providing tick data, as is Germany’s DTB, Singapore’s SIMEX, and many other exchanges.The form of the data provided is very similar, but all of them are incompatible with thetrading strategy software and’ require what is called data reduction. Because theexchanges have taken the policy that providing data facilitates trading on their markets,there is usually no cost for it; this gives you some latitude to pay a programmer to write aprogram to convert this into your own format.
Cieaning the Data - -
Occasionally, you will find that your strategy was immensely profitable, or lost muchmore than you expected. When you test 20 years of data, you may not look at each tradeUSE AND ABUSE Or fHE COMPUTER 557
 
but accept the results as correct and rationalize the outcome as a sequence of bad orgood trades. You might be surprised to learn that there can be many erroneous prices inyour data, It is an incredible waste of time to use data without first reviewing it to finderrors.
Tick data is notorious for including bad data with the good, simply because of themagnitude of the problem. Most daily data is already clean. The easiest approach tochecking your data is to display it as a chart and scan through all of the markets that youplan to use. Bad data points are obvious once seen on a screen. A low price of zero willshow up as a vertical line going to the bottom of the chart; this will result in the correctdata points being scaled down to look as though they are just a horizontal line. You canthen find the date of this point and correct it (provided your software has a correctionfeature),
If you can program the computer yourself, you will want to scan the data looking foroutliers, those data points that are at least 10% away from the previous data points. Thedata provider should have prescanned the data to be sure that there are no intraday pricesthat were outside the end-of-day high and low prices transmitted by the exchanges, but thisis not usually the case, and you will find that the most obvious errors have been over-looked. There are more subtle errors that can be found and corrected, but the largest onesmust be fixed before you use the data. For testing, even one error per year is enough tomake all of the tests useless. Don’t forget that, if you remove a price instead of correctingit, you must reduce the tick volume as well.
Data Updates
Iris not difficult to remember when all traders kept a copy of the Wall Street Journal andclipped the price pages so that data could be entered into their computer. These pages,kept in a file drawer, became their historic database and absolute reference. This doesn'tmake sense any more, Automatic end-of-day updates are available to anyone at very littlecost. They provide complete and accurate data as well as correcting any past data thatmight have been wrong. Some exchanges post daily prices on their Web sites and can beretrieved for free, but this requires added effort to convert that data into a form acceptableby your database. Most data providers will accommodate a number of different formats sothat data received each day simply gets added to your historic data and is immediatelyready to use. Don’t do it manually. Your time is best spent finding a profitable trading strat-egy rather than fiddling with data.
Archiving Data
Not only must you have a complete database of accurate, historic data, but you must keepthat data current. Even if you have a live data feed, the data management of your systemcannot store an unlimited amount of ticks or days before it begins to discard the oldest val-ues. Most software will have a specific number of data points that it will hold for intradaydata, and much more for daily data. For example, it is common to have only 3 months oftick or 15-minute data, or the computer may hold 4,000 entries each of 1-minute,5-minute, 15-minute, and daily data, which mean about 16 years of daily data but only 4months of 15-minute data. This design may be important when you choose the system youuse. By being aware of how much data is held, you will need to remember to archive the15-minute data before each 4-month period elapses; otherwise, you will lose the data andyour history will not be continuous.
Combining Standard Techniques into a SystemStrategy development programs provide a wide assortment of standard techniques in theform of indicators and systems. You can pick a stochastic oscillator, a moving average, or558 1 i CTICAL CONSIDERATIONS
 
