# Martingales

A bet is by definition risky : you bet and you can lose your bet or win, and earn a profit proportional to the bet placed. We are talking about

**ONE**bet here.A martingale is a **betting strategy upon several moves**, taking each time into account the previous results.
Matingales are a kind of staking plan.

The main goal is of course to increase its (chances of) gain. The ultimate goal is to win every time.

## Example of an ineffective martingale

To win the lottery, some say that it is wise to play numbers that have not come up much recently: as the probability of each number coming up is the same, numbers that have not come up very often "will have to catch up" and therefore in the days to come their expected probability will therefore be greater.This reasoning is erroneous and false. We can adapt it to a simpler case: heads or tails, with a balanced coin and the following martingale: I systematically wait until three successive Tails draws have taken place, after which I bet on Heads. (Of course I can change these three draws, by 10, or 100, draws). Since Heads and Tails have the same probability, they must come up on average equally often, and so after a certain number of consecutive heads tossed, the toss of heads becomes more likely. This reasoning is false, and this martingale (the game strategy) absolutely does not change the mathematical expectation: on average my gain will be zero.

## Classic (or doubling) martingale

Actually, previous example is quite general: in a game of chance, using a martingale does not change the probabilities and the mathematical expectation.You have to adapt the bets for this.

The classical martingale (or doubling martingale) is a betting strategy for a game where you bet and, if you win, you double your bet (odds = 2), and you totally lose your stake when you lose.

I bet at the very beginning for example 1 euro, then at each game:

- if I win, I am just happy, and I stop there, the martingale is over
- if I lose, I bet in the next round double stakes of the previous round.

- I bet one euro: if I win, I earn 2 euros, that is 1 euro net profit.
- if I lose, I then bet two euros in the next round, so 3 euros in total from the beginning:
- if I now win, I earn 2×2 = 4 euros, so 1 euro net profit.
- if I lose, I then bet 4 euros in the next round, so 7 euros total bet from the beginning:
- if I win, I earn 4×2 = 8 euros, so always 1 euro net profit.
- if I lose, I now bet 8 euros, ...

**this gain is assured**. In other words, with this strategy, I definitely win 1 euro.

Unfortunately, this strategy is a dangerous one because my total bet increases very quickly (after 10 bets for example, I will have invest total bets of more than 1000 euros...), and if I stop before having postponed my euro of net profit, I ultimately simply lost all my investment.

Full mathematical details can be found for example on this page and where we see (after complicated mathematical calculations of probabilities) that the sum to invest to respect our martingale and finally win can not only quickly become astronomical, but also that the probability of this happening is not so low.

## Building a winning martingale

We must be clear, a martingale, like any bet, any strategy, involving chance**remains risky**!

However, the classic martingale can be adapted to make it effective.

Two parameters make the classic martingale particularly risky:

- The gain is "only" doubled in the event of a victory, which requires you to bet increasingly larger to compensate for the losses from previous defeats and therefore quickly increases the total stake.
- the probability of winning is low, one chance in two here

Here are several possibilities as examples: