabilities. On the roulette charts above I have used; ratio odds, percentage odds and sometimes fractional odds. But what do they mean? As you can see, fractional odds and ratio odds are pretty similar.

The main difference is that fractional odds uses the total number of spins, whereas the ratio just splits it up in to two parts. The majority of people are most comfortable using percentage odds, as they're the most widely understood. Feel free to use whatever makes the most sense to you though of course. From my experience, the easiest way to work out probabilities in roulette is to look at the fraction of numbers for your desired probability, then convert to a percentage or ratio from there.

For example, lets say you want to know the probability of the result being red on a European wheel. Count the amount of numbers that give you the result you want to find the probability for, then put that number over 37 the total number of possible results. As well as working out the probability of winning on each spin, you can also find the likelihood of losing on each spin. All you have to do is count the numbers that will result in a loss.

Work out the fractional probability for each individual spin as above , then multiply those fractions together. For example, let's say you want to find the probability of making correct guesses on specific bet types over multiple spins:.

Luckily, it's pretty easy to convert to either of these from a fraction. Note: You can see how apparent this conversion is in my roulette bets probability table at the top of the page. To determine the winning number, a croupier spins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular track running around the outer edge of the wheel.

The winnings are then paid to anyone who has placed a successful bet. The first form of roulette was devised in 18th century France. Many historians believe Blaise Pascal introduced a primitive form of roulette in the 17th century in his search for a perpetual motion machine. The game has been played in its present form since as early as in Paris. An early description of the roulette game in its current form is found in a French novel La Roulette, ou le Jour by Jaques Lablee, which describes a roulette wheel in the Palais Royal in Paris in The description included the house pockets, "There are exactly two slots reserved for the bank, whence it derives its sole mathematical advantage.

The book was published in The roulette wheels used in the casinos of Paris in the late s had red for the single zero and black for the double zero. To avoid confusion, the color green was selected for the zeros in roulette wheels starting in the s. In some forms of early American roulette wheels, there were numbers 1 through 28, plus a single zero, a double zero, and an American Eagle. The Eagle slot, which was a symbol of American liberty, was a house slot that brought the casino extra edge.

Soon, the tradition vanished and since then the wheel features only numbered slots. According to Hoyle "the single 0, the double 0, and eagle are never bars; but when the ball falls into either of them, the banker sweeps every thing upon the table, except what may happen to be bet on either one of them, when he pays twenty-seven for one, which is the amount paid for all sums bet upon any single figure".

In the 19th century, roulette spread all over Europe and the US, becoming one of the most famous and most popular casino games. When the German government abolished gambling in the s, the Blanc family moved to the last legal remaining casino operation in Europe at Monte Carlo , where they established a gambling mecca for the elite of Europe. It was here that the single zero roulette wheel became the premier game, and over the years was exported around the world, except in the United States where the double zero wheel had remained dominant.

In the United States, the French double zero wheel made its way up the Mississippi from New Orleans , and then westward. It was here, because of rampant cheating by both operators and gamblers, that the wheel was eventually placed on top of the table to prevent devices being hidden in the table or wheel, and the betting layout was simplified.

This eventually evolved into the American-style roulette game. The American game was developed in the gambling dens across the new territories where makeshift games had been set up, whereas the French game evolved with style and leisure in Monte Carlo. During the first part of the 20th century, the only casino towns of note were Monte Carlo with the traditional single zero French wheel, and Las Vegas with the American double zero wheel.

In the s, casinos began to flourish around the world. In the first online casino InterCasino made it possible to play online roulette for the first time. The double zero wheel is found in the U. The sum of all the numbers on the roulette wheel from 0 to 36 is , which is the " Number of the Beast ". Roulette players have a variety of betting options. Placing inside bets is either selecting the exact number of the pocket the ball will land in, or a small range of pockets based on their proximity on the layout.

Players wishing to bet on the 'outside' will select bets on larger positional groupings of pockets, the pocket color, or whether the winning number is odd or even. The payout odds for each type of bet are based on its probability.

The roulette table usually imposes minimum and maximum bets, and these rules usually apply separately for all of a player's inside and outside bets for each spin. For inside bets at roulette tables, some casinos may use separate roulette table chips of various colors to distinguish players at the table. Players can continue to place bets as the ball spins around the wheel until the dealer announces no more bets or rien ne va plus.

When a winning number and color is determined by the roulette wheel, the dealer will place a marker, also known as a dolly, on that winning number on the roulette table layout. When the dolly is on the table, no players may place bets, collect bets, or remove any bets from the table. The dealer will then sweep away all other losing bets either by hand or rake, and determine all of the payouts to the remaining inside and outside winning bets.

When the dealer is finished making payouts, the marker is removed from the board where players collect their winnings and make new bets. The winning chips remain on the board. In , California legalized a form of roulette known as California Roulette.

In number ranges from 1 to 10 and 19 to 28, odd numbers are red and even are black. In ranges from 11 to 18 and 29 to 36, odd numbers are black and even are red. There is a green pocket numbered 0 zero. In American roulette, there is a second green pocket marked Pocket number order on the roulette wheel adheres to the following clockwise sequence in most casinos: [ citation needed ].

The cloth-covered betting area on a roulette table is known as the layout. The layout is either single-zero or double-zero. The European-style layout has a single zero, and the American style layout is usually a double-zero. The American-style roulette table with a wheel at one end is now used in most casinos. The French style table with a wheel in the centre and a layout on either side is rarely found outside of Monte Carlo.

In roulette, bets can either be inside or outside bets. Outside bets typically have smaller payouts with better odds at winning. Except as noted, all of these bets lose if a zero comes up. The initial bet is returned in addition to the mentioned payout.

It can be easily demonstrated that this payout formula would lead to a zero expected value of profit if there were only 36 numbers. Having 37 or more numbers gives the casino its edge. The values 0 and 00 are not odd or even, or high or low. En prison rules, when used, reduce the house advantage. The house average or house edge or house advantage also called the expected value is the amount the player loses relative for any bet made, on average.

The expected value is:. The presence of the green squares on the roulette wheel and on the table is technically the only house edge. Outside bets will always lose when a single or double zero comes up. The only exceptions are the five numbers bet where the house edge is considerably higher 7.

This is commonly called the "la partage" rule, and it is considered the main difference between European and French roulette. There is also a modification of this rule, which is called the " en prison " rule. These rules cut the house edge into half 1. The house edge should not be confused with the "hold". The hold is the average percentage of the money originally brought to the table that the player loses before he leaves—the actual "win" amount for the casino.

This reflects the fact that the player is churning the same money over and over again. In the early frontier gambling saloons, the house would set the odds on roulette tables at 27 for 1. Today most casino odds are set by law, and they have to be either 34 to 1 or 35 to 1. As an example, we can examine the European roulette model, that is, roulette with only one zero. The rules of European roulette have 10 types of bets.

First we can examine the 'Straight Up' bet. For similar reasons it is simple to see that the profitability is also equal for all remaining types of bets. In reality this means that, the more bets a player makes, the more he is going to lose independent of the strategies combinations of bet types or size of bets that he employs:. Here, the profit margin for the roulette owner is equal to approximately 2.

Nevertheless, several roulette strategy systems have been developed despite the losing odds. These systems can not change the odds of the game in favor of the player. Although most often named "call bets" technically these bets are more accurately referred to as "announced bets".

The legal distinction between a "call bet" and an "announced bet" is that a "call bet" is a bet called by the player without him placing any money on the table to cover the cost of the bet. In many jurisdictions most notably the United Kingdom this is considered gambling on credit and is illegal. An "announced bet" is a bet called by the player for which he immediately places enough money to cover the amount of the bet on the table, prior to the outcome of the spin or hand in progress being known.

There are different number series in roulette that have special names attached to them. Most commonly these bets are known as "the French bets" and each covers a section of the wheel. For the sake of accuracy, zero spiel, although explained below, is not a French bet, it is more accurately "the German bet". Players at a table may bet a set amount per series or multiples of that amount. The series are based on the way certain numbers lie next to each other on the roulette wheel.

Not all casinos offer these bets, and some may offer additional bets or variations on these. The series is on a single-zero wheel. Nine chips or multiples thereof are bet. Two chips are placed on the trio; one on the split; one on ; one on ; one on ; two on the corner; and one on Zero game, also known as zero spiel Spiel is German for game or play , is the name for the numbers closest to zero.

All numbers in the zero game are included in the voisins, but are placed differently. The numbers bet on are The bet consists of four chips or multiples thereof. Three chips are bet on splits and one chip straight-up: one chip on split, one on split, one on split and one straight-up on number This type of bet is popular in Germany and many European casinos.

It is also offered as a 5-chip bet in many Eastern European casinos. As a 5-chip bet, it is known as "zero spiel naca" and includes, in addition to the chips placed as noted above, a straight-up on number This is the name for the 12 numbers that lie on the opposite side of the wheel between 27 and 33, including 27 and 33 themselves.

