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Last Updated: August 19, 2016

Deconstructing Hot Roll

Introduction

Hot Roll is a bonus feature added to various 3-reel slot machines by maker IGT. In the case of this analysis, it is based on the classic game Triple Double Diamond. If the player gets the Hot Roll icon on all three reels, then he will play a craps-based bonus game. In this game, the player will keep throwing two dice and winning money, until he rolls a seven, ending the bonus.

I played 284 spins of this game at the Golden Nugget on January 2, 2014. As I played, I recorded my play and then uploaded the video to YouTube. This page documents my results and attempts to reverse engineer the game to show you how it might have been programmed. For this type of game, with weighted reels, 284 spins is not enough to know exactly how IGT programmed it. What you see in this page is my best educated guess.

Rules

Hot Roll is a 20-line 3-reel slot machine.

Following are the rules for the base game.

  1. The player must play all 20 lines.
  2. The player may bet one to ten coins per line.
  3. The pay table for each line is as follows:

    Hot Roll Pay Table

    Event Pays
    Three triple diamonds (20th payline) 10,000
    Three triple diamonds (payline 1 to 19) 1,200
    Any three wilds 1,000
    Three red sevens 100
    Three purple sevens 80
    Mixed sevens 50
    Three 3-bar 30
    Three 2-bar 20
    Three 1-bar 10
    Three cherries 10
    Three mixed bars 5
    Any two cherries 5
    Any one cherry 2

    Three double diamonds would be scored as any three diamonds.

    The win of 10,000 is on a "for one" basis, relative to the total amount bet. The pay table itself mentions a win of 100,000 coins, but this is based on a ten-coin bet. If the player bets less than 10 credits per payline, then he will win 1,200 only for three Triple Diamonds on the 20th payline.

  4. The Double Diamond and Triple Diamond symbols are wild and may substitute for any other symbol, except the Hot Roll or another wild, on the same pay line. Note that the wilds may not substitute for a cherry unless there is a natural cherry on the same line.
  5. A double diamond will double any win on that pay line. Likewise, a triple diamond will triple any win on that pay line.
  6. A combination of two wilds will both multiply. To be specific, two Double Diamonds will multiply a win by 4, one Double Diamond and one Triple Diamond will multiply a win by 6, and two Triple Diamonds will multiply a win by 9.
  7. All wins for three diamonds do not get multiplied.
  8. If the player gets three Hot Reel symbols anywhere on the screen, then he shall play the bonus game.
  9. If the player gets three Triple Diamonds on the 20th pay line, and makes a maximum bet of 200 credits, then he shall be paid 10,000 for 1 on that line, instead of the usual 1,200 for 1 for three Triple Diamonds.
  10. The pay lines are drawn as follows:

    Hot Roll Paylines

    Line Reel 1 Reel 2 Reel 3
    1 middle middle middle
    2 top top top
    3 bottom bottom bottom
    4 top middle bottom
    5 bottom middle top
    6 middle top middle
    7 middle bottom middle
    8 bottom middle bottom
    9 top middle top
    10 top middle middle
    11 bottom middle middle
    12 middle top top
    13 middle bottom bottom
    14 top top middle
    15 bottom bottom middle
    16 middle middle top
    17 middle middle bottom
    18 top bottom top
    19 bottom top bottom
    20 top top bottom

Rules for the bonus game.

  1. The player shall keep rolling a pair of dice until he rolls a seven.
  2. Per Nevada law, the outcome of each die is independent and each side has a 1/6 probability, as with real dice.
  3. If the player gets a seven on the first roll, then he shall win 7 times the total amount bet.
  4. Otherwise, the player shall win the amount in the following table. Wins are based on the total amount bet. The player will keep all wins until he rolls a bonus-ending seven.

    Hot Roll Bonus

    Roll Pays
    2 or 12 10
    3 or 11 6
    4 or 10 4
    5 or 9 3
    6 or 8 2

Data

Based on my 284 spins, I put together the order of the reel stripping and count how often each reel stopped on each position. The following table shows my results.

