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Deconstructing Cleopatra
Introduction
In my continued effort to pull the veil of secrecy away and show how slot machines work, I offer the following deconstruction of the popular IGT game Cleopatra. This is a simple fivereel video slot with a free spin bonus.
The version this analysis is based on is the one that can be played for free at .
The Reel Strips
Our friends at were kind enough to provide the reel strips for the game at Vegas Slots Online. I believe they pieced them together from a lot of playing, the same way I did other slot deconstructions, like the one of Lionfish.
The way video slots like this one work is to choose a random number for each reel, from 1 to the number of positions in the reel, and map the random number to the corresponding position on the reel. In other words each position on a given reel is equally likely. The number of each symbol on each reel is carefully chosen to achieve a desired return. Following are the reel strips for Cleopatra. Note that reel 5 is longer than first four reels.
Cleopatra Reel Strips
Position  Reel 1  Reel 2  Reel 3  Reel 4  Reel 5 

1  eye  Q  K  K  gold 
2  Q  J  9  9  10 
3  10  gold  eye  eye  J 
4  gold  Q  10  A  flower 
5  J  A  gold  flower  K 
6  A  Cleopatra  K  J  10 
7  crook and flail  Q  J  scarab  Cleopatra 
8  10  J  Cleopatra  10  K 
9  Q  eye  Q  crook and flail  9 
10  flower  9  10  J  scarab 
11  9  Q  gold  gold  10 
12  10  gold  9  Q  eye 
13  Sphinx  K  J  eye  A 
14  Q  scarab  scarab  J  J 
15  J  J  A  10  Sphinx 
16  Cleopatra  flower  10  crook and flail  9 
17  Q  10  crook and flail  A  A 
18  A  scarab  A  Q  crook and flail 
19  flower  9  K  Sphinx  Q 
20  K  Sphinx  scarab  10  A 
21  scarab  A  9  K  crook and flail 
22  Q  crook and flail  10  gold  10 
23  eye  10  flower  J  eye 
24  K  K  Q  crook and flail  Q 
25  crook and flail  flower  9  9  gold 
26  10  J  crook and flail  A  9 
27  scarab  Q  K  flower  Cleopatra 
28  9  eye  eye  9  J 
29  gold  J  9  A  flower 
30  10  crook and flail  Sphinx  Cleopatra  Q 
31  K  
32  eye  
33  J  
34  10  
35  gold  
36  Q  
37  crook and flail  
38  9  
39  scarab  
40  A  
41  9 
For an example, consider this screenshot above. Assume that the random number chosen is associated with the symbol appearing in the top row. This configuration would occur if the random numbers chosen were as follows:
 Reel 1: 2
 Reel 2: 19
 Reel 3: 7
 Reel 4: 5
 Reel 5: 14
In the event the random number chosen is last or second to last on the reel, then the reel would wrap around, with position 1 following the last position on the reel.
Symbol Frequency
The next table shows the total number of each symbol on each reel.
Cleopatra Symbol Distribution
Symbol  Reel 1  Reel 2  Reel 3  Reel 4  Reel 5 

Cleopatra  1  1  1  1  2 
Scarab  2  2  2  1  2 
Flower  2  2  1  2  2 
Gold  2  2  2  2  3 
crook and flail  2  2  2  3  3 
eye  2  2  2  2  3 
A  2  2  2  4  4 
K  2  2  4  2  3 
Q  5  5  2  2  4 
J  2  5  2  4  4 
10  5  2  4  3  5 
9  2  2  5  3  5 
Sphinx  1  1  1  1  1 
Total  30  30  30  30  41 
Pay Table
The pay table is easily seen from the rule screens, which I also show in the table below. Cleopatra is wild and may substitute for any symbol except the Sphinx. In addition, one or more Cleopatra symbols will double any win, except wins based only on Cleopatras. In the event the player qualifies for two possible wins on the same payline, because it started with two to four wilds, the game will choose the higher of the two wins. In that event results in a tie, I count it as an all Cleopatra win in the tables of winning combinations later on. For example, Cleopatra  Cleopatra  K  Q J, which can be scored as 10 for either two Cleopatras or a doubled three kings, is counted as two Cleopatra win.
Cleopatra Pay Table
Symbol  5 pays  4 pays  3 pays  2 pays 

