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166 Run of Probability Expected Number TotoJ Length of Occurrence of Occurrences Af>fteoror>ces ofRed ~ % 32 32 2 M» 16 32 3 !«2 8 24 4 4 16 5 1« 2 10 h. \ 6 Total appearances: 120 A run of greater thatt 6 is not expeaed to occur. Notice, however, that the total appearances of red is only 120, short by 8. These 8 appearances could increase any of the rum fh)m 1 through 6, or become a run of 7 or 8. The likelihood of a run greater than 6 is calculated using a geomciric progression to get the sum of all probabilities greater than 6, or (l)"*whe e256«>7! 1  256 There is a single chance that there will be a run greater than 6 in 256 tries. The average length of a run greater than 6 turns out to be 8, based on the decreasing probability of occurrences of longer runs. The average length of all runs greater than n will be n + 2. That makes the table of runs complete, with the number of occurrences of red equal to 128. Martingales and AntiMartingales The classic application of the Theory of Runs is called Martingales. In the simple version of this spproach, an initial bet is doubled each time a loss occurs; whenever there is a win, the betting begins again at the initial size. To demonstrate how this works, it is necessarj to use a table of uniform random numbers (which can be found in Appendix 1). These numbers varj from 0 ttirough 9. Let all those numbers beginning with digits 0 through 4 be assigned to red and 5 through 9 to black. Figure 224, read left to right, where open squares are red and solid squares are blade shows the first 257 assignments according to Appendix 1. Assuming that we bet on black, losses will depend on the longest run of red. We must decide the size of the initial bet in advance, which should be as large as possible and still withstand the longest run of red that is likely to occur. By using the results from the analjsis of the length of runs we find that for every 256 coups, it is likely that only one run greater than 6 will occur, and that run would mosl probably be 8 in length. The probability of a run of 9 is (2)", or 1 in 1,024. In 256 coups, the odds are about 3 to 1 against a run of 9 occurring. Having decided that capitalization must withstand a run of 8, we calculate that a bet of $1 doubled 8 times is $128. Divide the maximum amount of monej to be ridded by 128 and the result is the size of the initial bet. Each $1,000 divided by 128 gives 7.8125, which must be rounded down to $7; therefore, on the eighth consecutive occurrence of red, the bet will be $897. Counting the occurrences of runs on the simulated roulette table (Table 222), il is interesting to see that a run of 8, but not 7, appears. The results are within expectations, and no runs greater than 8 occurred in either red or black. For the pmposes of this exanple, the betting would then proceed as shown in Figure 2 25, using an initial bet of $ 1. The numbers in the squares represent winning bets. FIGURE 224 Sequence of random numbers representing occurrences of red and blad;.
COUPS GENERATED FROM RANDOM I 2 3 4 S 6 7 8 9 10 II INUMBERS 12 13 14 IS □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ I □ □ I □ I □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ OB □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ 1 □ □ □ I □ □ I □ □ I □ □ I □ □ □ □ □ □ □ □ □ □ □ □ □ I □ I □ □ I □ □ □ 1 □ □ □ G TABLE 222 Simulated Runs Block 35 runs ot 1 12 runs of 2 7 runs of 3 7 runs of4 2 runs of 5 2 runs of 6 I run of 8 138 ratal occurrences 35 runs of I 14 runs of 2 11 runs of 3 3 runs of 4 2 runs of 5 118 total occurrences Every occurrence of black is a winner. Although $64 is actually won on a single black coup, each sequence of runs nets only the initial bet because of the accumulated losses during that sequence. For the 256 coups shown in Table 222, there were a total of 65 distinct runs resulting in a profit of $65 (or $455 based on a $7 initial bet), or about onehalf of the initial capitalization ($ 128 or $ 1,000, respectively). Instead of using Martingales, if we had used the benchmark strategy of betting equal amounts on black to win on every coup, we would have lost $20 on a $1 bet. The Martingales method, therefore, has a good chance of winning a reasonable amount. AntiMartingales The antiMartingales approach offers a smaller chance of winnmg a large amount; it is exactly the opposite of applying Martingales. Instead of doubling each losing bet, the winners are doubled until a goal is reached. Because there is an excellent chance of I runof 6 in256 coups and a similar chance of a longer run, this method wins if the long run occurs in the first half of the number of coups to be played (256 in this case). Once a run of 6 occurs, you must immediately stop playing. We alreadj know that a run of 6 retums $32 on a bet of $1, and a run of 8 nets $128. This method would have lost if it had been applied to black in the test sequence; because there were 138 red coups with no black runs greater than 5, there would have been a loss of $138. If the bet had been on red, looking for a run of 6, there would have been
three wins, each for $94, and a loss of $118 for that many appearances of blad;. Waiting for a run of 8 would have won $128 and lost $117 on black by stopping right after the win. The success of antiMartingales depends on how soon the long run appears, in 4,096 coups, a run of will occur once, rehiming $1,024 on abet of $ 1. In the same 4,096 spins there will The lJa.tii,eBleE  acl, is likely to w tk if the player c .uld witl,.tai,d a E"f ll.butci ts Eiqi,ibcantiyleEEtl,an2 tm.eE FIGURE 225 Betting pattem.  12 3 4 5  6 7 8 9 10  11 12 1314 IS   1 2 4 8 fS  1 E 1  2 4 16 )   1 *  1 IS 1 2 4  { 1 2   4 8 16 33 g]  < 12 • 2  1 2   4 1 2  1 I  1 2 4 8   fS 1 I  2  1 1 2 
be 2,048 losses, showing that if the long run happens m the middle you win. the method breaks even; if it occurs sooner, The Theory of Runs Applied toTrading Before applying either Martingales or antiMartingales to the markets, it must be determined whether the movement of prices up or down is as uniform as in the case of roulette. A simple test was performed on a combined set of 21 diverse futures markets. The combined results of all up and down runs are shown in Table 223. The expected occurrences of both up and down were twice the probability of either a red or black coup occurring. Some differences between the random pattem previously used and actual results can be seen in Table 223, These futures markets show fewer runs of 1 and more frequent runs of 2; their tendency to fluctuate in a narrowing pattem around the expected (random) length of a run (Figure 226) may be due to the smaller sample of longer runs. The probabilities found here might be used to decide the chances of the next trading day continuing in the direction of the prior days frend, noting that a 3 to 4day run is as long as might be expected. The most interesting applications are in the betting strategies of the Martingales and antiMartingales methods, applied to futures markets. The following sections show two possibilities. Trading Daily Sequences By combining the Theory of Runs with the direction of the frend, the chances of being on the correct side of the longest run are increased., and, the size of the price move in the direction of the trend may also tend to be larger. To give yourself the best chance of identifjing the trend, it should be longterm. Assuming that the trend is up as determined from a chart or moving average, enter a long position on the close of a day in which the price moves lower. If Martingales is used, double the position each day that prices move adversely. A single day in which prices move up should recoup all losses and net a reasonable profit. If the frend is sustained, there should be fewer dajs down and more dajs up, thereby yielding a better retum to capitalization ratio. A variation on Martingales might include holding the trade for 2 consecutive dajs in the profitable direction because of the bias shown in the disfribution of run Another variation would take profits while prices are moving in a favor TABLE 223 Expected Occurrences of Ups and Downs for a Diverse Set of Markets
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