# Martingale Probability Theory

Martingale probability began as a popular betting theory in 18th century France. The basic premise of the theory was simple enough: In a game of coin flips that pays 2:1 if the coin comes up heads, but takes the bet money if the coin comes up tails, you should bet double on every loss so that you would automatically win back any losses.

## Problems with the Initial Model

• Clearly, the game assumes that the player has no limit on financial resources or time. In a practical setting, this game does not work, because as the player bets on each subsequent iteration, he exponentially reach poverty. Although the game does break even over a long enough time line, there is no way to be certain that this will happen quickly enough for the player to adequately recover his losses. However, the idea led to several other theories.

## Proof Against Betting Theories

• Paul Peiree Levy did much of the work toward proving that successful betting theories were impossible to create. The idea was to illustrate that betting games, in general, are fools’ games. There is no way to create a theory that will allow the player to win a majority of the time. Before his work in fields like Martingale Probability, it was not commonly accepted that gambling was essentially stacked against the player.

## Exponential Nature of Losses

• The main interest that mathematicians still have in Martingale Probability is the exponential rate of loss. The idea that can be inferred from the equations that define a Martingale set is that the expected value of the next number in a set of observations can be assumed to be equal to the last observation in the set. In other words, in a fair game, a gambler can assume his losses will be roughly between plus or minus the square root of the number of steps.

## Polya’s Urn Model

• George Polya came up with an example to explain this concept using a jar (or urn) containing red and blue marbles. The urn randomly and unbiasedly expels a marble of a given color. That marble is put back into the jar with another marble of the same color, which essentially has the same mathematical model as doubling down the gambler’s bet on any given game. The problem is that it has the false illusion of affecting the outcome.
• SOURCE:http://www.ehow.com/info_8577327_martingale-probability-theory.html