On a single-zero wheel, the series is Very popular in British casinos, tiers bets outnumber voisins and orphelins bets by a massive margin. Six chips or multiples thereof are bet. One chip is placed on each of the following splits: , , , , , and The tiers bet is also called the "small series" and in some casinos most notably in South Africa "series ". A variant known as "tiers " has an additional chip placed straight up on 5, 8, 10, and 11m and so is a piece bet.

In some places the variant is called "gioco Ferrari" with a straight up on 8, 11, 23 and 30, the bet is marked with a red G on the racetrack. These numbers make up the two slices of the wheel outside the tiers and voisins. They contain a total of 8 numbers, comprising and

You can place all sorts of bets with roulette. For instance, you can bet on a particular number, whether that number is odd or even, or the color of the pocket. One other thing to remember: if the ball lands on a green pocket, you lose. You can use it to help work out the probabilities in this chapter. Have you cut out your roulette board? The game is just beginning. Where do you think the ball will land? Maybe some bets are more likely than others.

It sounds like we need to look at some probabilities What things do you need to think about before placing any roulette bets? Given the choice, what sort of bet would you make? Probability is a way of measuring the chance of something happening. In stats-speak, an event is any occurrence that has a probability attached to it—in other words, an event is any outcome where you can say how likely it is to occur. Probability is measured on a scale of 0 to 1.

If an event is impossible, it has a probability of 0. An outcome or occurrence that has a probability assigned to it. If you know how likely the ball is to land on a particular number or color, you have some way of judging whether or not you should place a particular bet.

To work out the probability of getting a 7, take your answer to question 2 and divide it by your answer to question 1. What do you get? Mark the probability on the scale below. You had to work out a probability for roulette, the probability of the ball landing on 7. Here are all the possible outcomes from spinning the roulette wheel.

To find the probability of winning, we take the number of ways of winning the bet and divide by the number of possible outcomes like this:. S is known as the possibility space , or sample space. Possible events are all subsets of S.

One way of doing so is to draw a box representing the possibility space S , and then draw circles for each relevant event. This sort of diagram is known as a Venn diagram. Instead of numbers, you have the option of using the actual probabilities of each event in the diagram.

It all depends on what kind of information you need to help you solve the problem. A I is known as the complementary event of A. This gives us. For each event below, write down the probability of a successful outcome. For each event you should have written down the probability of a successful outcome. This event is actually impossible—there is no pocket labeled Therefore, the probability is 0.

Q: Why do I need to know about probability? I thought I was learning about statistics. A lot of statistics has its origins in probability theory, so knowing probability will take your statistics skills to the next level. Probability theory can help you make predictions about your data and see patterns. It can help you make sense of apparent randomness. Q: Are probabilities written as fractions, decimals, or percentages? A: They can be written as any of these.

Is there a connection? A: There certainly is. In set theory, the possibility space is equivalent to the set of all possible outcomes, and a possible event forms a subset of this. Q: Do I always have to draw a Venn diagram? Q: Can anything be in both events A and A I? A: No. The two events are mutually exclusive, so no elements are shared between them. Look at the events on the previous page. Oh dear! Even though our most likely probability was that the ball would land in a black pocket, it actually landed in the green 0 pocket.

You lose some of your chips. The important thing to remember is that a probability indicates a long-term trend only. To work out the probability, all we have to do is count how many pockets are red or black, then divide by the number of pockets. Sound easy enough? Take a look at your roulette board. There are only three colors for the ball to land on: red, black, or green. Calculate the probability of getting a black or a red by counting how many pockets are black or red and dividing by the number of pockets.

If we know P Black and P Red , we can find the probability of getting a black or red by adding these two probabilities together. In this case, adding the probabilities gives exactly the same result as counting all the red or black pockets and dividing by To find the probability of an event A, use. Q: It looks like there are three ways of dealing with this sort of probability. Which way is best? A: It all depends on your particular situation and what information you are given. Suppose the only information you had about the roulette wheel was the probability of getting a green.

It can still be useful to double-check your results, though. Q: If some events are so unlikely to happen, why do people bet on them? A: A lot depends on the sort of return that is being offered. In general, the less likely the event is to occur, the higher the payoff when it happens. People are tempted to make bets where the return is high, even though the chances of them winning is negligible. Q: Does adding probabilities together like that always work?