an N-day breakout and combine them in various ways to evaluate markets over differenttime horizons, such as a'10-day and 40-day smoothing, or even vary the frequency of datausing 30-minute, 1-hour, and daily bars. In most cases, you can write your own formulas tocreate customized indicators. If you experiment with these software features, you mightfind that they provide the ability to create a trading strategy that matches the time frameand risk profile that you want. For example, a long-term trend can be found with a stan-dard moving average, weighted average, exponential smoothing, N-day breakout, or linearregression. All of these techniques are already available. You may want to combine thatwith a band to avoid whipsaws. That is also easily done. If you want to alter your timing sothat you enter on a pullback after a trend signal, then you can add a stochastic or RSI toyour rules. Before you begin specialized programming, there is a lot to be learned by usingthese standard tools. It is not at all clear that you can create a trend that is any better thanthe ones provided in your system. It may be that the choice of trend period is much moreimportant than the trend technique.
Choosing Hardware
In an era of remarkable advances in technology, the hardware costs no longer seem an obsta-cle. As each new, faster chip is introduced, the previous model computers are reduced inprice. This process seems to be happening at least once each year. The cost of a computerwith a maximum configuration is small compared with your objectives, and not much greaterthan one trading loss. Because hardware has a shorter shelf life than only a few years ago,buying the biggest and best equipment is the only way to extend the life of your purchase.
The process of testing your strategies by optimization is a computationally intensivemethod for the computer. You'll want to have the fastest processor. In addition, testingmay still take one or more uninterrupted hours of data crunching; therefore, you’ll wantto have a large RAM to allow that process to go on in the background while you use thecomputer to do some other tasks. Institutional users will find it more ‘practical to have oneor more machines dedicated to testing. This will allow the calculations to continue at a pre-dictable pace, rather than being interrupted by demands from other programs.
Programming the System
Computer programming talent is no longer rare, but in high demand. In addition, under-standing a trading strategy is a special talent that requires training. Fortunately, the mostarcane parts of system development are already part of any trading development softwareyou buy; therefore, it is only necessary to code the actual strategy, rather than how toretrieve and store prices from the exchange data feed.
The prevailing philosophy about software development is to try to use as much third-party software as possible. If you need to see the results of a TradeStation optimization,save the results to a spreadsheet, then sort, organize, analyze, and graph the results usingyour favorite spreadsheet. You can compare one set of tests with another if you save themall in the same format. For the more popular platforms, such as TradeStation and Meta-Stock, there are a number of user’s groups, support groups, and newsletters that can be ofgreat help. Some companies provide alternate statisti€al measurements for your tests,while others have an add-in portfolio program. It is a good idea to check to see if these ser-vices satisfy your needs before attacking the programming yourself.
Selecting a Programmer
If you are not going to program the computer yourself, try to find someone with experiencein a language similar to the one you'll be using. It would be helpful if the programmer hadsome knowledge of trading, which might avoid errors caused by misunderstanding con-USE AND ABUSE © »HE COMPUTER 559
 
cepts, but that is not likely to happen. Given a choice, a scientific background is better thanbusiness programming experience.
Getting a programmer to say how much it will cost or how long it will take to solve aproblem is going to be difficult—they just don’t know. Programmers are incurable opti-mists, always thinking that the problem will be done at any moment. Even competent man-agers of programmers have difficulty in scheduling a project, based on the uncertainty ofthe debugging, or checking-out process. If it is necessary to contract the work or hire a pro-grammer, try to get a fixed-price quote and avoid the problems that will occur later.
Define the Steps
To keep the work on schedule, you will need to know what you want to do in advance. Youcan take a chart and mark the places where you believe the system will buy and sell. It isimportant that you have an actual example of what is expected so that you can verify theprogrammed results against the example.
Write down all the rules in outline form. You must be able to be precise about everyrule. The computer cannot deal with ambiguities. It is very likely that a programmer canwork directly from your rules to create the final strategy. This is also an opportunity to besure that your mules are clear and complete. Using your rules, select a price series and markthe trading signals that will be used to verify the program output. Check the results care-fully. If one trade doesn’t match your expectations, don’t stop until you find the reason.
Looking into the Past
We have already thoroughly discussed the advantages of using as much data as possible—the results reflect a broader variety of market conditions and are, therefore, more repre-sentative. There is often a concern that older data does not represent current conditions.While there may be some truth ‘in that, there is usually more to be gained with a system thatcan survive those years and still perform well in current markets. Unless there has been astructural change that makes the old data invalid, you should always favor longer tests andmore data. Even if you are convinced that the older years represent a period of low volatil-ity that will never be seen again, your system should, at the very least, not post losses forthat period. -
Paper Trading
With all good intentions, when you develop a new strategy you plan to hold the mostrecent data aside for out-of-sample testing. Somehow, before you’re done, that data hasbeen used a number of times. Therefore, as careful as you have tried to be, it is possiblethat you have overfitted the data by fine tuning the rules and parameters.
The only alternative is some period of paper trading. By monitoring the program daily,you can see how it is performing under real market conditions. Before doing that, how-ever, you should write down the system expectations, for example, the size and frequencyof the profits and losses. It is always a relief when the system posts its first profit, but it isthe overall profile that is most important. When paper trading, assume that your fill is theworst price of the 5-minute period immediately following an intraday signal, or the high-est price of the closing range if your trades come at the end of the day. A touch of realitycan start you thinking and save a lot of money. Don’t start trading a new system withoutmonitoring its performance first.
Dealing with the Unexpected .
By the time a computer and database have been acquired and installed, you have pro-grammed or had a programmer implement your ideas, debugged and tested the programs,560 PRACTICAL CONSIDERATIONS
 
