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Last Updated: November 26, 2017

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 five-reel video slot with a free spin bonus.

The version this analysis is based on is the one that can be played for free at Vegas Slots Online.

The Reel Strips



Our friends at Casino Guru 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 pay-line, 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 — Non-Doubled



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.

Non-Doubled 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 error-prone 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 non-doubled wins. The lower right cell shows a hit frequency per line of 1.85% for non-doubled wins.

Probabilities — Non-Doubled



Non-Doubled 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 — Non-Doubled



The next table shows the product of each winning probability and what it pays for non-doubled 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 non-doubled line pays.

Non-Doubled 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 non-doubled 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 pay-line. Scatter pay wins are based on the total amount bet, as opposed the bet per pay-line.

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 re-triggering 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



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%



Acknowledgements






Written by: Michael Shackleford

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