# Fable II Pub Games – Keystone Statistical Analysis

### Xbox 360

## Keystone Statistical Analysis by kentp1

**Version:** 0.75 | **Updated:** 10/20/2008

Statistical Analysis of Keystone, a component of Fable II Pub Games Version 0.75 authored by Kent E. Pryor kentp1@att.net XBox360 gamertag: ComradeNapoleon VERSION HISTORY Version 0.75 - Initial release, no analysis of Bloodstone subtype (10/20/2008) COPYRIGHT NOTICE Copyright 2008 Kent E. Pryor This may be not be reproduced under any circumstances except for personal, private use. It may not be placed on any web site or otherwise distributed publicly without advance written permission. Use of this guide on any other web site or as a part of any public display is strictly prohibited, and a violation of copyright. ABSTRACT The true statistical odds of all possible bets in the game of Keystone were calculated and compared to in-game payouts. Arch bet odds were determined by a Monte Carlo simulation of 200 million games of Keystone, both with and without the Jackpot feature. Inside bet odds were calculated analytically. There are no inside bets that favor the player over the house. However, in contrast to most casino games, there are several bets that pay out their true odds, and do not favor the house--these are bets on 5, 10, 11, 16, Run, and Keystone. For bets with a house advantage, this advantage is minimized at higher-limit tables. In general, the arch bets carry an advantage for the player. This advantage is maximized in Jackpot games relative to Standard games. Some arch bets are better than others, however, and even in the high-limit Jackpot game, arch bets on 9 and 12 carry an advantage for the house. INSIDE BETS Even at the highest-limit tables, most of the inside bets carry an advantage for the house. However, several of the bets pay true even odds. At the high- limit tables, the worst bets, by a substantial margin, are the bets on individual triples (i.e., roll 3, triple 2s, triple 3s, triple 4s, triple 5's, roll 18). In order to pay the true odds, these bets would need to pay 4300 : 20, but they only pay 4000 : 20. At the low-limit tables, there are some real sucker bets, but several bets still carry no house advantage. The table below shows the house advantage for each bet. A dash indicates that the bet pays true odds, and carries no advantage for the house or the player. The highest-limit table is shown next to the lowest-limit table. Many of the odds are the same, but odds that are even worse at the low-limit table are marked by an exclamation point. The 4-9/12-17, red/black, and oval/diamond bets are really, really bad for the player on the low-limit table! House House Advantage Advantage Bet (50-200) (5-10) 3 6.944% 6.944% 4 1.389% 1.389% 5 ---- ---- 6 2.778% 2.778% 7 2.778% 2.778% 8 2.778% 2.778% 9 0.463% 7.407% ! 10 ---- ---- 11 ---- ---- 12 0.463% 8.000% ! 13 2.778% 2.778% 14 2.778% 2.778% 15 2.778% 2.778% 16 ---- ---- 17 1.389% 1.389% 18 6.944% 6.944% Doubles 2.222% 11.111% ! Trips 2.778% 2.778% Run ---- ---- 4-9/12-17 1.852% 25.926% ! Keystone ---- ---- Red/black 1.852% 25.926% ! Oval/diamond 1.852% 25.926% ! ARCH BETS The house and player advantages for the arch bets on the 50-200 game of Standard Keystone are shown below. As you can see, most of the arch bets carry an advantage for the player, with the largest player advantage coming from 3 and 18. *Standard* House Player Archstone advantage advantage 3 2.755% 4 1.314% 5 1.327% 6 1.403% 7 1.117% 8 0.942% 9 3.071% 10 1.198% 11 1.198% 12 3.071% 13 