Introduction to Martingale Theory
Martingale Theory is a simple mathematical model which models sequence of a fair game [1]. It is a stochastic process on some probability space {Ω, F, P}. Originally it was used as a betting strategy during 18th century in France. In 1934, Paul Lévy introduced concept of Martingale in probability theory. It is named after ‘La Grande Martingale,’ which means strategy for even odd bets where bets are doubled every time we lose. Idea behind this betting strategy is tht one cannot expect gains without taking risks.
To understand Martingale, we need to understand Filtration first.
Filtration
Available information is modelled by a sub-σ-algebra F. A sequence of σ-algebras Fn such that F0 ⊂ F1 ⊂…⊂ F. This
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Every time the person lost he had to double his bet so that the first win would help him regain from the previous losses plus win profit equal to the original stake [2]. If X1, X2,…, is a sequence of independent and identically distributed random variables with P(Xn = 1) = 1/2 , P(Xn = -1)= ½. Filtration (F_n )_n Fn = σ(X1,…,Xn). Then sequence (s_n )_(n=1)^∞(simple random variable walk on Z) is martingale w.t.r. (F_n )_n as
E(Sn | Fn-1) = E(Sn-1 + Xn | Fn-1) = Sn-1 + E(Xn | Fn-1) = Sn-1 + E(Xn) = Sn-1 [1] Poyla’s Urn – A container has balls of two color, say red and blue (r and b). We pick out one ball, observe its color and place it back in the container along with another ball of the same color. Draw another ball from the same container. Probability of drawing red ball in first try is r/(r+b) and that of blue ball is b/(r+b). In second draw, probability of red ball being drawn if a red ball was drawn first time will be (r+1)/(r+b+1). Probability of red ball being drawn if a blue ball was drawn first time will be r/(r+b+x).
Probability of a red ball being drawn on second draw
P(Red : Draw =2) = ((r+1)/(r+b+1))*(r/(r+b))+((r/r+b+1)*(b/r+b))
= r/(r+b) [3] Example of Submartingale, Supermartingale : In a coin toss we have three coins. One unbiased and two biased. In the two biased coins, one has P(HEADS) = ¾ and the other has P(HEADS) = ¼. After tossing the coin if heads come, we win, tails, we lose. If we
Martin lived in a period of economic prosperity. Businesses were booming. However, one major problem of the time was the formation of
(b) E If n = 0, no radioactive coins found. If n = 1, test the coin. If n > 1, split the bag into two approximately equal bags. Try to find the radioactive coin in the first bag. If not found, try to find the
mix of calls from Exhibit 3. This calculation assumes that any time between calls is
The staking method is very effective and that's the main part of the system. The authors of this system improved martingale betting system in a new version that works. What that means? The system works because the punter increasing stakes after a losing day. It's a 4 game chase system where you double up each time when you lose.
want to bet, yet other players have to meet or raise the previous bet. However this advantage is
Gambling, whether for life or money, is risky and success is not guaranteed, even if you ‘win’. Two different tales both involving risk.
This means that in your stake, you will have to make a payment of seven times the value of your first unit stake. Take an example, when you intended to stake $1 on a Patent, your actual stake will be $7, one dollar each for all the seven bets that make up the bet. To win a Jackpot in a Patent Bet, you will have to win in all your three selections. Considering that you have a selection of 8/1 and they all end up trumps in Patent Bet, with a total combined stake of $7 your profit margin will roughly be around $ 1000. This high returns are as a result of winning the Three Single Stakes, Threes Double Bets, and the highly profitable Treble. For instance, landing a Treble given these example odds results in a total of over $700 in profits for your $1 stake. This explains the huge potential one is exposed to by taking a Full Coverage Multiple
Duration 20 15 10 5 20 30 15 20 5 15 30 15 25 15 5 15 Predecessor(s) None 3 4 4 5, 6 4 7, 9 10 11 12 5, 6 15 5, 6 7, 15 7, 15 13, 16, 17, 19FS 1 25 days, 20FS 1 15 days FS = Finish to Start lag
For example: the colors red and black in the roulette. For instance, if the ball keeps landing on a color black for the 3rd time in the roulette, then surely the next one will be red.
3) Evaluate the expected value for the discrete possibility variable. (1/18). Where x value begins from 1 to 6.
Pascal’s wager does a tremendous job in pitting the ideas of infinite gain and infinite loss against each other. Pascal’s wager goes as follows, if one accepts God and God exists, then a person will gain an infinite reward, but if God does not exist one will suffer some finite loses or if one does not accept God and God exists, then a person will suffer infinite loss, but if God does not exist one will gain some finite rewards. This argument can be expressed mathematically for the acceptance of God as well, if the probability of God existing is any number greater than zero, then the infinite multiplier from God existing would result in the utility being infinitely positive. The other aspect for if God does not exist, would result in the multiplication of the minute probability of God existing,
Prediction involves making a statement concerning the likely value of an event or action uncertain or unknown at the time of the statement. Since the theory of probability, (inaugurated by the French mathematicians Blaise Pascal and Pierre Fermat in 1654), was developed to quantify uncertain events in terms of their likelihood of occurrence, formal prediction is now viewed as a mathematical topic involving probabilistic modeling. Indeed, the mathematician Karl Pearson said in 1907 that the fundamental problem in statistics is prediction. Prediction, however, is usually not an end goal itself, but rather means to put probabilistic bounds on the relative frequency or likelihood of occurrence of future uncertain
The player should hold off from bets when the running count is below 0. It is safe that way.
The very first thing you should know about this interesting system is that the development it uses is determined by what size your initial guess is. Presuming you guess ?1 to start out with it appears the following: 1,3,5,7,9
In a paper I also discuss some ideas for what became the rational expectations approach is more prevalent in many disciplines in recent years. One idea is that the theory of rational expectations tends to provide opportunities for confirmation of a novel approach. I would argue that these opportunities are mainly a result of the nature of this mathematical approach.