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craigGA
craigGA
craigGA
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November 29th, 2009 at 5:03:15 PM permalink
"Lock & roll" slot machines are found in certain gaming jurisdictions which require an element of skill. The help screen on these machines show how often each symbol occurs on each of the 3 virtual reels. The skill comes in after the first spin, during which time the player has the option to "lock" any combination of the three reels, and then re-spin. The symbols on the payline after this two spin process map to a pay table like all other slot machines.

Given the following reel and pay table information for a 25cent machine, what is the process for determining the optimal strategy and rate of return? Also, how much does the progressive change the rate of return?

After spending a large portion of my Sunday trying to do this myself, I humbly concede...I was able to calculate the return after the first spin. I just couldn't figure out how to factor in the lock and re-spin aspect of the game.

Thanks!


Reel Weights

Reel 1 Symbols Reel 2 Symbols Reel 3 Symbols
blank
81
83
83
Cherry
4
6
8
Single Bar
19
5
11
Double Bar
14
17
5
Triple Bar
7
8
11
7 Bar
1
5
6
Red Seven
1
3
3
5x Pay
1
1
1
Total Stops
128
128
128



Pay Table

Reel 1 Symbol Reel 2 Symbol Reel 3 Symbol Payout for 1 credit
5x Pay 5x Pay 5x Pay 5000 *see note2
Red Seven Red Seven Red Seven 60
7 Bar 7 Bar 7 Bar 40
Any 7 Any 7 Any 7 30
AnyTwo 5x 25
Triple Bar Triple Bar Triple Bar 20
Double Bar Double Bar Double Bar 10
Cherry Cherry Cherry 10
Single Bar Single Bar Single Bar 5
AnyTwo Cherry 5
Any Bar Any Bar Any Bar 2


Pay Table Notes
  1. Player may bet between 1 and 5 quarters.
  2. 5 coins bet qualifies for progressive jackpot. ($8668 during last visit)
  3. 5x symbol multiplies 5 times a winning combination.
  4. Two 5x symbols multiplies 25 times a winning combination.
  5. All wins multiplied by number of coins bet.
dk
dk
dk
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November 30th, 2009 at 4:53:26 PM permalink
I think I can solve this, but I'm pretty unfamiliar with slots, so I have a couple questions:

1) Does the 7 Bar count towards the "Any Bar" payout?

2) Does the 5x Pay act as a Wild (e.g., Red 7, Red 7, 5x => 60*5 = 300 and Red 7, 5x, 5x => 60*25 = 1500)?

I assume both are true.
The ratio of people to cake is too big.
dwheatley
dwheatley
dwheatley
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November 30th, 2009 at 4:59:57 PM permalink
They are both true
Wisdom is the quality that keeps you out of situations where you would otherwise need it
marksolberg
marksolberg
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November 30th, 2009 at 5:26:35 PM permalink
Quote: dk

I think I can solve this, but I'm pretty unfamiliar with slots, so I have a couple questions:

1) Does the 7 Bar count towards the "Any Bar" payout?



Just to be clear, it also counts towards an any 7 payout.
dk
dk
dk
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November 30th, 2009 at 5:33:11 PM permalink
Removed.
Last edited by: dk on Dec 1, 2009
The ratio of people to cake is too big.
craigGA
craigGA
craigGA
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November 30th, 2009 at 8:43:19 PM permalink
DK - You and I are pretty close! I calculated the probability of winning something after one spin = 0.032257557.

Just cant seem to get past the second spin aspect...I think the first step is deriving the EV of each possible combination after 1 spin, and then building a strategy based on that...and then calculating the probability of winning after each optimal strategy situation...and that adds up to A LOT of scenarios.

