Martingale is an ancient betting strategy, older than football itself. It was originally practiced on tossing the coins, where each side has an equal chance of winning, matching ** Bernoulli process** (set of random variables, where each other has no correlation with previous or later, having same probability of ending with result 1(out of 2 options=0,1) – here 0.5).

The strategy forces gambler to double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. This is easily and nearly in same way applicable to red-black roulette problem (of course only nearly as number 0 loses every time and there is need to consider that as well).

So let’s imagine easy Martingale roulette game (warning – do not try this in real, casinos do not allow playing martingale or they are limiting maximum stake for players doing so):

Bet + amount | Result | Balance |

10$ on black | Red = loss | -10$ |

20$ on black | Zero = loss | -30$ |

40$ on black | Red = loss | -70$ |

80$ on black | Black = win | 10$ |

Sometimes player decides to repeat this again = this is called *Repeated martingale.*

A Bettor may ask the question – what is then the catch, this looks like sure money! Unfortunately this is not true, there is some small statistical risk of losing entire bank or ending with big loss frightened or stopped by casino.

## Math behind casino example

Imagine that your bank is 1000$.

**Question 1**: If my initial stake is 10$, after how many wrong tips I am bankrupted?

**Answer**: let be *s*=1+2+4+… then 10(*s*) <= 1000, so *s* <= 100 where *s* = 2^{n}-1.

From this we can use linear relaxation and have equation for a while as 2^{n}=101. Then obviously n = log(101) / log(2), thus **6.65**.

That means we cannot afford 7^{th} bet and we are ending with loss of 630 USD after 6 tosses with not pleasant outcome.

**Question 2:** What is the probability that I will lose like in first example?

**Answer: **If we are not playing roulette, but tossing a coin then: 1/2^{n} = **1.56%.** In American double-zero roulette, there is 10/19 probability of losing after single try (zero means no-win no matter if black/red was chosen), for this roulette, must be probability of bankrupting slightly higher. How big? **2.13%**

## Application to Football

In sports betting we can often see some bettors are trying to chase money by playing Martingale. Here bettors usually do not bet on even-priced events like red/black or toss a coin. Match after match the odds are varying, for instance we can use example of betting on **Hyde in 2014-2015** to win again (at that time English 6^{th} league club that was not doing well and ultimately they were relegated) .

Because sometimes odds can be 2, sometimes 4 or 7, we need to know how much to bet to get back the desired money.

Let me describe it on punter scenario when on beginning of 2014/2015 one might expect Hyde to do better that previous season in 5^{th} league from which they got relegated and in one tier below they were expected to do some better. Let’s imagine punter want to earn 10$ netto. So if odd is O and desired amount he want to earn is N, formula for calculating stake is **N/(O-1)**. If cumulating lost from before then N will contain sum of previous losses plus actual N for game that in our example is 10$.

**Game 1 – home against Oxford City – odd 2.05, stake 9.52. Result: LOSS (0:1). Balance: -9.52**

**Game 2 – away against North Ferriby – odd 4.5, stake 5.57. Result: LOSS (3:0). Balance: -15.09**

**Game 3 – away against Barrow – odd 8, stake 3.58. Result: LOSS (3:1). Balance: -18.67**

**Game 4 – home against Colwyn Bay – odd 3.4, stake 11.95. Result: LOSS (2:4). Balance: -30.62**

**Game 5 – away against Bradford PA – odd 5.25, stake 9.56. Result: LOSS (3:2). Balance: -40.18**

**Game 6 – home against Tamworth – odd 4.33, stake 15.07. Result: LOSS (2:2-draw). Balance: -55.25**

**Game 7 – away against Gloucester – odd 5, stake 16.31. Result: LOSS (1:1-draw). Balance: -71.56**

**Game 8 – home against Hednesford – odd 3.4, stake 33.98. Result: LOSS (0:1). Balance: -105.54**

Here the situation get serious…8 games without win and loss above 100$ can stop from betting nearly anyone. But let’s go on as situation progressed.

**Game 9 – away against Brackley – odd 2.4, stake 82.53. Result: LOSS (1:0). Balance: -188.87**

And after colossal loss again nearly everyone would stop.

**Game 10 – home against Lovestoft – odd 3.25, stake 88.39. Result: WIN (5:1). Balance: +10**

Aaaaah, finally in 10^{th} game they won! One may say “it was obvious that they will win finally”. My answer is – oh really, was it so pleasant to be close to minus 200 and risking another 88 to end with 10 bucks in positive numbers?

From my perspective this is the demonstration how Martingale should not be used for betting and if yes, then to have enough money, no limits for maximal stake and patience (a lot of patience) 🙂

And by the way, do you want to hear how story of Hyde continued?

Well: another 10 losses in row and finally in 21^{st} game 2^{nd} win on St.Nicholas Day…that is having after 4 months of betting hardly 20 bucks earned, but 10000 hair lost. I think no one wants to try that 🙂