Cleopatra  10000  2000  200  10 
Scarab  750  100  25  2 
Flower  750  100  25  2 
Gold  400  100  15  0 
crook and flail  250  75  10  0 
eye  250  50  10  0 
A  125  50  10  0 
K  100  50  5  0 
Q  100  25  5  0 
J  100  25  5  0 
10  100  25  5  0 
9  100  25  5  2 
The Sphnix is a scatter pay. This means the player is paid according to the number that appear on the screen anywhere. The player is paid a multiple of his total amount bet and the number of diver symbols. Following is the scatter pay table.
 5 Pays: 100
 4 Pays: 20
 3 Pays: 5
 2 Pays: 2
Line Pay Math
Winning Combinations — NonDoubled
The next table shows the number of combinations for each possible win when no wilds are involved in the win, except all wild wins. In other words, wins that are not multiplied. Cells are left blank if they don't produce a win.
NonDoubled Line Pay Combinations — Not Doubled
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Cleopatra  2  39  1,118  19,516  20,675 
Scarab  16  296  9,184  132,840  142,336 
Flower  16  296  4,428  137,760  142,500 
Gold  48  576  8,856  9,480  
crook and flail  72  864  8,528  9,464  
eye  48  576  8,856  9,480  
A  128  1,120  8,200  9,448  
K  96  1,152  17,712  18,960  
Q  400  3,500  55,350  59,250  
J  320  2,800  20,500  23,620  
10  600  4,080  42,640  47,320  
9  300  2,040  21,320  118,080  141,740 
Total  2,046  17,339  206,692  408,196  634,273 
Winning Combinations — Doubled
The next table shows the number of combinations for each possible win when at least one Cleopatra involved in the win, except all wild wins. In other words, wins that are doubled.
Doubled Line Pay Combinations — Doubled
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Scarab  196  1,628  20,664  132,840  155,328 
Flower  196  1,591  14,391  137,760  153,938 
Gold  352  2,232  19,926    22,510 
crook and flail  463  2,880  19,188  22,531  
eye  352  2,232  19,926  22,510  
A  676  3,430  18,450  22,556  
K  564  3,600  26,568  30,732  
Q  1,526  7,735  60,885  70,146  
J  1,270  6,475  31,775  39,520  
10  1,892  8,024  47,970  57,886  
9  1,184  5,168  29,848  118,080  154,280 
Total  8,671  44,995  309,591  388,680  751,937 
Be warned that the math is tedious and errorprone to calculate these combinations by hand. If you're up to the challenge and you can code, then I recommend coding five nested loops and cycle through all 30^4*41=33,210,000 possible outcomes.
The next table shows the probability of each possible win for the nondoubled wins. The lower right cell shows a hit frequency per line of 1.85% for nondoubled wins.
Probabilities — NonDoubled
NonDoubled Line Pay Probabilities
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Cleopatra  0.00000006  0.00000117  0.00003366  0.00058765  0.00062132 
Scarab  0.00000048  0.00000891  0.00027654  0.00400000  0.00427654 
Flower  0.00000048  0.00000891  0.00013333  0.00414815  0.00428148 
Gold  0.00000145  0.00001734  0.00026667  0.00026667  
crook and flail  0.00000217  0.00002602  0.00025679  0.00025679  
eye  0.00000145  0.00001734  0.00026667  0.00026667  
A  0.00000385  0.00003372  0.00024691  0.00024691  
K  0.00000289  0.00003469  0.00053333  0.00053333  
Q  0.00001204  0.00010539  0.00166667  0.00166667  
J  0.00000964  0.00008431  0.00061728  0.00061728  
10  0.00001807  0.00012285  0.00128395  0.00128395  
9  0.00000903  0.00006143  0.00064198  0.00355556  0.00419753 
Total  0.00006161  0.00052210  0.00622379  0.01229136  0.01851515 
Probabilities — Doubled
The next table shows the probability of each possible win for the doubled wins. The lower right cell shows a hit frequency per line of 2.11% for doubled wins.
Doubled Line Pay Probabilities
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Scarab  0.00000590  0.00004902  0.00062222  0.00400000  0.00462222 
Flower  0.00000590  0.00004791  0.00043333  0.00414815  0.00458148 
Gold  0.00001060  0.00006721  0.00060000  0.00060000  
crook and flail  0.00001394  0.00008672  0.00057778  0.00057778  
eye  0.00001060  0.00006721  0.00060000  0.00060000  
A  0.00002036  0.00010328  0.00055556  0.00055556  
K  0.00001698  0.00010840  0.00080000  0.00080000  
Q  0.00004595  0.00023291  0.00183333  0.00183333  
J  0.00003824  0.00019497  0.00095679  0.00095679  
10  0.00005697  0.00024161  0.00144444  0.00144444  
9  0.00003565  0.00015562  0.00089877  0.00355556  0.00445432 
Total  0.00026110  0.00135486  0.00932222  0.01170370  0.02102593 
Returns — NonDoubled
The next table shows the product of each winning probability and what it pays for nondoubled wins. Each cell is the product of the probability of winning and the win. The lower right cell shows that the player can expect to win 8.17% of his bet per spin from nondoubled line pays.
NonDoubled Line Pay Returns
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Cleopatra  0.000602  0.002349  0.006733  0.005877  0.012609 
Scarab  0.000361  0.000891  0.006914  0.008000  0.014914 
Flower  0.000361  0.000891  0.003333  0.008296  0.011630 
Gold  0.000578  0.001734  0.004000  0.004000  
crook and flail  0.000542  0.001951  0.002568  0.002568  
eye  0.000361  0.000867  0.002667  0.002667  
A  0.000482  0.001686  0.002469  0.002469  
K  0.000289  0.001734  0.002667  0.002667  
Q  0.001204  0.002635  0.008333  0.008333  
J  0.000964  0.002108  0.003086  0.003086  
10  0.001807  0.003071  0.006420  0.006420  
9  0.000903  0.001536  0.003210  0.007111  0.010321 
Total  0.008455  0.021454  0.052400  0.029284  0.081684 
Returns — Doubled
The next table shows the product of each winning probability and what it pays for doubled wins. Each cell is the product of the probability of winning, the win, and 2. The lower right cell shows that the player can expect to win 40.89 % of his bet per spin from doubled line pays.
Doubled Line Pay Returns
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Scarab  0.008853  0.009804  0.031111  0.016000  0.065768 
Flower  0.008853  0.009581  0.021667  0.016593  0.056693 
Gold  0.008479  0.013442  0.018000  0.039921  
crook and flail  0.006971  0.013008  0.011556  0.031534  
eye  0.005300  0.006721  0.012000  0.024020  
A  0.005089  0.010328  0.011111  0.026528  
K  0.003397  0.010840  0.008000  0.022237  
Q  0.009190  0.011646  0.018333  0.039169  
J  0.007648  0.009749  0.009568  0.026965  
10  0.011394  0.012081  0.014444  0.037919  
9  0.007130  0.007781  0.008988  0.014222  0.038121 
Total  0.082304  0.114980  0.164778  0.046815  0.408877 
Returns — Combined
The final line pay table shows the combined return between doubled and nondoubled wins for each type of win. The lower right cell shows that the player can expect to win 52.05% of his bet per spin from line pays.
Combined Line Pay Returns
Symbol  5 in a row  4 in a row  3 in a row  2 in a row  Total 