Here is the table I have so far:


Paytable Win P After 1 Spin P after 2 spins Return after 1 Spin Return after 2 Spins
Three 5x 5000 4.76837E-07 1.80649E-06 0.002384186 0.009032439
Three Red7 60 4.29153E-06 0.000257492
Three 7bar 40 1.43051E-05 0.000572205
Any Three 7s 30 6.86646E-05 0.002059937
any two 5x 25 0.000117779 0.002944469
Three 3bar 20 0.000293732 0.005874634
Three 2bar 10 0.000567436 0.005674362
Three Cherry 10 9.15527E-05 0.000915527
Three 1Bar 5 0.000498295 0.002491474
any two Cherry 5 0.006023407 0.030117035
Any Three Bar 2 0.021206856 0.042413712
Two 5x + Red7 1500 3.33786E-06 0.00500679
One 5x + Two Red7 300 1.00136E-05 0.003004074
Two 5x + 7bar 1000 5.72205E-06 0.005722046
One 5x + Two 7bar 200 3.14713E-05 0.00629425
One 5x + Two Any 7s 150 2.38419E-05 0.003576279
Two 5x + 3bar 500 1.23978E-05 0.006198883
One 5x + Two 3bar 100 0.00012064 0.01206398
Two 5x + 2bar 250 1.71661E-05 0.004291534
One 5x + Two 2bar 50 0.000114441 0.005722046
Two 5x + 1bar 125 1.66893E-05 0.002086163
One 5x + Two 1bar 25 0.000152111 0.003802776
Two 5x + Cherry 250 8.58307E-06 0.002145767
One 5x + Two Cherry 50 6.10352E-05 0.003051758
One 5x + Two Any Bar 10 0.002793312 0.027933121
0.032257557 0.1866045


Note that the returns are based on a 1 coin bet. Also note that this is my first attempt at anything like this...so all of the above may be totally wrong. :)

I calculated the second spin probability of the jackpot by assuming that the player would only go after the jackpot after any first spin combination of 5Xs and blanks. However, without having the full optimal strategy, I am not sure if that is even correct.
dk
dk
dk
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December 1st, 2009 at 8:40:39 AM permalink
We differ on 8 of the "P After 1 Spin" numbers.

"Any Three 7s", it looks like you made the same error I made the first time. I think you counted "Three Red7" and "Three 7bar" as "Any Three 7s", but these are already accounted for. If you subtract these out, your "Any Three 7s" "P After 1 Spin" should be 5.00679E-05.

The other 7 we differed on were the "One 5x + ..." numbers except for "One 5x + Two Any 7s".

Here is how I calculated the "One 5x + Two Red7":
5x on Reel1 = 1*3*3/1283 = 9/2,097,152
5x on Reel2 = 1*1*3/1283 = 3/2,097,152
5x on Reel3 = 1*1*3/1283 = 3/2,097,152
Summing these 3 numbers, we get 15/2,097,152 ~ 7.15256E-06

I used the same process for the other "One 5x + ..." numbers. Here are the numbers I got:
+ Two 7bar 1.95503E-05
+ Two 3bar 0.000105381
+ Two 2bar 0.000187397
+ Two 1bar 0.000171185
+ Two Cherry 4.95911E-05
+ Two Any Bar 0.001396656 (Note: Be sure to subtract out any possibilities already counted elsewhere).

I'm still working on the second spin...
The ratio of people to cake is too big.
dk
dk
dk
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December 1st, 2009 at 9:56:34 AM permalink
Another question...Is 5x, Cherry, Blank worth 0 or 10?

If 0, I get a first spin payback of 17.086%.
The ratio of people to cake is too big.
dk
dk
dk
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December 1st, 2009 at 12:41:53 PM permalink
I finished my analysis. It's not pretty. I get an overall payback of 75.5363% with optimal play (without the progressive).

Is that even possible/legal?
The ratio of people to cake is too big.
JB
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JB
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December 1st, 2009 at 4:21:29 PM permalink
Note: Figures revised 12/2/2009 4:07 PM

The total return based on the initial spin (the "deal") is 467428/(128^3) = 22.2887%

The total return after the final spin (the "draw") is 3960964784580/(128^6) = 90.0619% when the jackpot is at its minimum (reset) value.