and finally analyzed the results, there has been a substantial investment made. Hard workhas resulted in proving one of four things:
1. The system is profitable.
2. The system is not profitable.
3. The system cannot be implemented effectively.4, The whole process is too expensive.
If you’ve gotten to the end and your result is not satisfactory, you might want to reex-amine the process that you used. Trading strategies should be built slowly; each piece mustbe proved to be sound before you go on to the next step. You should never have gotten tothe end without knowing that the final product was going to be profitable. If the perfor-mance was deteriorating as you added new rules and features, you should have stoppedand reconsidered these features and modified each step to be more robust.
You may also find that your rules were too subjective, that you substituted a rule thatwas not well thought out, because you were not able to clearly define one part of yourStrategy. Knowing that you have been subjective about a trading decision is a valuable dis-covery. Many speculators believe that they are very clinical about their trading policies; theyknow exactly what will be done in every circumstance and; therefore, think that theirapproach can be well defined in writing. But computerization cannot be done successfullyif there are any “maybes,” “wait and see,” or any other exceptions.
One common problem js that the system does not work using either the rules or thevariables imposed by the program. If moving average periods ranging from 20 through 50were first used unsuccessfully, it is likely that you would test intervals of less than 20, Ifstop-loss points were too big or too small (judging by the size of frequency of the losses),you'll try other sizes. Systems do not always work as expected and perhaps only a slightchange to the rules is necessary. Negative results are important feedback. They can helpcreate a better strategy before it is too late. After all, if the answers were known in advance,this would all be unnecessary. If the results are good, but different from what wasexpected, you should take the time to understand why. Don’t accept the results withoutcareful review, This is an opportunity to either increase your understanding of how themarket works, or find that you have been making a mistake all along.
Acommon result is that the system works in some markets but not in others. This canbe the product of using fixed values rather than a rule. For example, a stop-loss of 4 tickswould make sense for Eurodollars trading at 4% yield, but not the S&P at 1100. Try tolocate the problem and make the rules more adaptable to changing conditions. Once thisis done, if the problem persists, you must face the issue of robustness.
Expectations
In some work, defining your expectations can bias the process and give poor results. Forthe development of a trading system, you must first have a sound idea before you start.Along with this idea is whether it works best in the short or long term, the frequency ofprofitable trades, and even the size of the expected profits and losses. By defining yourexpectations you can apply special rules and techniques that seem consistent with the phi-losophy of the method. These expectations of the profit profile are particularly importantto keep you focused throughout the development and testing of the strategy. If you expectto trade one each day and capture about 50% of the trading range, then signals that occuronly once each week should immediately cause you to look for an error. If profits are toosmall or losses too big, then you can carefully study how the rules apply to a series of pricesand understand whether you have omitted some rules, applied the rules incorrectly, orhave made a mistake in your thinking. Without setting down clear expectations in advance,you have no benchmark to help you through the development process.USE AND ABUSE OF 1 HE COMPUTER 561
AWinner in Disguise
One important twist on performance should not be overlooked: A consistently bad per-former can become a consistent winner by doing just the opposite. Finding a profitabletrend system in the range of shorter time intervals may seem to be a hopeless effort. Onthe other hand, systematic losses mean that the price movement fails to trend, and everybuy signal and sell signal is actually an overbought and oversold condition, predictive ofprice change. This can happen in markets that are heavily traded using systems; whatappears at first to be a trend change is a temporary distortion due to massive buy and sellorders.
Consistent losses do not always mean that the reverse trade will work. The net lossesmay be less than the total transaction costs; therefore, the opposite action will show evenlarger losses. The successful reverse system is one in which the net losses far exceed thetransaction costs.
Simple or Complex?
It is not necessary to create or acquire a system that is exceptionally complex to be suc-cessful. Simple systems, such as a dual moving average crossover, can be profitable as well.The difference between the performance of a simple system and a well-developed, com-plex method is the risk/reward ratio and the magnitude of the equity swings. :
Simple systems will catch the trends and take as much profit from the big moves as themore sophisticated approaches. It is the other patterns that make distinctions. Added fea-tures should reduce the losses and modify the trading patterns during nontrending peri-ods. This may be accomplished by identifying a trading range, using positions of varyingsize, or by the inclusion of economic data. Sometimes, more complex systems are justmore complex, without improving results. In that case, use the simple system, but retainlarger reserves than normal. ,
Just Because It Doesn’t Work in Practice Doesn’t Mean It Won’t Work in TheoryYou thought that you followed all the rules by
. Precisely defining the system
. Using enough test data
. Evaluating performance by forward extrapolation
. Investing enough to survive the worst losing streak. Trading a diversified portfolio
. Following every signal
DAW RW D
And still you lost all your money. . . . What happened?
Most likely you skipped some part of the well-planned development process. Carefulreassessment will usually show that some corners were cut. For example, new tradersoften only test their ideas in general by spot checking a few selected markets during yearsthat seem to have interesting or typical movement. They_assume that these shortcuts donot affect the results of the system. The advantage of computerized testing is that al/ yearscan be tested easily—the procedure necessary to produce a robust system that works inmost markets. -
There is always the temptation to force a system to work if it failed because of one ortwo large losses. An analyst can find something unique about these losing trades and cre-ate a rule that eliminates just those items. Some traders may just assume that circum-stances would have caused them to pass the trade rather than buy or sell in an obviouslyadverse market. This type of tampering doesn’t work and the reason for many of theseproblems is explained in the next section on price shocks.562 PRACTICAL CONSIDERATIONS
 