0.942% 14 1.117% 15 1.403% 16 1.327% 17 1.314% 18 2.755% The same chart is given below for the 50-200 game of Jackpot Keystone. See the Discussion and Methods section for a discussion of where the differences come from relative to the Standard game. In every case, the advantage swings in the player's favor. *Jackpot* House Player Archstone advantage advantage 3 4.976% 4 4.277% 5 3.521% 6 3.181% 7 0.396% 8 2.388% 9 1.671% 10 2.460% 11 2.460% 12 1.671% 13 2.388% 14 0.396% 15 3.181% 16 3.521% 17 4.277% 18 4.976% DISCUSSION AND METHODS --Inside Bets-- The inside bets all depend on the chances of rolling particular combinations of three dice in a single roll. There are 6 x 6 x 6 = 216 different ways of rolling 3 dice at a time. Some inside bets can only be satisfied by a single combination of dice. For instance, an inside bet on rolling an 18 can only win if the first die is a 6, the second die is a 6, and the third die is a 6, also. Thus, the chances of rolling an 18 are 1 in 216 (approximately 0.463%), corresponding to odds of 215:1. Other inside bets can be satisfied by multiple different die rolls. For instance, there are 3 different ways to roll a 17: 6-6-5, 6-5-6, and 5-6-6. This means that the chances of rolling a 17 are 3 in 216 (approximately 1.389%), corresponding to odds of 213:3. The number of combinations resulting in a win for all other bets were determined similarly, and are presented in the table below. The payout from 216 bets column is calculated based on the number of times a bet is expected to hit in 216 bets (column 2) times the payout for a winning bet given in the payout odds column. For example, in 216 rolls, a bet on "4" is expected to win 3 times, each time paying out 71 gold, for a total of 213 gold. Since those 216 bets would have cost a total of 216 gold, the house kept 3 gold more than you bet. 3/216 = 1.389%, which is the house advantage, reported in the last column. Bet True chance True Payout from House (out of 216) chance(%) True odds Payout odds 216 bets Advantage 3 1 0.4630% 215 : 1 200 : 1 201 6.944% 4 3 1.3889% 213 : 3 70 : 1 213 1.389% 5 6 2.7778% 210 : 6 35 : 1 216 ---- 6 10 4.6296% 206 : 10 20 : 1 210 2.778% 7 15 6.9444% 201 : 15 13 : 1 210 2.778% 8 21 9.7222% 195 : 21 9 : 1 210 2.778% 9 25 11.5741% 191 : 25 7 : 1 200 7.407% 10 27 12.5000% 189 : 27 7 : 1 216 ---- 11 27 12.5000% 189 : 27 7 : 1 216 ---- 12 25 11.5741% 191 : 25 7 : 1 200 7.407% 13 21 9.7222% 195 : 21 9 : 1 210 2.778% 14 15 6.9444% 201 : 15 13 : 1 210 2.778% 15 10 4.6296% 206 : 10 20 : 1 210 2.778% 16 6 2.7778% 210 : 6 35 : 1 216 ---- 17 3 1.3889% 213 : 3 70 : 1 213 1.389% 18 1 0.4630% 215 : 1 200 : 1 201 6.944% Doubles 96 44.4444% 120 : 96 1 : 1 192 11.111% Trips 6 2.7778% 210 : 6 34 : 1 210 2.778% Run 24 11.1111% 192 : 24 8 : 1 216 ---- 4-9/12-17 80 37.0370% 136 : 80 1 : 1 160 25.926% Keystone 54 25.0000% 162 : 54 3 : 1 216 ---- Red/black 80 37.0370% 136 : 80 1 : 1 160 25.926% Oval/diamond 80 37.0370% 136 : 80 1 : 1 160 25.926% --Arch Bets-- Since the likelihood of having a particular keystone disappear changes after each roll of the dice during any given game, there is no simple analytical method to calculate the exact probability of having a keystone disappear like the inside bets. Instead, we must use Monte Carlo simulation methods. The Monte Carlo method simply means that we play the game over and over and observe the results. If we play the game enough times, the results we get can give us a very good approximation of the real probability of certain outcomes. A simulation