With a jackpot of $8,668 the return was 90.5295%. The break-even jackpot amount is 42,746 times the bet = 213,730 coins = $53,432.50.

Here is the strategy for when the jackpot is at its lowest:

Reel 1 Reel 2 Reel 3 Hold Reel 1? Hold Reel 2? Hold Reel 3?
Blank Blank Blank - - -
Blank Blank Cherry - - Yes
Blank Blank Single Bar - - Yes
Blank Blank Double Bar - - Yes
Blank Blank Triple Bar - - Yes
Blank Blank Seven Bar - - Yes
Blank Blank Seven - - Yes
Blank Blank Wild - - Yes
Blank Cherry Blank - Yes -
Blank Cherry Cherry - Yes Yes
Blank Cherry Single Bar - Yes -
Blank Cherry Double Bar - Yes -
Blank Cherry Triple Bar - Yes -
Blank Cherry Seven Bar - Yes -
Blank Cherry Seven - Yes -
Blank Cherry Wild - Yes Yes
Blank Single Bar Blank - Yes -
Blank Single Bar Cherry - - Yes
Blank Single Bar Single Bar - Yes Yes
Blank Single Bar Double Bar - Yes Yes
Blank Single Bar Triple Bar - Yes Yes
Blank Single Bar Seven Bar - Yes Yes
Blank Single Bar Seven - Yes -
Blank Single Bar Wild - Yes Yes
Blank Double Bar Blank - Yes -
Blank Double Bar Cherry - - Yes
Blank Double Bar Single Bar - Yes Yes
Blank Double Bar Double Bar - Yes Yes
Blank Double Bar Triple Bar - Yes Yes
Blank Double Bar Seven Bar - Yes Yes
Blank Double Bar Seven - Yes -
Blank Double Bar Wild - Yes Yes
Blank Triple Bar Blank - Yes -
Blank Triple Bar Cherry - - Yes
Blank Triple Bar Single Bar - Yes Yes
Blank Triple Bar Double Bar - Yes Yes
Blank Triple Bar Triple Bar - Yes Yes
Blank Triple Bar Seven Bar - Yes Yes
Blank Triple Bar Seven - Yes -
Blank Triple Bar Wild - Yes Yes
Blank Seven Bar Blank - Yes -
Blank Seven Bar Cherry - - Yes
Blank Seven Bar Single Bar - Yes Yes
Blank Seven Bar Double Bar - Yes Yes
Blank Seven Bar Triple Bar - Yes Yes
Blank Seven Bar Seven Bar - Yes Yes
Blank Seven Bar Seven - Yes Yes
Blank Seven Bar Wild - Yes Yes
Blank Seven Blank - Yes -
Blank Seven Cherry - - Yes
Blank Seven Single Bar - Yes -
Blank Seven Double Bar - - Yes
Blank Seven Triple Bar - - Yes
Blank Seven Seven Bar - Yes Yes
Blank Seven Seven - Yes Yes
Blank Seven Wild - Yes Yes
Blank Wild Blank - Yes -
Blank Wild Cherry - Yes Yes
Blank Wild Single Bar - Yes Yes
Blank Wild Double Bar - Yes Yes
Blank Wild Triple Bar - Yes Yes
Blank Wild Seven Bar - Yes Yes
Blank Wild Seven - Yes Yes
Blank Wild Wild - Yes Yes
Cherry Blank Blank Yes - -
Cherry Blank Cherry Yes - Yes
Cherry Blank Single Bar Yes - -
Cherry Blank Double Bar Yes - -
Cherry Blank Triple Bar Yes - -
Cherry Blank Seven Bar Yes - -
Cherry Blank Seven Yes - -
Cherry Blank Wild Yes - Yes
Cherry Cherry Blank Yes Yes -
Cherry Cherry Cherry Yes Yes Yes
Cherry Cherry Single Bar Yes Yes -
Cherry Cherry Double Bar Yes Yes -