Suppose you are not guilty of any of these infractions, that the system was developedwithout making exceptions, and all the rules were followed. Then, the most commonproblem is to misjudge the volatility of the market or some systematic change that hasoccurred. You might see that the program was more profitable in the earlier years, and thatthe average profit has been declining and the average loss rising. This might be due toincreased participation and more market noise. As this occurs, it is necessary to increasethe period of calculation to compensate for the noise. For example, in the early 1980s, theenergy markets were very smooth, and a 3-day moving average generated good profits. Asvolume increased, the market appeared more erratic, and a 10- to 20-day moving averagewould have been needed. In the early 1990s, the trend speed would have increase to 30 to40 days, all due to changing market noise. When there is higher noise, it takes longer toidentify the trend. Even when you sit still, the market will continue to change.
Start Slowly
After paper trading, to begin trading with an exceptionally small portfolio, one that can belost.10 times over, will not hurt at the beginning. There must be a chance to see if the sys-tem performs the way it appeared in testing. This will be the first opportunity to find outthe cost of execution. The slippage may be the factor that separates profits from losses. Youmay find that, during quiet periods, when the market is moving sideways, there is smallslippage but no profits; during fast market, when you expect the largest profits, the slip-page also increases. The size and frequency of the losses should be checked as well as theequity cycle. When everything passes inspection, the investment can be increased slowly.Patience and thoroughness are important ingredients for success.
Once the system has been checked out, it is even safer to turn the day-to-day execu-tions over to someone else. It may be the only certain way to follow the rules precisely. Tryto stay detached and objective, but monitor the results carefully to be sure there are no sur-prises. Even a large profit may be an early warning of a problem. Large losses often followlarge profits.
PRICE SHOCKS
Price shocks represent the most significant obstacle in the effort to close the gap betweentest results and actual trading, or expectations and reality. A price shock is, by circum-stance, an unexpected event. Exceptionally volatile price moves are caused by actual newsthat differs from expectations, such as the Fed raising rates by 1% when only 4% was antic-ipated, or, it may be a significant, unexpected political event, such as the abduction of Gor-bachev in 1991, The key word to remember is “unexpected.”
If an event was expected, then there would be no price change. The market wouldhave already moved to the anticipated level. Therefore, we cannot expect to profit from aprice shock by clever planning, but only by chance. We should never assume that we wouldbe on the right side of a large, unexpected move in more than 50% of those events. Unfor-tunately, when we back-test a trading system using historic data, we tend not to identifyspecific price shocks and treat them as normal, predictable events. We choose systems thatperform best over a set of parameters, without regard.t6 specific trades that may have beenthe result of shocks. We judge results by higher profits, lower risk, or a combination of sta-tistical values. The results chosen as the best performance often have been the greatestbeneficiary of these unpredictable price shocks.
Making Money with Hindsight
We should already be aware of the problems of back-testing. While there are no otheralter-natives for validating a proposed trading strategy, it is necessary to look at the problemsPRICE SHOCKS ~*~ 563
 