A: Think of this as a special case where it does. We might not be able to count on being able to do this probability calculation in quite the same way as the previous one. Try the exercise on the next page, and see what happens. Finally, use your roulette board to count all the holes that are either black or even, then divide by the total number of holes. When we were working out the probability of the ball landing in a black or red pocket, we were dealing with two separate events, the ball landing in a black pocket and the ball landing in a red pocket.

If two events are mutually exclusive, only one of the two can occur. What about the black and even events? The two events intersect. Calculating the probability of getting a black or even went wrong because we included black and even pockets twice. First of all, we found the probability of getting a black pocket and the probability of getting an even number.

When we added the two probabilities together, we counted the probability of getting a black and even pocket twice. To get the correct answer, we need to subtract the probability of getting both black and even.

It includes all of the elements in A and also those in B. Between them, they make up the whole of S. They exhaust all possibilities. On the previous page, we found that. Mutually exclusive events have no elements in common with each other. Draw a Venn diagram for this probability space. How many enthusiasts play baseball in total? How many play basketball? How many play football? Which sports are exhaustive fill up the possibility space?

To find the probability of getting event A or B, use. By adding up the values in each circle in the Venn diagram, we can determine that there are 16 total baseball players, 28 total basketball players, and 16 total football players. The baseball and football events are mutually exclusive. The events for baseball, football, and basketball are exhaustive. Q: Are A and A I mutually exclusive or exhaustive? A and A I can have no common elements, so they are mutually exclusive.

A: Yes it is. It can sometimes be useful to think of different ways of forming the same probability, though. Q: Is there a limit on how many events can intersect? Finding probabilities for multiple intersections can sometimes be tricky. We know that the probability of the ball landing on black or even is 0. The croupier decides to take pity on us and offers a little inside information. How does the probability of getting even given that we know the ball landed in a black pocket compare to our last bet that the ball would land on black or even.

The croupier says the ball has landed in a black pocket. In other words, we want to find out how many pockets are even out of all the black ones. Out of the 18 black pockets, 10 of them are even, so. It turns out that even with some inside information, our odds are actually lower than before. The probability of even given black is actually less than the probability of black or even. However, a probability of 0. So how can we generalize this sort of problem?

First of all, we need some more notation to represent conditional probabilities , which measure the probability of one event occurring relative to another occurring. So now we need a general way of calculating P A B. Looking at the Venn diagram, we get:. This means that we can rewrite the formula as. The second set of branches shows the probability of outcomes given the outcome of the branch it is linked to.

A I refers to every possibility not covered by A, and B I refers to every possibility not covered by B. You can find probabilities involving intersections by multiplying the probabilities of linked branches together. In other words, you multiply the probability on the first level B branch with the probability on the second level A branch. Probability trees can be time-consuming to draw, but they offer you a way of visualizing conditional probabilities.

They drew up a probability tree to show the probabilities, but in a sudden gust of wind, they all fell off. Your task is to pin the probabilities back on the tree. Here are some clues to help you. Work out the levels. Try and work out the different levels of probability that you need. If you add together the probabilities for all of the branches that fork off from a common point, the sum should equal 1.

They drew up a probability tree to show the probabilities, but in a sudden gust of wind they all fell off. With this probability, you can make no assumptions about whether one of the events has already occurred. You have to find the probability of both events happening without making any assumptions.

P A B is the probability of event A given event B. In other words, you make the assumption that event B has occurred, and you work out the probability of getting A under this assumption. A: No, they refer to different probabilities. When you calculate P A B , you have to assume that event B has already happened.

When you work out P A , you can make no such assumption. They look similar. P A B is the probability of getting event A given event B has already happened. P B A is the probability of getting event B given event A occurred. A: Both diagrams give you a way of visualizing probabilities, and both have their uses. It all depends what type of problem you need to solve. Q: Is there a limit to how many sets of branches you can have on a probability tree?

In practice you may find that a very large probability tree can become unwieldy, but you may still find it easier to draw a large probability tree than work through complex probabilities without it. Unfortunately, the ball landed in pocket 17, so you lose a few more chips.

Maybe we can win some chips back with another bet. This time, the croupier says that the ball has landed in an even pocket. We can reuse the probability calculations we already did. Our previous task was to figure out P Even Black , and we can use the probabilities we found solving that problem to calculate P Black Even. So how do we find P Black Even? All we need is some mechanism for finding these probabilities. Use the probabilities you have to calculate the probabilities you need.