Cleopatra  0.000602  0.002349  0.006733  0.005877  0.015560 
Scarab  0.009214  0.010696  0.038025  0.024000  0.081934 
Flower  0.009214  0.010473  0.025000  0.024889  0.069576 
Gold  0.009058  0.015176  0.022000  0.046234  
crook and flail  0.007513  0.014959  0.014123  0.036596  
eye  0.005661  0.007588  0.014667  0.027916  
A  0.005571  0.012014  0.013580  0.031165  
K  0.003686  0.012575  0.010667  0.026927  
Q  0.010394  0.014280  0.026667  0.051341  
J  0.008612  0.011856  0.012654  0.033123  
10  0.013201  0.015152  0.020864  0.049217  
9  0.008034  0.009316  0.012198  0.021333  0.050881 
Total  0.090759  0.136435  0.217177  0.076099  0.520470 
Scatter Pay
The Sphinx is known as a "scatter pay" symbol. It pays based on the number that appear anywhere on the screen. They do not have to be left aligned or on the same payline. Scatter pay wins are based on the total amount bet, as opposed the bet per payline.
The following table shows the probability and contribution to the return for each number of scatter pays. The bottom row shows a hit frequency on the scatter pay of 7.36% and a contribution to the total return of 17.51%.
Scatter Pay — Fixed Wins
Scatters  Pays  Combinations  Probability  Return  

