In order to be successful, forex traders need to know the basic mathematics of probability. Many traders use a combination of black box indicators to develop and implement trading rules. Probability and statistics are the key to developing, testing and profiting from forex trading.

The most basic tool of probability in forex trading is the concept of normal distribution. This is the sort of distribution that would result from artificially spreading objects as evenly as possible across an area, with a uniform amount of spacing between them. Normal distribution offers forex traders predictive power regarding the likelihood that a currency-pair price will reach a certain level during a certain time frame.

Computers use a random-number generator to calculate the means averages of forex prices in order to determine their normal distribution.

If a large number of sample prices are checked, the normal distribution will form the shape of a bell curve when plotted graphically. The greater the number of samples, the smoother the curve will be.

The rules of simple averages are helpful to traders, yet the rules of normal distribution offer more useful predictive power. Yet, the normal distribution can also tell the trader the likelihood that a certain daily price move will fall between 30 and 50 pips, or between 50 and 70 pips. Finally, there is a Normal distribution and standard deviation functions in expert advisors EA and trading systems help forex traders assess the probability that prices may move a certain amount during a given period of time.

Yet, traders should be cautious when using the concept of normal distribution alone for purposes of risk management. When modelling normal distribution curves, the amount and quality of input price data is very important.

So, for testing a forex-trading strategy by estimating the results from sample trades, the system developer must analyze at least 30 trades in order to reach statistically-reliable conclusions regarding the parameters being tested.

Likewise, the results from a study of trades are more reliable than those from an analysis of only 50 trades. For forex traders, the most important characteristics of a distribution are its mathematical expectation and dispersion.

Mathematical expectation for a series of trades is easy to calculate: Just add up all the trade results and divide that amount by the number of trades. If the trading system is profitable, then the mathematical expectation is positive. If the mathematical expectation is negative, the system is losing on average.

The relative steepness or flatness of the distribution curve is shown by measuring the spread or dispersion of price values within the area of mathematical expectation. Typically, the mathematical expectation for any randomly-distributed value is described as M X. Dispersion and standard deviation are critically important for risk management in forex trading systems. The higher the value of the standard deviation, the higher will be the potential drawdown, and the higher the risk. Likewise, the lower the value for standard deviation, the lower will be the drawdown while trading the system.

However, the standard deviation is high, so in order to earn each dollar the trader is risking a much larger amount; this system carries significant risk. This is the mean value M X for all the trades. Thus far, the system looks promising. The sum is divided by 29, which is the total number of trades minus 1. The same calculation is performed for each trade in the test series.

In this example, the dispersion over the series equals 9, Thus the forex trader sees that the risk for this particular system is fairly high: This risk may be acceptable, or the trader may choose to modify the system in search of lower risk. Beyond the riskiness of a particular trading system, forex traders can also use normal distribution and standard deviation to calculate the Z-score, which indicates how often profitable trades will occur in relation to losing trades.

Some traders may assume that the system will win over time, as long as there is an average of at least one profitable trade for each four losing trades. Yet, depending upon the distribution of wins and losses, during real-world trading this system may draw down too deeply to recover in time for the next winner. A positive Z-score represents a value above the mean, and a negative Z-score represents a value below the mean. To obtain this value, the trader subtracts the population mean from an individual raw value then divides the difference by the population standard deviation.

Z represents the distance between the population mean and the raw score, expressed in units of the standard deviation. So, for a forex trading system:. N is the total number of trades during a series; R is the total number of series of winning and losing trades; P equals 2 x W x L W is the total number of winning trades during a series L is the total number of losing trades during a series. R counts the number of such series. Z can offer an assessment of whether a forex trading system is operating on-target, or how far off-target it may be.

Just as importantly, a trader can use Z-score to determine whether a trading system contains fewer or greater series of winners and losers than expected from a random sequence of trades— In other words, whether the outcomes of consecutive trades are dependent upon each other. If the Z-score is near 0, then the distribution of trade results is near the normal distribution.

The score of a sequence of trades may indicate a dependency between the results of those trades. Whether the Z value is positive or negative will inform the trader about the type of dependence: A positive Z value indicates that the profitable trade will be followed by a loser.

And, positive Z indicates that the profitable trade will be followed by another profitable one, and a loser will be followed by another loss. This observed dependency lets the forex trader vary the position sizes for individual trades in order to help manage risk.

The Sharpe Ratio, or reward-to-variability ratio, is one of the most valuable probability tools for forex traders. As with the methods described above, it relies on applying the concepts of normal distribution and standard deviation.

It gives traders a method to check the performance of a trading system by adjusting for risk. Likewise, HPR can be calculated by dividing the after-trade balance amount by the before-trade amount. AHPR by itself produces an arithmetic average which may not properly estimate the performance of a forex trading system over time.

The concepts of normal distribution, dispersion, Z-score and Sharpe Ratio are already incorporated into the logarithms of EAs and mechanical trading systems, and their usefulness is invisible to most traders.

Yet, by knowing how these basic probability tools work, forex traders can have a deeper understanding of how automated systems perform their functions, and thereby enhance the probability of winning trades.

Normal distribution The most basic tool of probability in forex trading is the concept of normal distribution. Reliability of analysis depends on quantity and quality of data When modelling normal distribution curves, the amount and quality of input price data is very important.

Dispersion and mathematical expectation to estimate risk For forex traders, the most important characteristics of a distribution are its mathematical expectation and dispersion.

For example, below is a sample risk assessment for a test of a forex trading system: Z-score Beyond the riskiness of a particular trading system, forex traders can also use normal distribution and standard deviation to calculate the Z-score, which indicates how often profitable trades will occur in relation to losing trades. The basic standard score calculation for a raw score designated as x is: So, for a forex trading system: Sharpe Ratio The Sharpe Ratio, or reward-to-variability ratio, is one of the most valuable probability tools for forex traders.

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