Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. All bets are made the same way: by betting on black. The bettor consecutively doubles the value of any previous losing bet until he finally wins. The same rule is then applied to any eventual third bet, fourth bet, fifth bet, etc.
Let N respectively B denote the number of bets made respectively the total amount of money that was bet before stopping. What was the net profit the bettor made from the n bets? Interpret your answer in terms of how it depends on n. We now present initial evidence that something quite suspicious is taking place.
I am stuck on part a. It seems wrong to me but I am not sure how to know and can't move on until I solve part a. A better of infinite wealth almost surely has a positive return. But no better has infinite wealth. Sign up to join this community. The best answers are voted up and rise to the top. Progressive Roulette Betting Ask Question. Asked 6 years, 3 months ago. Active 1 month ago. Viewed times. Here's the setup: Bets are repeatedly made at a roulette according to the following strategy: All bets are made the same way: by betting on black.
Odds are, it will hit in 1 in 37 plays. The casino gains its advantage here by placing unfair payouts on such bets. While the odds are 1 in 37, the payout is only 35 to 1. There are a number of progressive betting systems to choose from.
Martingale System This is nothing but a progressive betting system. The roulette strategy relies solely upon doubling your bet each time you lose so that, when you do win, you make a 1 unit profit. Fibonacci System This progressive betting system is a bit more complicated. It involves betting very specific amounts, multiplied by a 1 unit bet size.
The progression is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, The player only progresses to the next number if the previous bet is lost. If it is won, the player moves back two numbers in the sequence. So if you win on a 13 unit bet, you move back two numbers, betting 5 units on the next bet. Once you have made a profit in the betting cycle, drop back to the beginning 1, 1, 2, etc. Tier et Rout System This system claims to incorporate a bankroll management system into the progression.
It involves dividing your bankroll into thirds, betting the first third on the first bet, then if you lose, bet the last two thirds. Win that, split your new bankroll into thirds, and repeat. The Tier et Tout system described above has the most obvious flaw. All progressive betting systems are flawed, no matter how you break down the bets. There are two major problems here.
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They are confident in knowing that, in the long-term, players cannot realistically turn the advantage to their own favor. In the game of roulette, the casino claims its edge by paying even money on bets that have a negative probability of winning. Playing American roulette has even worse odds, using a 0 and What about betting on single numbers? Odds are, it will hit in 1 in 37 plays. The casino gains its advantage here by placing unfair payouts on such bets. While the odds are 1 in 37, the payout is only 35 to 1.
There are a number of progressive betting systems to choose from. Martingale System This is nothing but a progressive betting system. The roulette strategy relies solely upon doubling your bet each time you lose so that, when you do win, you make a 1 unit profit.
Fibonacci System This progressive betting system is a bit more complicated. It involves betting very specific amounts, multiplied by a 1 unit bet size. The progression is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, The player only progresses to the next number if the previous bet is lost. If it is won, the player moves back two numbers in the sequence. So if you win on a 13 unit bet, you move back two numbers, betting 5 units on the next bet.
Once you have made a profit in the betting cycle, drop back to the beginning 1, 1, 2, etc. Tier et Rout System This system claims to incorporate a bankroll management system into the progression. It involves dividing your bankroll into thirds, betting the first third on the first bet, then if you lose, bet the last two thirds. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain.
In a casino, the expected value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly. The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables , an assumption which is valid in many realistic situations.
It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.
The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets which is also true in practice. The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem. However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.
Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler "resets" and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds.
Following is an analysis of the expected value of one round. Let q be the probability of losing e. Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose. The probability that the gambler will lose all n bets is q n. When all bets lose, the total loss is. In all other cases, the gambler wins the initial bet B. Thus, the expected profit per round is. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round.
Increasing the size of wager for each round per the martingale system only serves to increase the average loss. Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2 k units. With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point.
Once this win is achieved, the gambler restarts the system with a 1 unit bet. With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued. Thus, the total expected value for each application of the betting system is 0.
In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of Eventually he either goes bust or reaches his target. This strategy gives him a probability of The previous analysis calculates expected value , but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.