5  100  243  0.000007  100  0.000732 
4  20  11,826  0.000356  20  0.007122 
3  5  228,906  0.006893  5  0.034463 
2  2  2,204,496  0.066380  2  0.132761 
Total  127  2,445,471  0.073637  0.175078 
Bonus
If the player gets three or more scatter symbols, then he will play a bonus round. The bonus round consists of an initial 15 free spins. Free spins use the same reel stripping as initial spins. All wins, except for five Cleopatras, are tripled. If the player gets three or more scatter symbols in a free spin, then he will get an additional 15 free spins.
As shown in the scatter section above, the probability of three or more scatters is 0.007256, or 1 in 138. After considering the probability if retriggering the bonus in a free spin, the final number of free spins averages 16.832022. The rules say there is a cap of 180 free spins but that probability of that is truly negligible.
We can see from the final return table that the return from line pays is 0.586609. From the scatter section, we see the return from scatter pays is 0.175078. If it weren't for the rule of the jackpot (five Cleopatras) not tripling, the value per free spin would be 3×(0.520470+0.175078) = 2.086643. We need to subtract from that two times the value of the jackpot, because the jackpot is not tripled, which is 2× 0.000602 = 0.001204. Thus, the value of a free spin is 2.086643  0.001204 = 2.085439.
Multiplying the average number of free spins and the average win per free spin gives us the average bonus win of 16.832022 × 2.085439 = 35.102153.
The overall return from the bonus is the bonus probability times the average bonus win, which equals 0.007256 × 35.102153 = 0.254704.
Summary of Return
The final table below shows the return of each win category in Cleopatra. The lower right cell shows a total return for the game of 95.025%. The rule screens on the online game claim a return of 95.02% (perhaps they round down). Now you understand where it comes from.
Cleopatra Summary
Type of Win  Return 

Line pays in initial spins  52.047% 
Scatter pays in initial spins  17.508% 
Bonus  25.470% 
Total  95.025% 
Hit Frequency and Standard Deviation
The following table shows the hit frequency and standard deviation for all possible number of paylines bet from 1 to 20. The standard deviation is expressed two ways, as follows:
 Total standard deviation: This is the total standard deviation if betting one credit per line.
 Relative standard deviation: This is the ratio of total standard deviation to the amount bet. In other words, the standard deviation relative to the total amount bet.
The table below is based on a 20 random simulations, one for each number of lines bet, of 100 million spins in each simulation. The data was kindly provided by Casino Guru.
Hit Frequency and Standard Deviation
Paylines  Hit Frequency 
Total Std. Dev. 
Relative Std. Dev. 

1  11.36%  13.45  13.45 
2  14.92%  19.32  9.66 
3  18.56%  25.37  8.46 
4  21.66%  31.08  7.77 
5  24.70%  36.02  7.20 
6  25.62%  39.79  6.63 
7  26.55%  43.81  6.26 
8  29.36%  48.43  6.05 
9  31.81%  53.26  5.92 
10  32.52%  57.42  5.74 
11  33.20%  61.09  5.55 
12  33.87%  65.78  5.48 
13  34.63%  69.88  5.38 
14  35.26%  75.00  5.36 
15  35.86%  79.10  5.27 
16  35.86%  84.37  5.27 
17  35.87%  89.56  5.27 
18  35.84%  93.16  5.18 
19  35.85%  98.59  5.19 
20  35.88%  103.51  5.18 
Internal Links
 My main slot machine page.
 Deconstructing Jackpot Party analysis of the video slot machine.
 Deconstructing Megabucks.
 Deconstructing the Atkins Diet slot machine. (PDF)
 Analysis of Lucky Larry's Lobstermania.
 Deconstructing Hexbreaker.
 Deconstructing Lionfish.
Acknowledgements
 for the reel strips as well as the data for calculating the standard deviation and hit frequency.
 for the free game, from which my screenshots were taken.
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