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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 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 FreeCasinoGames.net.

## 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

## 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 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.

### 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

- Casino Guru for the reel strips as well as the data for calculating the standard deviation and hit frequency.
- FreeCasinoGames.net for the free game, from which my screenshots were taken.