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The nature of the Martingale strategy

Mathematics in Martingale strategyThe classic Martingale system is one of the most widespread of casino betting systems in modern days', but its founding roots are French from the mid 18th century. The master plan was adopted from a simple coin toss game, where the risk-taker wins the bet when a coin falls heads up and loses the bet when the coin falls tails up. According to the strategy, the action that follows is: the gambler doubles the bet if there is a loss, and lowers to the initial stake after a win so that his very first win would cover all prior losses plus win a profit equivalent to the original stake. 

This strategy can be implemented broadly in different games at all casinos, mainly in roulette, where the prospect of hitting either red or black is close to 50% ( in the 5th paragraph of this article you will find a chart with the exact percentage). 

Mathematics in Martingale system

Like every bother strategy, The Martingale strategy also has a catch: in the chase of your personal “bonanza”, you face the risk of losing not only part but possibly your total capital. (Be aware of the feasible negative mathematical expectation at a distance as well). 

With this feature, we will try to examine the mathematical aspects of the system, as well to assess the pros and cons of using it. 

Roulette example using Martingale

Minimum bet 10 CAD. Permitted betting 1:1. Single bet. You check your luck on red.

  • If the ball falls on red - win, you should return to your initial bet 10 CAD!
  • If the ball falls on black - lose, you should double whatever your last bet was!
  • With the two charts below we are showing you

10 consecutive wins

Roulette Bet with Round Win Clean profit
Round 1 10 CAD 20 CAD 10 CAD
Round 2 10 CAD 20 CAD 10 CAD
Round 3 10 CAD 20 CAD 10 CAD
Round 4 10 CAD 20 CAD 10 CAD
Round 5 10 CAD 20 CAD 10 CAD
Round 6 10 CAD 20 CAD 10 CAD
Round 7 10 CAD 20 CAD 10 CAD
Round 8 10 CAD 20 CAD 10 CAD
Round 9 10 CAD 20 CAD 10 CAD
Round 10 10 CAD 20 CAD 10 CAD

10 consecutive losses

Roulette Bet with Round lost Clean lost
Round 1 10 CAD 20 CAD 10 CAD
Round 2 20 CAD 20 CAD 30 CAD
Round 3 40 CAD 40 CAD 70 CAD
Round 4 80 CAD 80 CAD 150 CAD
Round 5 160 CAD 160 CAD 310 CAD
Round 6 320 CAD 320 CAD 630 CAD
Round 7 640 CAD 640 CAD 1270 CAD
Round 8 1280 CAD 1280 CAD  2550 CAD
Round 9 2560 CAD 2560 CAD 5110 CAD
Round 10 5120 CAD 5120 CAD 10230 CAD

 

10 consecutive wins are easy to comprehend: 100 CAD clean profit;

10 consecutive loses however show some drawbacks, the most alarming of which is:

  • The big bankroll that you need to prepare before even enter the game, and
  • The possibility that next bet might be restricted by pre-settled table limit - (for instance in the second chart at round 10)
  • The Martingale strategy requires you to make a bet of 5120 CAD, but the casino pre-arranged the limit to be - 5000 CAD, so you cannot complete this system round.
  • Play European roulette

Calculating the mathematical expectation

Let’s go back to the game of flipping coin. The probability of falling tails or falling heads is 0.5 (50% - half of 1). We choose tails as a win.  According to The Martingale system, when heads are dropped, we lose, so we need to double our bet. Let’s assume that we have a roughly unlimited budget in a series of "N" bets on the tails. In this example, the losing risk will be 0.5 by the power of “N”. The calculation for the winning is 1 - (0.5 at the power of “N”).

In numbers that will say:
  • Initial bet 1CAD. 
  • Our budget then is put together as “N”=10 bets, by doubling every round = total sum of 1023 CAD.