more carefully to see a solution. No market is more evident than crude oil during the GulfWar, discussed in the previous chapter (see Figures 22-1, 22-2, and 22-3). From August 1,1990, through the end of the War in February 1991, oil prices were driven by news. But notall news is a surprise. In the agricultural markets, a crop freeze in orange juice or coffee isoften anticipated by a change in weather. Before a freeze can occur, the temperature mustdrop. And that low temperature must be sustained to cause damage. This weather changecauses processors to protect themselves by buying forward contracts or futures. In turn,prices move up.
Because Iraq had been moving troops near the Kuwait border, their intentions werenot a surprise; however, diplomacy failed. The threat of an oil supply disruption caused oilprices to slowly rise. Many systematic traders and commercials would have been long onAugust 7, 1990, when Iraq invaded Kuwait.
Identifying Price Shocks
If we can’t predict price shocks, and they seriously impact our historic tests, then we must
_ identify those shocks so that we can correct for their effects. In the end, this may mean thatwe need to look at the individual trades, compare them against a list of price shocks ofwhich we are aware, and decide which of the profits were realistic. Even more important,we must decide which risk was understated. There may not be a safe way of changing thisfinal process. It may always be necessary to manually review the results for credibility.Some analysts may find it comforting to think that the computer may not be able to doeverything on its own!
 
FIGURE 22-1 A far-reaching price shock. A vivid reminder of price shocks occurred onAugust 16, 1991, and again 3 days later when Russian premier Gorbachev was abducted andthen released. Most financial markets were affected simultaneously, as were the energymarkets. In the case of the currency markets, many traders profited from the first shock,which occurred on a Sunday, then gave it all back on the following Wednesday.
Rita12/3191 C=61.350 GCG O=61.420 H=61.510 L=61.190
| { ‘56.000
Gorbachev is abducted .‘on Sunday, August 18. $5.000Market plunges on open
of August 19, then recovers
‘on Wednésday, August 21, ~
when crisis ends.
ad ‘lt ‘hy | | "iM il. qn
‘Crisis ends onPrices gap ! Wednesdayloweron - . .Monday —> |
 
 
‘aug Sep
 
Source: Chart created with TradeStation® by Omega Research, Inc.564 PRACTICAL CONSIDERATIONS.pe
—FIGURE 22-2 A more common shock.The election of conservatives in Britain in January1992 was a surprise, but the effect was relatively local. {
 
arenejO2r2892 C=160,500 -.640, O=160.920 H=161 200 _
a Mt _slection shock
yen 1992 ©
ight
(ite . Pics wank an uptrend ,
in anticipation of electionresults
 
Source: Chart created with TradeStation® by Omega Research, Inc.
The process of identifying price shocks can be speeded up with a small amount ofautomation. A price shock will be defined as a day with a true range greater than the aver-age true range by a multiplier, called the shock factor. This would be coded asshock = 0; |truerange = high - low;if high - close{1]> truerange then truerange = high - close[1);if close[1] - low > trusrange then truerange = close[1} - low;volatility = @average(truerange(1],vperiod); \if truerange > volatility*shockfactor then shock = truerange;
Having coded the identification of a price shock, we can write a record to a file or print logto create a list of all price shocks, the date they occurred, and the size of the initial shock.It might also be useful to count the number of periods until prices return to a relativelynormal level. It may be unrealistic to assume that the volatility resulting from a price shockwill completely disappear; therefore, the point at which some degree of normal mightbegin could be when volatility declines to one-half the level of the initial price shock. Thiswould be found using the code
if truerange < shock/2 then shock = 0; —
With this simple method of identifying both the beginning and end of a price shock, we canstudy the effects of price shotks on testing performance. A simple test was performed-inwhich an optimized trend system was inspected for periods of price shocks, where a shockwas defined as a price move of three times normal volatility. The profits and losses fromthese periods was removed from the net performance over a 10-year period, with theresults shown itt Table 22-1, The reduction in. profits was more than 50% when priceshocks were eliminated. This means that the selection of parameters heavily favored profits that were generated by large, probably unpredictable, price jumps.GAMBLING TECHNIQUE—THE THEORY OF RUNS 565Fun TSS nn EEEaEESEEEEESSEERiemaeeeee ed
 
