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Suppose that someone applies the martingale betting system at an American roulette table, with 0 and 00 values; a bet on either red or black will win 18 times out of each 38. If the player's initial bankroll is $160 and the betting unit is $10, the player will make a win in approximately 96% of sessions, gaining an average of $4.30 from each winning session. In the remaining 4% of sessions, the player will "bust", exhausting his bankroll, for a loss of $160. It follows then that the average session losses of a gambler employing this strategy are $2.27. Given a larger bankroll, the odds of making a win before running out of cash increase; however, the average winnings grow more slowly than the average losses, so the game remains a losing proposition. Modern casinos generally have table minimums and maximums to prevent players from doubling their bets more than five or six times, rendering the martingale system obsolete.
The betting strategy seems intuitively to work. We think of long streaks as impossible, and they are the only thing that could actually bankrupt the betting party. However, they're not actually impossible, just unlikely. One can easily demonstrate that they are possible enough to keep the balance at the casino. Let's say that a very large number of people are each gambling $1 on the flip of a coin in their hand. About half will win, and about half will lose, and thus the casino will pull in as much as it puts out. For the next flip, it branches off and half bet $1, half bet $2, in line with the strategy. Remember that previous flips won't affect future flips, so half of the people betting $1 will win and half will lose. Half of the people betting $2 will win and half will lose. The casino will take in as much as it loses, again. This simply continues to branch out. Half betting $4 win on the next round, half lose. Even out at round 25, when there are people betting $33,554,432 the casino will not become off-balance for winnings and losings, because half of the people betting that much will win and half will lose. Let's now suppose that the game being played results in 51% losing and 49% winning. This branches in the same way, except the casino will always take in 2% of the total money gambled. Group them by how much they bet, and we find that in each of those categories the casino makes 2%, so in total they will make 2%. Interestingly, the casino would make less money per round if everyone continued to bet $1 each time, instead of increasing the betting pool -- but each individual would play for a greater number of rounds before quitting or going bankrupt. The casino still takes in the same 2% of the total money gambled.
One activity where money management based on an Anti-Martingale approach has a recognized value [1] is speculation and trading. Many markets have some cyclical component to them, and the approach of an individual speculator or trader may only be appropriate for one portion of that cycle. Using an anti-martingale risk management scheme will increase profits during time periods when a trading approach is working well, while automatically decreasing exposure during portions of the cycle where trading is unprofitable. This is believed to decrease the risk of ruin for trading.
These are common misunderstandings that arise in everyday reasoning about probabilities, many of which have been studied in great detail. Many people lose money while gambling due to their erroneous belief in this fallacy. Put simply, the chances of something happening the next time are not necessarily related to what has already happened, especially in many gambling games. This is known in probability theory as the memoryless property.
The gambler's fallacy can be illustrated by considering the repeated toss of a coin. With a fair coin the chances of getting heads are exactly 0.5 (one in two). The chances of it coming up heads twice in a row are 0.5*0.5=0.25 (one in four). The probability of three heads in a row is 0.5*0.5*0.5= 0.125 (one in eight) and so on. Now suppose that we have just tossed four heads in a row. A believer in the gambler's fallacy might say, "If the next coin flipped were to come up heads, it would generate a run of five successive heads. The probability of a run of five successive heads is (1 / 2)5 = 1 / 32; therefore, the next coin flipped only has a 1 in 32 chance of coming up heads." This is the fallacious step in the argument. If the coin is fair, then by definition the probability of tails must always be 0.5, never more (or less), and the probability of heads must always be 0.5, never less (or more). While a run of five heads is only 1 in 32 (0.03125), it is 1 in 32 before the coin is first tossed. After the first four tosses the results are no longer unknown, so they don't count. The probability of five consecutive heads is the same as four successive heads followed by one tails. Tails is no more likely. In fact, the calculation of the 1 in 32 probability relied on the assumption that heads and tails are equally likely at every step. Each of the two possible outcomes has equal probability no matter how many times the coin has been flipped previously and no matter what the result. Reasoning that it is more likely that the next toss will be a tail than a head due to the past tosses is the fallacy. The fallacy is the idea that a run of luck in the past somehow influences the odds of a bet in the future. As an example, the popular doubling strategy (start with $1, if you lose, bet $2, then $4 etc., until you win) does not work; see Martingale (betting system). Situations like these are investigated in the mathematical theory of random walks. This and similar strategies either trade many small wins for a few huge losses (as in this case) or vice versa. With an infinite amount of working capital, one would come out ahead using this strategy; as it stands, one is better off betting a constant amount if only because it makes it easier to estimate how much one stands to lose in an hour or day of play. A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an airplane, a man decides to always bring a bomb with him. "The chances of an airplane having a bomb on it are very small," he reasons, "and certainly the chances of having two are almost none!" - this joke was also written into Blackadder Goes Forth - when Baldrick is carving his name into a bullet, Edmund Blackadder asks why and is told "because if I have the bullet with my name on it, I can't get shot with it!". Some claim that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic.
1) Bet 1 unit and lose: -1
unit
So, first of all I'd like to warn you that all the calculations are based on odd 1.91 and the winning % should be not less than 52.38% to make profit. There are many systems (Matingale, Kelly etc.) to get the sum you might put in particular bet, but they're still about making the winning percent higher and not about reaching the loseless poing. So we consider flat-betting (each bet 5% from bank) as the most logical system. The most unlogical systems proposes to raise or lower bets when you're on a such called "winning/loosing row". That's because if you're on a winning streak, and logicaly will win, then you must bet all the money on such an event , and if you're on a loosing streak then why to put money on a loose. Anyway looseless is much more important that winning/loosing streaks. If your winning percent is about 56% (for odds 1.91) then after making >200 bets 17% of total time you will be winning less that 50% of all bets. In other words even if you win 50% of all then after making 200 bets of 5% each you will be loosing 45% of total bank (200*5%-100*5%*1.91=45%). So, Miller says that bets higher than 2% of total bank will probably lead to financial disaster. You may wonder how to make profit winning total of 56% bets (this % was mentioned above), but don't forget about finance turnover, i.e. Miller makes 1000 bets a year, 1% from total bank each, in a year it's 1%*1000=1000%. So the bank turns over 10 times. Consider the winning 56%, we have a 7.6$ from every invested 100$. And that's a profit of 76% a year. Let's leave the sum of each bet and the overal amouth of bank for the next article. And now in general. Miller makes every bet of 1% from initial bank (consider it 10000$) till time it raised on 25%. After that the amouth of money put in a single bet is recalculated according to the the bank sum. So, if initially 1%=100$, after recalculating it will be 12500*1%=125$. Then raising another 25% another recalculation. Such a financial method will make profit of 100% from initial bank if winning percentage = 56%.
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