of the Keystone game was constructed and run either with or without the Jackpot feature. I chose to use 200,000,000 simulated games simply because much more than that and some of the numbers the program was tracking would cause overflow errors. Two hundred million simulations is considerable overkill for this relatively simple simulation exercise, however, and the results from 1,000,000 games were essentially identical. One nice thing about this particular simulation is that there are a number of internal checks on the results that we can run. For instance, we calculated the probabilities of individual dice rolls of 3-18 in the section above. We can check to make sure that the simulation returns results consistent with those calculated probabilities. The results of this check are shown below: Standard: 1,647,989,493 dice rolls in 200,000,000 games Die Roll Chance in 216 Expected Found %difference 3 1 7629581 7628813 -0.0101% 4 3 22888743 22883113 -0.0246% 5 6 45777486 45769659 -0.0171% 6 10 76295810 76304263 0.0111% 7 15 114443715 114448378 0.0041% 8 21 160221201 160201205 -0.0125% 9 25 190739525 190742323 0.0015% 10 27 205998687 206000926 0.0011% 11 27 205998687 206003975 0.0026% 12 25 190739525 190739326 -0.0001% 13 21 160221201 160209425 -0.0073% 14 15 114443715 114441821 -0.0017% 15 10 76295810 76315107 0.0253% 16 6 45777486 45778751 0.0028% 17 3 22888743 22892350 0.0158% 18 1 7629581 7630058 0.0063% Total 1647989496 1647989493 0.0000% Jackpot: 1,647,971,752 dice rolls in 200,000,000 games Die Roll Chance in 216 Expected Found %difference 3 1 7629499 7631334 0.0241% 4 3 22888497 22887230 -0.0055% 5 6 45776993 45782514 0.0121% 6 10 76294989 76287246 -0.0101% 7 15 114442483 114435061 -0.0065% 8 21 160219476 160233042 0.0085% 9 25 190737471 190739353 0.0010% 10 27 205996469 206015439 0.0092% 11 27 205996469 206004953 0.0041% 12 25 190737471 190724405 -0.0069% 13 21 160219476 160225523 0.0038% 14 15 114442483 114424071 -0.0161% 15 10 76294989 76285258 -0.0128% 16 6 45776993 45779783 0.0061% 17 3 22888497 22887720 -0.0034% 18 1 7629499 7628820 -0.0089% Total 1647971754 1647971752 0.0000% The simulated results are extremely close to the expected results. The greatest difference between the expected and simulated value is only about 1 part in 4000. Note also that the number of die rolls in 200,000,000 games in both simulations are very close to the same value, with both averaging 8.2399 rolls per game after rounding to the nearest ten-thousandth. This is consistent with the in-game hint that the average Keystone game is about 8 rolls long. A second check we can run is to compare the results from archstones 3-10 with the results from archstones 11-18 in each simulation. Because the game is symmetrical, the two halves of the game serve as an internal control for each other. The results for the Standard game and the Jackpot game are given below. The first column is the archstone number, the second column is the number of times that archstone was removed in 200,000,000 games, the third column is the difference (in absolute count) between the number of times the archstone was removed and the number of times its mirror image was removed (e.g., 3 and 18, 4 and 17), the fourth column is that difference expressed as a percentage, and the last column is the final estimated probability that each archstone will be removed from the board in any given game of Keystone. This estimated probability represents the average of the probabilities calculated for symmetrical keystones, yielding a