FIGURE 22-3 The invasion of Kuwait by Iraq. News of massing armies and troop movementsnear the Kuwait border helped the market anticipate the August 7, 1990, price jump in oil.At the beginning, many traders were on the right side of the market, although the move wasnot very dramatic,
 
irre{022082 C=44.290 +030 O=44.320 H=44.410 L=44150
U.S. retaliationwas stroriger thanexpected.
Vv
P Nen allNervous |market shows ‘yey 4volatile changes f
ih
"¢— Iraq's invasion of |rye Kuwait came whileoll prices were in an uptrend
 
Source: Chart created with TradeStation® by Omega Research, Inc.
How can we use these results if we cannot see the price shock until after it’s happened?The only way is to correct the optimization process so that the parameter selection is notbased on which calculation period is best tuned to get the most profit from these events. Ifyou remove the profits or losses during the initial price shock from the historic test results,as shown in Table 22-1, the optimization would then show the best performers, net of priceshocks. Because price shocks are unpredictable, selecting from a shock-adjusted resultshould yield a better trading system. Some traders might also prefer to use the identificationofa price shock period to change the current trading strategy to one of crisis management.
GAMBLING TECHNIQUE—THE THEORY OF RUNS
The application of Gambling Theory to high-leveraged or short-term trading satisfies twoimportant conditions. First, it presumes no statistical advantage in the occurrence of profits
 
TABLE 22-] Results of Removing Price Shocks from10-Year Historic Performance
 
Trend Adjusted for PercentSystem Shocks ChangeCrude oil 17.12 7.88 52%U.S. Treasury bonds 9.85. 421 ~57%Deutschemark 14.92 5.97 | 60%British pound 20.56 10.85 ATK
S&P 500 ~3.50 4.25 na
 
 566 PRACTICAL CONSIDERATIONS
 
and losses but concerns itSelf with the probable patcerns. Each trade could be treated as anoccurrence of red or black—up or down price moves or profits. Second, Gambling Theorystresses money management under adverse conditions. A successful professional gamblerand an active trader must both be highly disciplined and conserve capital. This section willlook at a gambler’s approach to money management and risk, using the Theory of Runs.
If the assumption is made that each trade is unrelated to the previous trade or thateach successive price has an equal chance of going up or down, the situation closelyresembles roulette. In Monte Carlo, the roulette wheel has 37 compartments: 18 black, 18red, and 1 white, assuring a loss of 2.7% in the same way that transaction costs are a hand-icap to trading. Continuously betting on only red or black will break even, less 2.7%, overlong betting periods. The only way to change the odds is by money management—vary-ing the size of the bets. Although the variables in trading are more complex, we will lookat this first.
The most well-known method for winning in a gambling situation is based on theprobability of successive wins, the Theory of Runs. On each spin of the wheel, the likeli-hood of the same color (red or black) reoccurring is 50%, or % (ignoring the white slot). Todefine a run of 3 reds, it is necessary to show a black on each side, that is,
Black-Red-Red-Red-Black
otherwise, the run may not be complete. If each color had a 50% chance of occurring, theprobability of a run of 3 is
Att (ay2°2°2°2°2°\2 -
One run of 3 can be expected for every 32 spins of the wheel (called coups). Extending thatto runs of 7 consecutive reds gives (%4)"*. For 256 coups, which is both a power of two andthe approximate number of trading days in a year (if you consider daily closing pricesrather than intraday trades), there are the following possibilities for runs of red:
 
Run of ProBability Expected Number Total .Length of Occurrence of Occurrences Appearances of Red.
| hs 32 32,
2 Ye 16 32
3 Ya 8 24
4 Ya 4 16
5 fae 2 10
6 ‘Ase I 6
Total appearances: 120
 
Arun of greater than 6 is not expected to occur. Notice, however, that the total appear-ances of red is only 120, short by 8. These 8 appearances could increase any of the runsfrom 1 through 6, or become a run of 7 or 8. The likelihood of a run greater than 6 is cal-culated using a geometric progression to get the sum of all probabilities greater than 6, or(4)"*?,