Hot Roll Data

Reel 1 Reel 2 Reel 3
Symbol Count Symbol Count Symbol Count
blank 1 blank 2 blank 1
Double Diamond 2 Double Diamond 4 Double Diamond 1
blank 13 blank 2 blank 1
Triple Diamond 1 Triple Diamond 4 Triple Diamond 4
blank 6 blank 3 blank 2
Purple 7 6 2-bar 5 1-bar 46
blank 4 blank 1 blank 9
Cherry 6 Cherry 5 Red 7 19
blank 5 blank 8 blank 14
3-bar 8 1-bar 19 2-bar 19
blank 19 blank 13 blank 11
3-bar 28 Purple 7 21 3-bar 14
blank 20 blank 21 blank 12
Hot Roll 28 3-bar 23 Hot Roll 22
blank 21 blank 22 blank 16
2-bar 20 Hot Roll 31 Purple 7 14
blank 29 blank 22 blank 12
Red 7 26 1-bar 30 1-bar 8
blank 21 blank 30 blank 47
3-bar 10 1-bar 11 1-bar 9
blank 3 blank 4 blank 3
1-bar 7 Red 7 3 Cherry 0
Total 284 Total 284 Total 284

Triple Double Diamond Analysis

The way that three-reel slot machines usually work is to pick one random number for each reel and then map it to a position on the reel strips, according to how each stop on each reel is weighted. It is not unusual for the total number stops to sum to an even power of 2. I don't know the total number of stops for this game but for the sake of example I will assume it is 28 = 256.

The following table is my best estimate of the actual reel weights. I combined trying to keep the proportions the same as the actual data and trying to achieve what I felt was a believable return for the game. You may recall that in my Las Vegas penny slot survey that the Golden Nugget came in 48th place out of 71, with an average return of 90.85%.

Hot Roll Data — Hypothetical Reel Weights

Reel 1 Reel 2 Reel 3
Symbol Count Symbol Count Symbol Count
blank 1 blank 2 blank 1
Double Diamond 2 Double Diamond 3 Double Diamond 1
blank 12 blank 2 blank 1
Triple Diamond 1 Triple Diamond 3 Triple Diamond 4
blank 5 blank 3 blank 2
Purple 7 5 2-bar 4 1-bar 41
blank 4 blank 1 blank 8
Cherry 5 Cherry 5 Red 7 17
blank 5 blank 7 blank 12
3-bar 7 1-bar 17 2-bar 17
blank 17 blank 12 blank 10
3-bar 25 Purple 7 19 3-bar 12
blank 18 blank 19 blank 11
Hot Roll 25 3-bar 21 Hot Roll 20
blank 19 blank 20 blank 14
2-bar 18 Hot Roll 28 Purple 7 13
blank 26 blank 20 blank 11
Red 7 24 1-bar 27 1-bar 7
blank 19 blank 27 blank 42
3-bar 9 1-bar 10 1-bar 8
blank 3 blank 3 blank 3
1-bar 6 Red 7 3 Cherry 1
Total 256 Total 256 Total 256

The way the game would be programmed, based on these weights, would be to choose three random integers from 0 to 255 (programmers always start counting at zero). It would then map those numbers to a specific stop on the reel according to the following ranges for each stop. The game would then stop each reel on the predestined position in the middle row.

Hot Roll — Reel Stop Ranges

Reel 1 Reel 2 Reel 3
Symbol Range Symbol Range Symbol Range
blank 0 blank 0 to 1 blank 0
Double Diamond 1 to 2 Double Diamond 2 to 4 Double Diamond 1
blank 3 to 14 blank 5 to 6 blank 2
Triple Diamond 15 Triple Diamond 7 to 9 Triple Diamond 3 to 6
blank 16 to 20 blank 10 to 12 blank 7 to 8
Purple 7 21 to 25 2-bar 13 to 16 1-bar 9 to 49
blank 26 to 29 blank 17 blank 50 to 57
Cherry 30 to 34 Cherry 18 to 22 Red 7 58 to 74
blank 35 to 39 blank 23 to 29 blank 75 to 86
3-bar 40 to 46 1-bar 30 to 46 2-bar 87 to 103
blank 47 to 63 blank 47 to 58 blank 104 to 113
3-bar 64 to 88 Purple 7 59 to 77 3-bar 114 to 125
blank 89 to 106 blank 78 to 96 blank 126 to 136
Hot Roll 107 to 131 3-bar 97 to 117 Hot Roll 137 to 156
blank 132 to 150 blank 118 to 137 blank 157 to 170
2-bar 151 to 168 Hot Roll 138 to 165 Purple 7 171 to 183
blank 169 to 194 blank 166 to 185 blank 184 to 194
Red 7 195 to 218 1-bar 186 to 212 1-bar 195 to 201
blank 219 to 237 blank 213 to 239 blank 202 to 243
3-bar 238 to 246 1-bar 240 to 249 1-bar 244 to 251
blank 247 to 249 blank 250 to 252 blank 252 to 254
1-bar 250 to 255 Red 7 253 to 255 Cherry 255