perfectly symmetrical probability table. In both the Standard and Jackpot simulations, the symmetry of the probabilities was extremely close before averaging (differences of less than 0.022% in every case), suggesting that the simulation is working correctly and has enough individual games to adequately sample all possibilities. *Standard* # of times difference difference estimated Archstone removed (absolute) (%) probability 3 42812220 -4525 -0.01057% 21.407% 4 63316308 -10184 -0.01608% 31.661% 5 85869189 -2971 -0.00346% 42.935% 6 105626616 -2607 -0.00247% 52.814% 7 120588391 -2136 -0.00177% 60.295% 8 129412280 -1113 -0.00086% 64.706% 9 127534439 -7017 -0.00550% 63.769% 10 148819237 -1580 -0.00106% 74.410% 11 148820817 74.410% 12 127541456 63.769% 13 129413393 64.706% 14 120590527 60.295% 15 105629223 52.814% 16 85872160 42.935% 17 63326492 31.661% 18 42816745 21.407% *Jackpot* # of times difference difference estimated Archstone removed (absolute) (%) probability 3 43744488 9313 0.02129% 21.870% 4 65173406 1125 0.00173% 32.586% 5 87733493 7473 0.00852% 43.865% 6 107482959 5406 0.00503% 53.740% 7 122436613 4752 0.00388% 61.217% 8 131273102 13395 0.01020% 65.633% 9 129385638 10459 0.00808% 64.690% 10 150676625 -537 -0.00036% 75.338% 11 150677162 75.338% 12 129375179 64.690% 13 131259707 65.633% 14 122431861 61.217% 15 107477553 53.740% 16 87726020 43.865% 17 65172281 32.586% 18 43735175 21.870% While it may be counterintuitive, archstones 9 and 12 are less likely to be removed, on average, than archstones 8 and 13, despite having 9 and 12 be rolled more often than 8 or 13. Why might this be? A difference in the "cascade" effect. It is less likely to have the 9 removed by "cascading" down from the 10 than it is to have 8 removed by "cascading" down from the 9 since having a "hole" at the 10 keystone makes the game much more likely to end sooner. A "hole" at the 9 archstone (not a keystone!) does not bring the game much closer to its end, so there are generally more chances for the "cascade" to take out the 8. The Jackpot feature has no impact on most games. However, every 1 in 108 games a 3 or an 18 will be rolled in the very first roll. In these cases, all arch bets pay out. This makes arch bets 4-17 1/108, or approximately 0.926% more likely to hit in Jackpot games relative to Standard games. Arch bets of 3 and 18 are 1/216, or approximately 0.463% more likely to hit in Jackpot games than Standard games, half as much as the other arch bets. This is because one of those two keystones would have hit in the Standard game anyway because it was rolled naturally, so it doesn't get counted "extra". The simulated results are entirely consistent with this expected advantage for the Jackpot games, further suggesting that the simulations are valid: Standard Jackpot Advantage for Archstone Probability Probability Jackpot from simulation 3 21.407% 21.870% 0.463% 4 31.661% 32.586% 0.926% 5 42.935% 43.865% 0.930% 6 52.814% 53.740% 0.926% 7 60.295% 61.217% 0.922% 8 64.706% 65.633% 0.927% 9 63.769% 64.690% 0.921% 10 74.410% 75.338% 0.928% 11 74.410% 75.338% 0.928% 12 63.769% 64.690% 0.921% 13 64.706% 65.633% 0.927% 14 60.295% 61.217% 0.922% 15 52.814% 53.740% 0.926% 16 42.935% 43.865% 0.930% 17 31.661% 32.586% 0.926% 18 21.407% 21.870% 0.463% The house and player advantages were calculated from the expected payout of a 50 gold bet. This is found by multiplying the payout for a successful 50 gold bet by the estimated probability of having the bet be successful. For instance, for archstone 3, the payout if successful is 190 + 50 = 240. 