Martingale trade sizing and the gambler's fallacy.

Over the years, forex has acquired such a bad reputation that there are books published sold with statements similar to the below:
"And you're surprised that we suggest gambling strategies for forex? But don't be, there is no way of knowing if a currency will go up or down, and the likelihood of guessing the next movement is as good as it is with guessing a coin toss. But don't be alarmed: coin tosses and similar gambling strategies have been perfected by experts for generations. In this book you'll find all you need on how to profit from trading forex with the only methods that work: betting strategies that experts use to beat each other."
Though such innovative approaches to forex are bound to bring a smile to our lips, and very short term movements in the forex market are indeed very difficult to predict, solving the uncertainties associated with the market with gambling methods is like throwing fuel on fire. You're unlikely to make much from such a brave, but ultimately futile effort.
Before looking at these strategies we must make an assumption that the outcome of each movement at a chosen time period (say 1-minute, 5-minute, 30 seconds) is independent of the preceding or following period. In other words, if there are three or four 5 minute periods (say P1,P2,P3,P4) and in one of them(say, P2) the price is up, this result has no bearing on the outcome of the other two 5 minute periods(P1,P3) which follow and precede it. The opposite of this assumption, which suggests that there’s memory in the trading process, that the outcomes influence each other, would change the nature of our trading entirely. If this were true,we could apply mathematical tools, such as the z-score which we will discuss later, to maximize profits and minimize losses.
Let us look at a number of these strategies:

Martingale

This strategy was first developed in 18th century France, and has its roots in the mathematical and scientific developments of the enlightenment era.
The martingale strategy involves increasing bet sizes with every losing coin toss. When the trader bets with amount x that a currency will go up at P1, and his bet fails, he will simply double the amount to 2x, and repeat his bet that the price will be up at P2. When the bet fails again, he will double the amount to 4x, and bet the price will be up at P3, and will go on with this until he wins what he wants, or he’s bankrupted as he receives a margin call.

Period	P1	P2	P3	P4
Outcome	Down	Down	Up	Down
Amount bet	1	2	2	4
Account Value	9	7	9	5

Now the point of this trade is simple. Even if the trade was a failure at P1, and a portion of original capital (amount x) was lost, a win with twice the original capital would cover the original losses, and register a profit. The same logic is valid at P3, P4, and so on.
In order to win, the martingale trader is making the assumption that short term trade results (coin tosses) are not independent of each other. That is, one coin toss, or one losing trade, is somehow influencing the outcome of the next trade in line, and making it less likely to be a loss as a result of mean regression.
Unfortunately for him, as we stated before, the outcome of coin tosses or trades at each period is independent, and there can be any large or small number of heads or tails (or a sequence of up P1, P2, P3, P4, P5) without constituting an anomaly. We will come back to this subject when discussing the gambler's fallacy.

Anti-Martingale

In the anti-martingale strategy, the trader does the opposite of what the martingale player does, but still reaches at the same outcome, because events are independent, and it's not possible to sustain a winning streak infinitely.
So what does he do? Instead of doubling on losing trades, he doubles on winning ones, and for instance, if he bets x at P1, and loses it at P2, he keeps betting x at P2, until he makes a profit at P3, when he doubles his bet to 2x, and goes on like that until the account is wiped out.
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Period	P1	P2	P3	P4
Outcome	Down	Down	Up	Down
Amount Bet	1	1	2	2
Account Value	9	8	9	7

It's of course very simple to see the problem with this method. What makes the trader double his risk on a winning trade at P3 if the result of P3 signals nothing about the winning potential at P4? And we have already made the statement that outcome at each period is independent of everything else.

Gambler's Fallacy

Gambler’s fallacy is the unjustified expectation that outcomes that constitute a rare series will regress to the mean in the future. Thus, the gambler (for example, the martingale strategist) believes that a streak of four losses at P1, P2, P3, and P4 implies a higher probability of success at P5, because of the deviation of the series from the mean. A simple example of this is the belief that four successive heads in a series of 5 tosses will make the last outcome being a tail likelier.
In fact, the probability of heads or tails will remain just ½ at every single toss if the coin tosser has an infinite amount of opportunities of repeating the toss. If the number of tosses allowed is limited however, the probability of getting a successful bet actually diminishes with every successive failure.
Note that the opposite is also false: This is the belief that in a random process (such as short-term fluctuations, or coin tosses) developments that occur one after another influence the outcome of the next to conform to the series, exemplified in the expectation that if a coin toss returns a tail at P1, P2, P3, P4, the result at P5 will also be a tail. Or, we can say using familiar terminology that in a micro trend (on a thirty second, five minute, or one minute chart) an up movement at P1, P2, P3 and so forth, will imply an up movement at P5.
The gambler’s fallacy is only valid in random processes where there’s no relationship of causality within members of a series, or in other words, successive outcomes are independent of each other. So, as in our previous example, if the coins were somehow shaped to increase the likelihood of tails at each successive throw (that is, if the coin tosser were cheating), the gambler would be justified in expecting tails eventually, and the outcomes would not be independent of each other. How do we judge if the streaks of wins and losses generated by our method are random or not (and independent of each other)? To decide on that we use the z-score, and the interested may read the related article on this website.