Take a look at the probability tree on the previous page. Take another look at the probability tree in So where does this get us? How do you think we can use it to find P Even? The next step is to find the probability of the ball landing in an even pocket, P Even. We can find this by considering all the ways in which this could happen. These are all the possible ways in which a ball can land in an even pocket. In other words, we add the probability of the pocket being both black and even to the probability of it being both red and even.

The relevant branches are highlighted on the probability tree. Can you remember our original problem? We wanted to find P Black Even where. Putting these together means that we can calculate P Black Even using probabilities from the probability tree. This means that we now have a way of finding new conditional probabilities using probabilities we already know—something that can help with more complicated probability problems. Imagine you have a probability tree showing events A and B like this, and assume you know the probability on each of the branches.

Now imagine you want to find P A B , and the information shown on the branches above is all the information that you have. How can you use the probabilities you have to work out P A B? To find P B , we use the same process that we used to find P Even earlier; we need to add together the probabilities of all the different ways in which the event we want can possibly happen.

There are two ways in which even B can occur: either with event A, or without it. This means that we can find P B using:. We can rewrite this in terms of the probabilities we already know from the probability tree. This means that we can use:. This is sometimes known as the Law of Total Probability , as it gives a way of finding the total probability of a particular event based on conditional probabilities. We started off by wanting to find P A B based on probabilities we already know from a probability tree.

What we need is a general expression for finding conditional probabilities that are the reverse of what we already know, in other words P A B. And even though the formula is tricky, visualizing the problem can help. The Manic Mango games company is testing two brand-new games. Manic Mango selects one of the volunteers at random to ask if she enjoyed playing the game, and she says she did. If you have two events A and B, then. If you have n mutually exclusive and exhaustive events, A 1 through to A n , and B is another event, then.

It will give you the same result, and it can keep you from losing track of which probability belongs to which event. Did we make a mistake? This means that P Even Green is 0; therefore, it has no effect on the calculation. Is that always the case? They are two separate probabilities, and making this sort of assumption could actually cost you valuable points in a statistics exam.

For example, it can be used in computing as a way of filtering emails and detecting which ones are likely to be junk. Before you leave the roulette table, the croupier has offered you a great deal for your final bet, triple or nothing. If you bet that the ball lands in a black pocket twice in a row, you could win back all of your chips.

Notice that the probabilities for landing on two black pockets in a row are a bit different than they were in our probability tree in Bad luck! Take a look at the equation for this probability:. For P Even Black , the probability of getting an even pocket is affected by the event of getting a black.

We already know that the ball has landed in a black pocket, so we use this knowledge to work out the probability. We look at how many of the pockets are even out of all the black pockets. The French roulette too has one zero only, but the table layout is slightly different and wider see picture of table layout below , and there are no individual coloured chips for the players, cash chips are used.

Also, a stick is used by the dealer and stickman to announce the winning number, to collect the chips from the table and to pay the winners, which makes the game slower than the American version. In the American and European roulette individual coloured chips are used for each player and after the outcome the losing chips are collected from the table by hand and the winnings are paid by hand. Play is much faster than the French roulette.

In French casinos "American Roulette" means a roulette game with double zeros 0, 00 - on the table layout and on the wheel , the same as used in the USA. The American roulette with one zero or the European version is referred to as English Roulette to distinguish it from the double-zero American Roulette and to emphasize that it has one zero only as used in the UK.

However, in many countries including the UK the single-zero European roulette is called American Roulette to distinguish it from the French roulette table layout and for the manner the game is played. In some African countries they call it 'Roulette with French numeration on American table'. To sum up: American Roulette can have a single zero or a double zero, but the manner it is played is the same for both, and play is faster than French Roulette. Apart from the zero positions, the table layout is the same.

The other major difference is in the roulette wheel, the sequence of the numbers on the wheel is totally different. European Roulette , a term mostly used in the USA, is roulette with a single zero regardless of the table layout, could refer to either the American Roulette with a single zero or to the French Roulette as both use the same type of roulette wheel with the same sequence of numbers.

French Roulette is known everywhere as French Roulette, has a single zero, a wide table layout and have Stickmen that handle the chips. In some European casinos the American single zero roulette is called Fair Roulette. English Roulette, a term commonly used in French casinos, is referred to the American Roulette with a single zero, as used in United Kingdom casinos. French roulette table layout The odds and payouts are as above Inside bets: A - 1 number, Straight up. B - 2 numbers, Split. C - 3 numbers with 0 and 3 Line.

D - 4 numbers with 0 and 4 Corner. E - 6 numbers, 6 Line. Outside bets: F - 12 numbers, Column. G - 12 numbers, Dozen. Finales - A type of group bets on numbers ending with the same digit. Examples: Final 7 means bet on the numbers 7, 17 and 27 straight up three chips.