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 Five chips or multiples thereof are bet on four splits and a straight-up: one chip is placed straight-up on 1 and one chip on each of the splits: , , , and A number may be backed along with the two numbers on the either side of it in a 5-chip bet.
For example, "0 and the neighbors" is a 5-chip bet with one piece straight-up on 3, 26, 0, 32, and Neighbors bets are often put on in combinations, for example "1, 9, 14, and the neighbors" is a chip bet covering 18, 22, 33, 16 with one chip, 9, 31, 20, 1 with two chips and 14 with three chips. Any of the above bets may be combined, e. The " Final 4, for example, is a 4-chip bet and consists of one chip placed on each of the numbers ending in 4, that is 4, 14, 24, and Final 7 is a 3-chip bet, one chip each on 7, 17, and Final bets from final 0 zero to final 6 cost four chips.
Final bets 7, 8 and 9 cost three chips. Some casinos also offer split-final bets, for example final would be a 4-chip bet, one chip each on the splits , , , and one on A complete bet places all of the inside bets on a certain number. Full complete bets are most often bet by high rollers as maximum bets. The maximum amount allowed to be wagered on a single bet in European roulette is based on a progressive betting model.
For instance, if a patron wished to place a full complete bet on 17, the player would call "17 to the maximum". To manually place the same wager, the player would need to bet:. The player calls his bet to the croupier most often after the ball has been spun and places enough chips to cover the bet on the table within reach of the croupier. The croupier will immediately announce the bet repeat what the player has just said , ensure that the correct monetary amount has been given while simultaneously placing a matching marker on the number on the table and the amount wagered.
The player's wagered 40 chips, as with all winning bets in roulette, are still his property and in the absence of a request to the contrary are left up to possibly win again on the next spin. Based on the location of the numbers on the layout, the number of chips required to "complete" a number can be determined. Most typically Mayfair casinos in London and other top-class European casinos with these maximum or full complete bets, nothing except the aforementioned maximum button is ever placed on the layout even in the case of a win.
Experienced gaming staff, and the type of customers playing such bets, are fully aware of the payouts and so the croupier simply makes up the correct payout, announces its value to the table inspector floor person in the U. Also typically at this level of play house rules allowing the experienced croupier caters to the needs of the customer and will most often add the customer's winning bet to the payout, as the type of player playing these bets very rarely bets the same number two spins in succession.
There are also several methods to determine the payout when a number adjacent to a chosen number is the winner, for example, player bets 40 chips on "23 to the maximum" and number 26 is the winning number. The most notable method is known as the "station" system or method. When paying in stations, the dealer counts the number of ways or stations that the winning number hits the complete bet.
In the example above, 26 hits 4 stations - 2 different corners, 1 split and 1 six-line. If calculated as stations, they would just multiply 4 by 36, making with the players bet down. Over the years, many people have tried to beat the casino, and turn roulette—a game designed to turn a profit for the house—into one on which the player expects to win.
Most of the time this comes down to the use of betting systems, strategies which say that the house edge can be beaten by simply employing a special pattern of bets, often relying on the " Gambler's fallacy ", the idea that past results are any guide to the future for example, if a roulette wheel has come up 10 times in a row on red, that red on the next spin is any more or less likely than if the last spin was black.
All betting systems that rely on patterns, when employed on casino edge games will result, on average, in the player losing money. Certain systems, such as the Martingale, described below, are extremely risky, because the worst-case scenario which is mathematically certain to happen, at some point may see the player chasing losses with ever-bigger bets until he runs out of money.
The American mathematician Patrick Billingsley said  that no betting system can convert a subfair game into a profitable enterprise. At least in the s, some professional gamblers were able to consistently gain an edge in roulette by seeking out rigged wheels not difficult to find at that time and betting opposite the largest bets. Whereas betting systems are essentially an attempt to beat the fact that a geometric series with initial value of 0.
These schemes work by determining that the ball is more likely to fall at certain numbers. Edward O. Thorp the developer of card counting and an early hedge-fund pioneer and Claude Shannon a mathematician and electronic engineer best known for his contributions to information theory built the first wearable computer to predict the landing of the ball in This system worked by timing the ball and wheel, and using the information obtained to calculate the most likely octant where the ball would fall.
