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--