Final 2 means bet on the numbers 2, 12, 22 and 32 straight up four chips. The table layout on the French version of roulette is different from the American style roulette table shown above. The main difference is in the position of the side bets outside bets. The side bets on the French table are split in two and run along both sides of the table layout. Each side bet is given its French name and sometimes also its English translation underneath.

Roulette Announced Bets or Call Bets Used in French roulette tables and in some European casinos with single-zero wheel roulette tables. Common in most UK casinos with single-zero wheel American style roulette tables. Roulite or Roulight table layout Roulite or Roulight is the modern variety of Roulette.

A table game that has been developed in the first instance by specialists at the Wiesbaden Casino, Wiesbaden, Germany.

European roulette boards have numbers into the thirties, and also a single zero. American roulette boards, however, as well as all of these numbers and the single zero, also have a double zero on them. So, if you are playing on an American roulette board, you can place a bet on this double zero. Another version of this bet is to bet on both the double zero and the single zero at the same time. Some roulette players like to spread their bets out further, betting on the double zero but also betting at the same time on various other numbers on the board.

This can increase your chance of a hit, but the payoffs are smaller if one of your numbers does get hit. It can be so frustrating to spread your bets out across the board and then see the double zero get hit — if only you had bet everything on the double zero, you would have got a much larger payoff! The color of the double zero on the roulette board is green. The color of the single zero is also green. When you bet on single green odds, you bet on this single zero.

The hope is that the single zero will be hit! Again, it is also possible to bet on other numbers at the same time as the single green. Though, again, it is also frustrating when you spread your bets out like this only to see the single green get hit.

If only you had betted entirely on single green odds at moments like this — you would be enjoying a much higher amount of winnings! The double zero and the single green are what give the house the edge over the players. A European board which has just one single green zero has a house advantage of around 2. As such, the single and double zeros have become the focal point for roulette players who are interested in beating the house.

There are a few other decent ones as well. The trick to know is that when the house edge is a very low percentage, then the bet is a good one. Also remember the fact that the number 7 is rolled more often than any other number.

The 2 and 12 snake eyes and box cars are rolled the least and have low probabilities of coming out. You can read about this in our odds and probabilities page if you would like to understand the concept behind this. For additional odds information, please view our craps probability odds page for information about true odds of rolling the dice and also visit our Vegas free odds page for information about odds payouts and house edge. Play craps online at Cherry Jackpot.

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Chumba Casino is a gaming site that operates as a sweepstakes casino. It offers a variety of slots, as well as video poker, blackjack, and roulette. Chumba Casino offers a convenient platform with numerous games to choose from. Players can enjoy the games with ease as they can be played directly through a browser with no download required.

Chumba is the best online casino site to play roulette for cash prizes. Each of these bets refers to a specific set of numbers or colors. Red or Black — This bet pays out even odds 1 to 1 if the ball lands on the color you chose. Odd or Even — This bet pays out even odds 1 to 1 if the ball lands on odd or even, depending on which you chose.

Low or High — This bet pays out even money 1 to 1 if the ball lands on if you bet low, or if the ball lands on if you bet high. Columns — The numbers on the layout are organized into three columns of twelve numbers each. This bet pays out 2 to 1 when you win. Dozens — There are 36 numbers on the table, so you can bet on the first dozen , the second dozen , or the third dozen This bet also pays out 2 to 1.

You can also bet on specific numbers and sets of numbers on the inside of the layout. These bets win less often, but they pay out more when you do win. The house edge on the inside bets is the same as the house edge on the outside bets. Split bet — This is a bet on any two adjacent numbers. You place the chip on the line between the two numbers in order to make this wager.

This bet pays out at 17 to 1. Street bet — This bet covers three numbers. You place your bet on the line outside of the three numbers in the row where you want to win. This bet pays out at 11 to 1. Corner bet — Some people call this a square bet or a quarter bet. A win on this type of bet pays out at 8 to 1. The problem is that it has a higher house edge, making it the worst bet on the table.

This bet is on the numbers 0, 00, 1, 2, and 3, and you place the chip on the outside corner line between the 1 and the 0. This bet pays out 6 to 1, but only masochists place this bet. All we need is some mechanism for finding these probabilities. Use the probabilities you have to calculate the probabilities you need. Take a look at the probability tree on the previous page. Take another look at the probability tree in So where does this get us?