240 * 21.407% = 51.377. The house advantage is 51.377/50 = -2.755%. Most of the house advantages are negative, implying that they are advantages for the player. The table below shows these calculations for the Standard Keystone 50-200 table. *Standard* Estimated Payout Payout from Expected House Archstone probability odds fair bet payout advantage 3 21.407% 190 : 50 50 51.377 -2.755% 4 31.661% 110 : 50 50 50.657 -1.314% 5 42.935% 68 : 50 50 50.664 -1.327% 6 52.814% 46 : 50 50 50.701 -1.403% 7 60.295% 32 : 50 50 49.442 1.117% 8 64.706% 28 : 50 50 50.471 -0.942% 9 63.769% 26 : 50 50 48.464 3.071% 10 74.410% 18 : 50 50 50.599 -1.198% 11 74.410% 18 : 50 50 50.599 -1.198% 12 63.769% 26 : 50 50 48.464 3.071% 13 64.706% 28 : 50 50 50.471 -0.942% 14 60.295% 32 : 50 50 49.442 1.117% 15 52.814% 46 : 50 50 50.701 -1.403% 16 42.935% 68 : 50 50 50.664 -1.327% 17 31.661% 110 : 50 50 50.657 -1.314% 18 21.407% 190 : 50 50 51.377 -2.755% --The Simulation-- (a.k.a. the really technical stuff) The Pascal program used to run these simulations is given below, with sample output at the bottom, in order for others to analyze and verify the logic I used in the simulation. The program was run with the constant "jackpot" set to both True and False to simulate Jackpot and Standard Keystone, respectively. Disclaimer: I make no representation that this program is anywhere close to elegant or efficient--this is the first Pascal program I have written in 20 years! My apologies to any programmers who are offended by the ugly code. (* Copyright 2008 Kent E. Pryor *) program Keystone; uses sysutils, math; const ngames : longint = 200000000; jackpot : boolean = True; var die1, die2, die3, roll, rollcount, minutes, seconds : integer; n3, n4, n5, n6, n7, n8, n9, n10, n11, n12, n13, n14, n15, n16, n17, n18, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, g, nrolls : longint; gameover, arch4, arch5, arch6, arch7, arch8, arch9, arch10, arch11, arch12, arch13, arch14, arch15, arch16, arch17 : boolean; starttime, endtime : TTimeStamp; mselapsed : comp; procedure Restart; begin gameover := False; arch4 := False; arch5 := False; arch6 := False; arch7 := False; arch8 := False; arch9 := False; arch10 := False; arch11 := False; arch12 := False; arch13 := False; arch14 := False; arch15 := False; arch16 := False; arch17 := False; roll := 0; rollcount := 0; end; procedure Roll3; begin n3 := n3+1; gameover := True; if jackpot then if rollcount = 1 then begin n4 := n4+1; n5 := n5+1; n6 := n6+1; n7 := n7+1; n8 := n8+1; n9 := n9+1; n10 := n10+1; n11 := n11+1; n12 := n12+1; n13 := n13+1; n14 := n14+1; n15 := n15+1; n16 := n16+1; n17 := n17+1; n18 := n18+1; end end; procedure Roll4; begin if arch4 then Roll3 else begin n4 := n4+1; arch4 := True; end end; procedure Roll5; begin if arch5 then Roll4 else begin n5 := n5+1; arch5 := True; end end; procedure Roll6; begin if arch6 then Roll5 else begin n6 := n6+1; arch6 := True; end end; procedure Roll7; begin if arch7 then Roll6 else begin n7 := n7+1; arch7 := True; end end; procedure Roll8; begin if arch8 then Roll7 else begin n8 := n8+1; arch8 := True; end end; procedure Roll9; begin if arch9 then Roll8 else begin n9 := n9+1; arch9 := True; end end; procedure Roll10; begin if arch10 then Roll9 else begin n10 := n10+1; arch10 := True; if arch11 then gameover := True; end end; procedure Roll18; begin n18 := n18+1; gameover := True; if jackpot then if rollcount = 1 then begin n3 := n3+1; n4 := n4+1; n5 := n5+1; n6 := n6+1; n7 := n7+1; n8 := n8+1; n9 := n9+1; n10 := n10+1; n11 := n11+1; n12 := n12+1; n13 := n13+1; n14 := n14+1; n15 := n15+1; n16 := n16+1; n17 := n17+1; end end; procedure Roll17; begin if arch17 then Roll18 else begin n17 := n17+1; arch17 := True; end end; procedure Roll16; begin if arch16 then Roll17 else begin n16 := n16+1; arch16 := True; end end; procedure Roll15; begin if arch15 then Roll16 else begin n15 := n15+1; arch15 := True; end end; procedure Roll14; begin if arch14 then Roll15 else begin n14 := n14+1; arch14 := True; end end; procedure Roll13; begin if arch13 then Roll14 else begin n13 := n13+1; arch13 := True; end end; procedure Roll12; begin if arch12 then Roll13 else begin n12 := n12+1; arch12 := True; end end; procedure Roll11; begin if arch11 then Roll12 else begin n11 := n11+1; arch11 := True; if arch10 then gameover := True; end end; procedure PlayGame; begin Restart; Repeat die1 := random(6) + 1; die2 := random(6) + 1; die3 := random(6) + 1; roll := die1+die2+die3; rollcount := rollcount + 1; nrolls := nrolls + 1; case roll of 3 : begin c3 := c3+1; Roll3 end; 4 : begin c4 := c4+1; Roll4 end; 5 : begin c5 := c5+1; Roll5 end; 6 : begin c6 := c6+1; Roll6 end; 7 : begin c7 := c7+1; Roll7 end; 8 : begin c8 := c8+1; Roll8 end; 9 : begin c9 := c9+1; Roll9 end; 10 : begin c10 := c10+1; Roll10 end; 11 : begin c11 := c11+1; Roll11 end; 12 : begin c12 := c12+1; Roll12 end; 13 : begin c13 := c13+1; Roll13 end; 14 : begin c14 := c14+1; Roll14 end; 15 : begin c15 := c15+1; Roll15 end; 16 : begin c16 := c16+1; Roll16 end; 17 : begin c17 := c17+1; Roll17 end; 18 : begin c18 := c18+1; Roll18 end; end; Until gameover; end; (* main program *) begin starttime := DateTimeToTimeStamp(Now); randomize; n3 := 0; n4 := 0; n5 := 0; n6 := 0; n7 := 0; n8 := 0; n9 := 0; n10 := 0; n11 := 0; n12 := 0; n13 := 0; n14 := 0; n15 := 0; n16 := 0; n17 := 0; n18 := 0; c3 := 0; c4 := 0; c5 := 0; c6 := 0; c7 := 0; c8 := 0; c9 := 0; c10 := 0; c11 := 0; c12 := 0; c13 := 0; c14 := 0; c15 := 0; c16 := 0; c17 := 0; c18 := 0; nrolls := 0; FOR g := 1 to ngames do PlayGame; endtime := DateTimeToTimeStamp(Now); mselapsed := TimeStampToMSecs(endtime) - TimeStampToMSecs(starttime); minutes := floor(mselapsed/60000); seconds := floor(mselapsed/1000 - 60*minutes); writeln('Processor time elapsed: ', minutes, ' minutes, ', seconds, ' seconds.'); if jackpot then writeln('Jackpots enabled in simulation.') else writeln('Jackpots disabled in simulation.'); writeln('Total number of rolls in ', ngames, ' games = ', nrolls); writeln('# of times 3 was rolled = ', c3); writeln('# of times 4 was rolled = ', c4); writeln('# of times 5 was rolled = ', c5); writeln('# of times 6 was rolled = ', c6); writeln('# of times 7 was rolled = ', c7); writeln('# of times 8 was rolled = ', c8); writeln('# of times 9 was rolled = ', c9); writeln('# of times 10 was rolled = ', c10); writeln('# of times 11 was rolled = ', c11); writeln('# of times 12 was rolled = ', c12); writeln('# of times 13 was rolled = ', c13); writeln('# of times 14 was rolled = ', c14); writeln('# of times 15 was rolled = ', c15); writeln('# of times 16 was rolled = ', c16); writeln('# of times 17 was rolled = ', c17); writeln('# of times 18 was rolled = ', c18); writeln('# of times 3 Archstone removed = ', n3); writeln('# of times 4 Archstone removed = ', n4); writeln('# of times 5 Archstone removed = ', n5); writeln('# of times 6 Archstone removed = ', n6); writeln('# of times 7 Archstone removed = ', n7); writeln('# of times 8 Archstone removed = ', n8); writeln('# of times 9 