**Probability play**

Trade win-loss statistics can be analyzed using ideas from binomial trails and distributions, similar to the way we count heads or tails when flipping a coin. A fair coin will have a 50% chance of a head or tail with any one toss, while trading typically will have an uneven win-loss percentage averaged over a series of trades. For example, our sample fade strategy has an approximate 60% winning percentage based on its performance history, meaning we expect to see six of 10 trades closed for a profit. We consider a trade strategy executed over a number of trades similar to counting the number of heads or tails realized over a number of tosses.

The well-known counting formula is:

The above equation gives the probability of realizing exactly r successes in N trials, where P is the probability of success of any one trial. For example, the probability of two heads (heads = success) from four tosses of a coin, where p equals 50% or ½, is:

This kind of counting formula is a practical tool for all systematic traders, and there are many varieties that can be applied to trading. However, when the goal is to determine the number and length of winning or losing trade runs over a trading period, a closed-form solution, or formula like the above, is difficult to determine. In this case, computer simulation can be used to model a large number of trades and count the wins and losses.

“Basic statistics of runs” (below) gives the average number and probability of various run lengths over 250 successive trades, using per-trade winning percentages of 30%, 50% and 70%. The 250 figure assumes one trade per day in a 250-trading-day year. The three winning percentages offer a range of trade execution, where 30% might represent conservative swing trading and 70% aggressive scalping. The 50% rate represents even win-loss trading similar to flipping a coin. The data were generated from 1 million simulations of the 250 trades using software random number generation.

The basic run statistics data show, for example, that over 250 trades, and with a per-trade win-loss percentage of 50%, the trader must endure six losses in a row 62% of the time. If we assume swing trading with a lower per-trade winning percentage of 30%, then the trader must endure eight losing trades in a row 73% of the time. About 98% of the time (really in every case), there will be five losses in a row and, on average, a five-loss run will occur 3.7, or almost four, times over 250 trades. The active trader can study the data here to become familiar with basic run-length statistics.

“Run distribution” (below) illustrates the probability distributions for the three per-trade winning percentages. As an example, it is informative to draw a cut-line at the 50% level to see the expected run lengths that will occur at least half the time when trading at a given per-trade winning percentage rate. (Contact the author if you would like a copy of the Windows/C++ program that generated the data.)