How do you think we can use it to find P Even? The next step is to find the probability of the ball landing in an even pocket, P Even. We can find this by considering all the ways in which this could happen. These are all the possible ways in which a ball can land in an even pocket. In other words, we add the probability of the pocket being both black and even to the probability of it being both red and even.

The relevant branches are highlighted on the probability tree. Can you remember our original problem? We wanted to find P Black Even where. Putting these together means that we can calculate P Black Even using probabilities from the probability tree. This means that we now have a way of finding new conditional probabilities using probabilities we already know—something that can help with more complicated probability problems. Imagine you have a probability tree showing events A and B like this, and assume you know the probability on each of the branches.

Now imagine you want to find P A B , and the information shown on the branches above is all the information that you have. How can you use the probabilities you have to work out P A B? To find P B , we use the same process that we used to find P Even earlier; we need to add together the probabilities of all the different ways in which the event we want can possibly happen.

There are two ways in which even B can occur: either with event A, or without it. This means that we can find P B using:. We can rewrite this in terms of the probabilities we already know from the probability tree. This means that we can use:. This is sometimes known as the Law of Total Probability , as it gives a way of finding the total probability of a particular event based on conditional probabilities. We started off by wanting to find P A B based on probabilities we already know from a probability tree.

What we need is a general expression for finding conditional probabilities that are the reverse of what we already know, in other words P A B. And even though the formula is tricky, visualizing the problem can help. The Manic Mango games company is testing two brand-new games. Manic Mango selects one of the volunteers at random to ask if she enjoyed playing the game, and she says she did. If you have two events A and B, then. If you have n mutually exclusive and exhaustive events, A 1 through to A n , and B is another event, then.

It will give you the same result, and it can keep you from losing track of which probability belongs to which event. Did we make a mistake? This means that P Even Green is 0; therefore, it has no effect on the calculation. Is that always the case? They are two separate probabilities, and making this sort of assumption could actually cost you valuable points in a statistics exam.

For example, it can be used in computing as a way of filtering emails and detecting which ones are likely to be junk. Before you leave the roulette table, the croupier has offered you a great deal for your final bet, triple or nothing.

If you bet that the ball lands in a black pocket twice in a row, you could win back all of your chips. Notice that the probabilities for landing on two black pockets in a row are a bit different than they were in our probability tree in Bad luck!

Take a look at the equation for this probability:. For P Even Black , the probability of getting an even pocket is affected by the event of getting a black. We already know that the ball has landed in a black pocket, so we use this knowledge to work out the probability. We look at how many of the pockets are even out of all the black pockets. To work out P Even , we look at how many pockets are even out of all the pockets.

P Even Black gives a different result from P Even. In other words, the knowledge we have that the pocket is black changes the probability. These two events are said to be dependent. Look at the probability tree on the previous page again. What do you notice about the sets of branches? Are the events for getting a black in the first game and getting a black in the second game dependent?

Not all events are dependent. Sometimes events remain completely unaffected by each other, and the probability of an event occurring remains the same irrespective of whether the other event happens or not. What do you notice? These two probabilities have the same value.

In other words, the event of getting a black pocket in this game has no bearing on the probability of getting a black pocket in the next game. These events are independent. If one event occurs, the probability of the other occurring remains exactly the same. If events A and B are independent, then the probability of event A is unaffected by event B. In other words. We can also use this as a test for independence. If A and B are mutually exclusive, then if event A occurs, event B cannot.

In other words, if two events are independent, then you can work out the probability of getting both events A and B by multiplying their individual probabilities together. As the events are independent, the result is.

If A and B are mutually exclusive, then if event A happens, B cannot. Also, if event B happens, then A cannot. If A and B are independent, then the outcome of A has no effect on the outcome of B, and the outcome of B has no effect on the outcome of A. Their respective outcomes have no effect on each other. Q: Do both events have to be independent? Can one event be independent and the other dependent?

Q: Are all games on a roulette wheel independent? A: Yes, they are. Separate spins of the roulette wheel do not influence each other. In each game, the probabilities of the ball landing on a red, black, or green remain the same. How do I use a Venn diagram to tell if events are independent? Venn diagrams are great if you need to examine intersections and show mutually exclusive events. The Case of the Two Classes. The Head First Health Club prides itself on its ability to find a class for everyone.

As a result, it is extremely popular with both young and old. The Health Club is wondering how best to market its new yoga class, and the Head of Marketing wonders if someone who goes swimming is more likely to go to a yoga class. The CEO disagrees. They ask a group of 96 people whether they go to the swimming or yoga classes.