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 = 1023

  • So, with 10 heads in a row (remember – we choose tails to win), the whole fund of 1023 CAD could be gone.
  • In varied ways, the ten tosses could fallout differently: not all heads, but all tails, or half coins tails, and half coins heads, etc. The full count of combinations is 1024. 
  • So in order to double our money we have to use Martingale: 1023 TIMES ("1023 CAD / 1 CAD").
  • In fact, every Martingale outcome chance in this example equals “1/1024”.
  • I.e. the probability to lose our capital with the worst-case scenario of falling 10 heads after another is also “1/1024”.
  • After the 10 toss-ups the winning chance, i.e. getting any other combination of outcomes (except for ten consecutive heads), is calculated as “1 - 1/1024 = 1023/1024”.
  • Therefore, the ratio likelihood of total budget loss or winning equals “1/1024”, divided by “(1 - 1/1024) = 1/1023”.

With the figures above you see, applying The Martingale system each winning round is practically giving you 1CAD, while the whole capital of 1023 CAD is in constant risk-exposure status.

If you play toss-up a countless number of times, it is fair to say that every 1024th time, you are going to lose all the winnings of the previous 1023 tosses. In other words, the mathematical expectation will be flat zero.

Some great mathematicians or analysts are striving to show that The Martingale strategy is working with higher than zero outcomes. Yes, but that subject is proven only hypothetically. After all, in reality, infinite series, funds, and unlimited bets do not exist

Is the Martingale System a fraudulent system?

Hustlers’ repeatedly consider the Martingale system as an ideal chance of guaranteed winnings at the online casinos. You can find different sites and videos, where it is stated that using Martingale can bring sizeable earnings of 100 CAD, 200 CAD per hour, and more. 

Bear in mind: all the small frequent winnings could be equal to one significant loss at a distance. But because of the zero in Roulette or dealer's Blackjack, the strategy will lead to negative for the player and positive for the casino. When betting on a color the expected value is 1*18/37 + -1*19/37 = -0.027. The house has a 2.7% edge. This means that on average a player will lose 2.7% when betting.

Probabilities

Martingale strategy: Pros and Cons

Pros:

  • The system lets you get minor winnings, and the loss may be massive but seldomly happens. (The chart at the beginning of this article with 10 consecutive lost rounds is extreme.) 
  • Some high rollers summarize that the chance of 6 or more consecutive rounds of bad luck is very improbable and that with tenacious sticking to the strategy, they will gradually build one solid bankroll.
  • There is a safe middle ground, where one can see the effectiveness of using Martingale. 
  • That is only betting on even-money outside bets – 1-18, 19-36, Red, Black, Even, and Odd. For they have the maximum chance of winning (near 50% odds), but the lowest payout of all – 1:1, cause you win the same size of money as your bet for one spin. All in all, in a game of roulette placing those bets, is the most secure game to play.

Cons:

  • No one has limitless bankroll, so keeping doubling the bet indefinitely it is unmanageable.
  • Basic mathematic: Just look at the betting progression. Yes, you get a lot of winning sessions - and that is fine. But the profit on them is really small. On the flip side, only very few losing sessions can bring you entire budget loss. Odds: less than 50:50. Even betting on Red or Black at roulette is slightly worse than 50%, because both colours lose when the ball lands on green zero.
  • To top it all, most roulette tables have betting control, i.e. you are extra burdened by their bet limits. You can expect to see your game obstructed by these limitations during some long losing series – thus not being able to double your betting just one more time, and win back the money already lost.

To try the Martingale strategy, or not to try?

Many players think that it is worth experimenting with this strategy. It can bring you success no doubt - but for a certain time. In the long term, because of its sharp progression, the strategy becomes extremely tricky. Eventually, every roulette player is having one or another Martingale episode –usually a dark one. Don’t forget: with one extended unlucky game, you can lose your whole deposit. So, some consider it speculative, nerve breaking, or even catastrophic.  Others, find this system exciting, thinking of personally modifying it, or even using it in other fields with betting (sports, market, etc.)

A group of experienced players describes it like jumping willingly in an endless money pit, and while you're falling you are praying that luck saves you, while you are completely out of control of the situation. Is it better not to jump at all? You decide for yourself. Hereby you have been warned!

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