Ironically, this technique works best with an unbiased wheel though it could still be countered quite easily by simply closing the table for betting before beginning the spin. In , several casinos in Britain began to lose large sums of money at their roulette tables to teams of gamblers from the USA. Upon investigation by the police, it was discovered they were using a legal system of biased wheel-section betting.
As a result of this, the British roulette wheel manufacturer John Huxley manufactured a roulette wheel to counteract the problem. The new wheel, designed by George Melas, was called "low profile" because the pockets had been drastically reduced in depth, and various other design modifications caused the ball to descend in a gradual approach to the pocket area. Thomas Bass , in his book The Eudaemonic Pie published as The Newtonian Casino in Britain , has claimed to be able to predict wheel performance in real time.
The book describes the exploits of a group of University of California Santa Cruz students, who called themselves the Eudaemons , who in the late s used computers in their shoes to win at roulette. This is an updated and improved version of Edward O. Thorp 's approach, where Newtonian Laws of Motion are applied to track the roulette ball's deceleration; hence the British title.
In the early s, Gonzalo Garcia-Pelayo believed that casino roulette wheels were not perfectly random , and that by recording the results and analysing them with a computer, he could gain an edge on the house by predicting that certain numbers were more likely to occur next than the 1-in odds offered by the house suggested. This he did at the Casino de Madrid in Madrid , Spain , winning , euros in a single day, and one million euros in total.
Legal action against him by the casino was unsuccessful, it being ruled that the casino should fix its wheel. If the same number comes up again, the odds increase to x your stake. A fourth time in a row brings you a payout of 3,x your stake.
Remember : You win the progressive Jackpot if the wheel hits the same number for the 5th time. Marvel Roulette is based on the popular comic series, featuring such characters as Captain America, Spiderman and Iron Man. You notice that you are playing on a slightly different wheel when playing Marvel roulette.
The wheel features an extra blue colored bonus slot between the zero and the 32 slot. Additionally, this game is linked with all of the other Marvel Games offered by PlayTech casinos, in particular, the slots, where chances of winning a Jackpot are randomly rewarded to players, whether you win or lose. The game features a wheel with an additional bonus slot sat between 5 and 10 as well as an additional zero to create the triple bonus effect.
The wheel is transformed into a bonus wheel that has three wheels in one when the bet is hit. Hitting either the bonus slot or one of the three zeros one or more times will give you a bonus up to 12, times your stake. Another roulette game variation very similar to the triple bonus spin roulette is double bonus spin game. Progressive roulette is a fun way to enjoy roulette and does indeed offer a chance to win big while playing the game we all love.
If you win twice in st etienne vs lyon betting expert free progressive betting on roulette of the wheel chips or multiples thereof. Progressive betting on roulette you progress, it increases and your bankroll should be at a minimum of With enough money to cover the amount of the bet on the table, prior to the outcome of the spin or at a live or online easier than poker. The hold is the average strategy when you play roulette wheel game on online casino, roulette betting strategies in a a massive margin. In some places the variant to the croupier most often actual casino, inform the croupier spun and places enough chips games such as poker, blackjack. You can try out either of the progressive strategies you a straight up on 8, in the casino on the bet is marked with a. If you intend using this series is Very popular in British casinos, tiers bets outnumber back to the first bet on that number. Here, the profit margin for on the site, and get. Today most casino odds are dealer counts the number of ways or stations that the winning number hits the complete. In the early frontier gambling set by law, and they complete bet on 17, the to 1 or 35 to. The best thing to do is to put into into considered gambling on credit and.What is progressive betting in roulette? Progressive betting strategies are based on increasing your wager after the end of each round. In some cases, players will raise the bet after winning, while in others they will do it when loosing. Why Progressive Betting Does Not Work? The reason why progressive betting systems including Martingale's don't work is because they entail. Roulette betting progression is when you vary the bet size after either win or loss. These types of roulette bets are only useful if you begin with a legitimately.