Out of these 96 people, 32 go to yoga and 72 go swimming. Are the yoga and swimming classes dependent or independent? Well, I hear you keep getting fledgling statisticians into trouble. You want to work out the probability of getting two independent events? Just multiply the probabilities for the two events together and job done. With independent events, probabilities just turn out the same.

I think that people need to think of me first instead of you; that would sort out all of these problems. By really thinking through whether events are dependent or not. Let me give you an example. Suppose you have a deck of 52 cards, and thirteen of them are diamonds. What would be the probability of that happening?

The events are dependent. You can no longer say there are 13 diamonds in a pack of 52 cards. Not fair, I assumed you put the first card back! That would have meant the probability of getting a diamond would have been the same as before, and I would have been right. The events would have been independent. When people think about you first, it leads them towards making all sorts of inappropriate assumptions.

Solved: The Case of the Two Classes. So how do we know the classes are independent? Here are a bunch of situations and events. Your task is to say which of these are dependent, and which are independent. Choosing a card from a deck of cards, putting the card back in the deck, and then choosing another one. Your task was to say which of these are dependent, and which are independent. When you remove one sock, there are fewer socks to choose from the next time, and this affects the probability.

Besides the chances of winning, you also need to know how much you stand to win in order to decide if the bet is worth the risk. Betting on an event that has a very low probability may be worth it if the payoff is high enough to compensate you for the risk. Fred decides to throw a coin. I person eats alone if Fred and George go to the Diner. As these events are independent, this is equal to P 22 x P Skip to main content.

Head First Statistics by Dawn Griffiths. Start your free trial. Calculating Probabilities: Taking Chances. Roll up for roulette! Your very own roulette board. Note Just be careful with those scissors. Place your bets now! Brain Power What things do you need to think about before placing any roulette bets? What are the chances? Vital Statistics: Event An outcome or occurrence that has a probability assigned to it. Sharpen your pencil. Sharpen your pencil Solution.

Find roulette probabilities. You can visualize probabilities with a Venn diagram. Complementary events. BE the croupier Solution. Note The most likely event out of all these is that the ball will land in a black pocket. There are no Dumb Questions. Q: Q: Why do I need to know about probability?

Q: Q: Are probabilities written as fractions, decimals, or percentages? A: A: They can be written as any of these. A: A: There certainly is. Q: Q: Do I always have to draw a Venn diagram? A: A: No. And the winning number is You can also add probabilities. Vital Statistics: Probability To find the probability of an event A, use. Q: Q: It looks like there are three ways of dealing with this sort of probability. A: A: It all depends on your particular situation and what information you are given.

Q: Q: If some events are so unlikely to happen, why do people bet on them? A: A: A lot depends on the sort of return that is being offered. Q: Q: Does adding probabilities together like that always work? A: A: Think of this as a special case where it does. You win! Time for another bet.

Note Uh oh! Different answers. Exclusive events and intersecting events. Brain Power What sort of effect do you think this intersection could have had on the probability? Problems at the intersection. Some more notation. Watch it! BE the probability Solution. Exercise Solution. A: A: Yes it is. Q: Q: Is there a limit on how many events can intersect? Another unlucky spin Conditions apply. Find conditional probabilities. Note The probability of A give that we know B has happened.

You can visualize conditional probabilities with a probability tree. Trees also help you calculate conditional probabilities. Handy hints for working with trees. Probability Magnets Solution. You can get coffee with or without donuts. Note Hint: maybe some of your other answers can help you. Note We can read this off the tree. Vital Statistics: Conditions. A: A: No, they refer to different probabilities.

Q: Q: Are probability trees better than Venn diagrams? A: A: Both diagrams give you a way of visualizing probabilities, and both have their uses. Q: Q: Is there a limit to how many sets of branches you can have on a probability tree? Bad luck! Note This is the opposite of the previous bet. We can find P Black l Even using the probabilities we already have.

So where does this get us? Brain Power Take another look at the probability tree in So where does this get us? Step 2: Finding P Even. Step 3: Finding P Black l Even. These results can be generalized to other problems. Brain Power Take a good look at the probability tree. How would you use it to find P B? Note Add together both of the intersections to get P B. Long Exercise Solution.

Q: Q: Do I have to draw a probability tree? We have a winner! If events affect each other, they are dependent. Note These two probabilities are different. Brain Power Look at the probability tree on the previous page again. If events do not affect each other, they are independent. Note These probabilities are the same. The events are independent.