Let's look at an example. Suppose the random numbers were as follows:
  • Reel 1: 222
  • Reel 2: 0
  • Reel 3: 175

From the above table, you can see the 222 for reel 1 gets mapped to the blank between the red 7 and the 3-bar. The 0 for reel 2 gets mapped to the first blank, above the double diamond and below the red 7. The reels wrap around so the red 7 at the bottom is above the blank at the top of the list. The 175 for reel 3 would be mapped to the purple 7. The outcome would then look as follows.

Based on the weights above, the following table shows the number of combinations of each win by its multiplier for reel 1.

Payline 1 Probability Combinations

Win Pays Natural x2 x3 x4 x6 x9 Total
Three triple diamonds 1,200 12 - - - - - 12
Any three wilds 1,000 78 - - - - - 78
Three red sevens 100 1,224 1,398 1,563 180 540 351 5,256
Three purple sevens 80 1,235 784 822 131 363 175 3,510
Mixed sevens 50 16,681 2,386 3,437 - - - 22,504
Three 3-bar 30 10,332 2,841 5,172 237 912 612 20,106
Three 2-bar 20 1,224 1,126 1,274 164 459 283 4,530
Three 1-bar 10 18,144 7,380 5,328 462 1,080 456 32,850
Three cherries 10 25 50 120 31 129 83 438
Three mixed bars 5 406,775 23,793 32,056 - - - 462,624
Any two cherries 5 8,715 9,949 15,849 - - - 34,513
Any one cherry 2 677,010 - - - - - 677,010
Total 1,141,455 49,707 65,621 1,205 3,483 1,960 1,263,431

The next table shows the return combinations for each win. Each cell in the main body of the table is the product of the win, multiplier, and number of combinations from the table above. The total number of possible combinations is 2563 = 16,777,216. Dividing the total return combinations in the lower right cell of 10,717,885 by the total possible combinations of 16,777,216 we get 63.88%. So, for the one credit bet on the center payline, the player can expect to get back 0.6388 credits, not counting the bonus.

Payline 1 Return Combinations

Win Pays Natural x2 x3 x4 x6 x9 Total
Three triple diamonds 1,200 14,400 - - - - - 14,400
Any three wilds 1,000 78,000 - - - - - 78,000
Three red sevens 100 122,400 279,600 468,900 72,000 324,000 315,900 1,582,800
Three purple sevens 80 98,800 125,440 197,280 41,920 174,240 126,000 763,680
Mixed sevens 50 834,050 238,600 515,550 - - - 1,588,200
Three 3-bar 30 309,960 170,460 465,480 28,440 164,160 165,240 1,303,740
Three 2-bar 20 24,480 45,040 76,440 13,120 55,080 50,940 265,100
Three 1-bar 10 181,440 147,600 159,840 18,480 64,800 41,040 613,200
Three cherries 10 250 1,000 3,600 1,240 7,740 7,470 21,300
Three mixed bars 5 2,033,875 237,930 480,840 - - - 2,752,645
Any two cherries 5 43,575 99,490 237,735 - - - 380,800
Any one cherry 2 1,354,020 - - - - - 1,354,020
Total 5,095,250 1,345,160 2,605,665 175,200 790,020 706,590 10,717,885

Since the reel stops are weighted, this analysis must be repeated for each payline. To prevent this page from getting too long for the other 19 paylines I will just present the return in the following table. Note the bottom right cell shows an average return for the base game of 68.69%.

Payline Returns

Payline Return
1 63.88%
2 71.95%
3 59.04%
4 75.87%
5 80.55%
6 58.29%
7 48.95%
8 76.68%
9 79.28%
10 101.65%
11 103.87%
12 46.34%
13 37.66%
14 91.58%
15 78.90%
16 50.32%
17 48.36%
18 60.83%
19 69.86%
20 69.87%
Average 68.69%

The next table shows the number of combinations for each type of win over all 20 paylines.