Archstone removed = ', n9); writeln('# of times 10 Archstone removed = ', n10); writeln('# of times 11 Archstone removed = ', n11); writeln('# of times 12 Archstone removed = ', n12); writeln('# of times 13 Archstone removed = ', n13); writeln('# of times 14 Archstone removed = ', n14); writeln('# of times 15 Archstone removed = ', n15); writeln('# of times 16 Archstone removed = ', n16); writeln('# of times 17 Archstone removed = ', n17); writeln('# of times 18 Archstone removed = ', n18); end. SAMPLE OUTPUT 1--Jackpots disabled (Standard Keystone): Processor time elapsed: 5 minutes, 41 seconds. Jackpots disabled in simulation. Total number of rolls in 200000000 games = 1647989493 # of times 3 was rolled = 7628813 # of times 4 was rolled = 22883113 # of times 5 was rolled = 45769659 # of times 6 was rolled = 76304263 # of times 7 was rolled = 114448378 # of times 8 was rolled = 160201205 # of times 9 was rolled = 190742323 # of times 10 was rolled = 206000926 # of times 11 was rolled = 206003975 # of times 12 was rolled = 190739326 # of times 13 was rolled = 160209425 # of times 14 was rolled = 114441821 # of times 15 was rolled = 76315107 # of times 16 was rolled = 45778751 # of times 17 was rolled = 22892350 # of times 18 was rolled = 7630058 # of times 3 Archstone removed = 42812220 # of times 4 Archstone removed = 63316308 # of times 5 Archstone removed = 85869189 # of times 6 Archstone removed = 105626616 # of times 7 Archstone removed = 120588391 # of times 8 Archstone removed = 129412280 # of times 9 Archstone removed = 127534439 # of times 10 Archstone removed = 148819237 # of times 11 Archstone removed = 148820817 # of times 12 Archstone removed = 127541456 # of times 13 Archstone removed = 129413393 # of times 14 Archstone removed = 120590527 # of times 15 Archstone removed = 105629223 # of times 16 Archstone removed = 85872160 # of times 17 Archstone removed = 63326492 # of times 18 Archstone removed = 42816745 logout [Process completed] SAMPLE OUTPUT 2--Jackpots enabled (Jackpot Keystone): Processor time elapsed: 5 minutes, 38 seconds. Jackpots enabled in simulation. Total number of rolls in 200000000 games = 1647971752 # of times 3 was rolled = 7631334 # of times 4 was rolled = 22887230 # of times 5 was rolled = 45782514 # of times 6 was rolled = 76287246 # of times 7 was rolled = 114435061 # of times 8 was rolled = 160233042 # of times 9 was rolled = 190739353 # of times 10 was rolled = 206015439 # of times 11 was rolled = 206004953 # of times 12 was rolled = 190724405 # of times 13 was rolled = 160225523 # of times 14 was rolled = 114424071 # of times 15 was rolled = 76285258 # of times 16 was rolled = 45779783 # of times 17 was rolled = 22887720 # of times 18 was rolled = 7628820 # of times 3 Archstone removed = 43744488 # of times 4 Archstone removed = 65173406 # of times 5 Archstone removed = 87733493 # of times 6 Archstone removed = 107482959 # of times 7 Archstone removed = 122436613 # of times 8 Archstone removed = 131273102 # of times 9 Archstone removed = 129385638 # of times 10 Archstone removed = 150676625 # of times 11 Archstone removed = 150677162 # of times 12 Archstone removed = 129375179 # of times 13 Archstone removed = 131259707 # of times 14 Archstone removed = 122431861 # of times 15 Archstone removed = 107477553 # of times 16 Archstone removed = 87726020 # of times 17 Archstone removed = 65172281 # of times 18 Archstone removed = 43735175 logout [Process completed] logout --end of file--

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