Win Combinations over all 20 Paylines

Win Pays Natural x2 x3 x4 x6 x9 Total
Three triple diamonds 10,000 15 - - - - - 15
Three triple diamonds 1,200 672 - - - - - 672
Any three wilds 1,000 3,045 - - - - - 3,045
Three red sevens 100 15,307 18,411 22,128 4,515 11,843 7,341 79,545
Three purple sevens 80 20,374 27,735 32,131 5,250 14,364 8,910 108,764
Mixed sevens 50 238,828 51,474 65,301 - - - 355,603
Three 3-bar 30 177,518 60,873 87,192 6,850 20,299 13,670 366,402
Three 2-bar 20 15,555 18,177 22,460 4,545 12,016 7,564 80,317
Three 1-bar 10 186,654 291,680 336,601 19,088 49,727 30,561 914,311
Three cherries 10 720 1,690 2,409 1,167 3,532 2,260 11,778
Three mixed bars 5 7,083,199 560,024 721,851 - - - 8,365,074
Any two cherries 5 185,229 296,674 371,898 - - - 853,801
Any one cherry 2 13,361,573 - - - - - 13,361,573
Total 21,288,689 1,326,738 1,661,971 41,415 111,781 70,306 24,500,900

The following table shows the expected number of each kind of win over all 20 paylines. The lower right cell shows the player can expect 1.46 wins per bet.

Expected wins over all 20 Paylines

Win Pays Natural x2 x3 x4 x6 x9 Total
Three triple diamonds 10000 0.000001 0.000001
Three triple diamonds 1200 0.000040 0.000040
Any three wilds 1000 0.000181 0.000181
Three red sevens 100 0.000912 0.001097 0.001319 0.000269 0.000706 0.000438 0.004741
Three purple sevens 80 0.001214 0.001653 0.001915 0.000313 0.000856 0.000531 0.006483
Mixed sevens 50 0.014235 0.003068 0.003892 0.021196
Three 3-bar 30 0.010581 0.003628 0.005197 0.000408 0.001210 0.000815 0.021839
Three 2-bar 20 0.000927 0.001083 0.001339 0.000271 0.000716 0.000451 0.004787
Three 1-bar 10 0.011125 0.017385 0.020063 0.001138 0.002964 0.001822 0.054497
Three cherries 10 0.000043 0.000101 0.000144 0.000070 0.000211 0.000135 0.000702
Three mixed bars 5 0.422192 0.033380 0.043026 0.498597
Any two cherries 5 0.011041 0.017683 0.022167 0.050891
Any one cherry 2 0.796412 0.000000 0.000000 0.796412
Total 1.268905 0.079080 0.099061 0.002469 0.006663 0.004191 1.460367

The following table shows the expected return from each kind of win over all 20 paylines. The lower right cell shows the player can expect 13.737592 credits from line pays per bet. Dividing that by a 20-unit bet, the return from the base game is 68.688%.

Expected return over all 20 Paylines

Win Pays Natural x2 x3 x4 x6 x9 Total
Three triple diamonds 10,000 0.008941 0.008941
Three triple diamonds 1,200 0.048065 0.048065
Any three wilds 1,000 0.181496 0.181496
Three red sevens 100 0.091237 0.219476 0.395679 0.107646 0.423539 0.393802 1.631379
Three purple sevens 80 0.097151 0.264502 0.459638 0.100136 0.410957 0.382376 1.714759
Mixed sevens 50 0.711763 0.306809 0.583836 1.602408
Three 3-bar 30 0.317427 0.217699 0.467734 0.048995 0.217785 0.219995 1.489635
Three 2-bar 20 0.018543 0.043337 0.080323 0.021672 0.085945 0.081153 0.330974
Three 1-bar 10 0.111254 0.347710 0.601889 0.045509 0.177838 0.163942 1.448143
Three cherries 10 0.000429 0.002015 0.004308 0.002782 0.012631 0.012124 0.034289
Three mixed bars 5 2.110958 0.333800 0.645385 3.090143
Any two cherries 5 0.055203 0.176831 0.332503 0.564537
Any one cherry 2 1.592824 0.000000 0.000000 1.592824
Total 5.345290 1.912179 3.571296 0.326741 1.328695 1.253391 13.737592

The next table shows the frequency of each win amount, after applying the multiplier, over all 20 paylines. The lower right cell shows a total win of 13.737592. Dividing this by 20, the total amount bet, results in a return for the base game of 68.688%.

Win Summary over all 20 Paylines

Win Count Expected Return
10,000 15 0.00000089 0.008941
1,200 672 0.00004005 0.048065
1,000 3,045 0.00018150 0.181496
900 7,341 0.00043756 0.393802
720 8,910 0.00053108 0.382376
600 11,843 0.00070590 0.423539
480 14,364 0.00085616 0.410957
400 4,515 0.00026911 0.107646
320 5,250 0.00031292 0.100136
300 22,128 0.00131893 0.395679
270 13,670 0.00081480 0.219995
240 32,131 0.00191516 0.459638
200 18,411 0.00109738 0.219476
180 27,863 0.00166076 0.298938
160 27,735 0.00165313 0.264502
150 65,301 0.00389224 0.583836
120 18,866 0.00112450 0.134940
100 66,781 0.00398046 0.398046
90 120,013 0.00715333 0.643800
80 24,919 0.00148529 0.118823
60 136,592 0.00814152 0.488491
50 238,828 0.01423526 0.711763
40 38,432 0.00229073 0.091629
30 516,528 0.03078747 0.923624
20 308,925 0.01841336 0.368267
15 1,093,749 0.06519252 0.977888
10 1,044,072 0.06223154 0.622315
5 7,268,428 0.43323207 2.166160
2 13,361,573 0.79641181 1.592824
0 311,043,420 18.53963256 0.000000
Total 335,544,320 20.00000000 13.737592

Bonus Analysis

The rules for the bonus are stated in the rules section above. Let's start the analysis of the bonus by solving for the average win per roll, assuming it isn't a seven. The table below answers that question. The bottom right cell shows an average win of 3.733333, assuming no seven.

Hot Roll Bonus Analysis

Total Win Weight Probability Return
2 10 1 0.033333 0.333333
3 6 2 0.066667 0.400000
4 4 3 0.100000 0.400000
5 3 4 0.133333 0.400000
6 2 5 0.166667 0.333333
8 2 5 0.166667 0.333333
9 3 4 0.133333 0.400000
10 4 3 0.100000 0.400000
11 6 2 0.066667 0.400000
12 10 1 0.033333 0.333333
Total 30 1.000000 3.733333

Next, what is the average number of rolls? If the probability of an event is p then it will take on average 1/p trials for it to happen. The probability of rolling a seven is 1/6, so it takes on average six rolls to happen. However, the player doesn't win anything for the actual roll of the seven, so there are five paying rolls before the seven.

There is also a consolation prize of 7 for rolling a seven on the first roll. The value of that is (1/6) × 7 = 1.166667. So, the average win per bonus is 1.166667 + 5 × 3.733333 = 19.833333.

As a reminder, the bonus is triggered if the player gets three Hot Roll symbols anywhere on the screen. To determine the probability of it occurring on each reel, we also need to examine the blank stops immediately above and below the Hot Roll symbol that touch the center payline. For reel 1 there are, 18 (blank) + 25 (Hot Roll) + 19 (blank) = 62 stops that when touching the center payline make the Hot Roll symbol appear anywhere in reel 1, yielding a probability of 62/256 = 0.242188.

The following table shows the probability of a Hot Roll symbol appearing on each of the three reels as well as the product. The lower right cell shows a bonus probability of 1.13%.

Hot Roll Bonus Analysis

Reel Probability
1 0.242188
2 0.265625
3 0.175781
Product 0.011308

The overall return from the bonus is the probability of the bonus times the average win. This product is 0.011308 × 19.833333 = 0.224279.

Final Analysis

After all that, we have shown the return from the base game is 68.688% and the return from the bonus is 22.428% for a total return of 91.116%. If the player bets less than 200 credits, thus losing the max coin incentive, the return drops by 0.039% to 91.077%.

I would like to emphasize that I am not claiming this is the exact return. This page is more for an exercise in slot machine design than solving for the exact return of that one game. To determine the exact return I would need to know the exact reel weights, which is information I do not have.

Video

Video of the 288 spins this analysis is based on.

Acknowledgments

My thanks to Miplet and tringlomane for their help verifying the math above.


Written by: Michael Shackleford

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