Cherry Cherry Triple Bar Yes Yes -
Cherry Cherry Seven Bar Yes Yes -
Cherry Cherry Seven Yes Yes -
Cherry Cherry Wild Yes Yes Yes
Cherry Single Bar Blank Yes - -
Cherry Single Bar Cherry Yes - Yes
Cherry Single Bar Single Bar - Yes Yes
Cherry Single Bar Double Bar Yes - -
Cherry Single Bar Triple Bar Yes - -
Cherry Single Bar Seven Bar Yes - -
Cherry Single Bar Seven Yes - -
Cherry Single Bar Wild Yes - Yes
Cherry Double Bar Blank Yes - -
Cherry Double Bar Cherry Yes - Yes
Cherry Double Bar Single Bar Yes - -
Cherry Double Bar Double Bar - Yes Yes
Cherry Double Bar Triple Bar Yes - -
Cherry Double Bar Seven Bar Yes - -
Cherry Double Bar Seven Yes - -
Cherry Double Bar Wild Yes - Yes
Cherry Triple Bar Blank Yes - -
Cherry Triple Bar Cherry Yes - Yes
Cherry Triple Bar Single Bar Yes - -
Cherry Triple Bar Double Bar Yes - -
Cherry Triple Bar Triple Bar - Yes Yes
Cherry Triple Bar Seven Bar Yes - -
Cherry Triple Bar Seven Yes - -
Cherry Triple Bar Wild Yes - Yes
Cherry Seven Bar Blank Yes - -
Cherry Seven Bar Cherry Yes - Yes
Cherry Seven Bar Single Bar Yes - -
Cherry Seven Bar Double Bar Yes - -
Cherry Seven Bar Triple Bar Yes - -
Cherry Seven Bar Seven Bar - Yes Yes
Cherry Seven Bar Seven - Yes Yes
Cherry Seven Bar Wild Yes - Yes
Cherry Seven Blank Yes - -
Cherry Seven Cherry Yes - Yes
Cherry Seven Single Bar Yes - -
Cherry Seven Double Bar Yes - -
Cherry Seven Triple Bar Yes - -
Cherry Seven Seven Bar - Yes Yes
Cherry Seven Seven - Yes Yes
Cherry Seven Wild Yes - Yes
Cherry Wild Blank Yes Yes -
Cherry Wild Cherry Yes Yes Yes
Cherry Wild Single Bar Yes Yes -
Cherry Wild Double Bar Yes Yes -
Cherry Wild Triple Bar Yes Yes -
Cherry Wild Seven Bar Yes Yes -
Cherry Wild Seven Yes Yes -
Cherry Wild Wild Yes Yes Yes
Single Bar Blank Blank Yes - -
Single Bar Blank Cherry - - Yes
Single Bar Blank Single Bar Yes - Yes
Single Bar Blank Double Bar Yes - Yes
Single Bar Blank Triple Bar Yes - Yes
Single Bar Blank Seven Bar Yes - Yes
Single Bar Blank Seven - - Yes
Single Bar Blank Wild - - Yes
Single Bar Cherry Blank - Yes -
Single Bar Cherry Cherry - Yes Yes
Single Bar Cherry Single Bar - Yes -
Single Bar Cherry Double Bar - Yes -
Single Bar Cherry Triple Bar - Yes -
Single Bar Cherry Seven Bar - Yes -
Single Bar Cherry Seven - Yes -
Single Bar Cherry Wild - Yes Yes
Single Bar Single Bar Blank Yes Yes -
Single Bar Single Bar Cherry Yes Yes -
Single Bar Single Bar Single Bar Yes Yes Yes
Single Bar Single Bar Double Bar Yes Yes Yes
Single Bar Single Bar Triple Bar Yes Yes Yes
Single Bar Single Bar Seven Bar Yes Yes Yes
Single Bar Single Bar Seven Yes Yes -
Single Bar Single Bar Wild Yes Yes Yes
Single Bar Double Bar Blank Yes Yes -
Single Bar Double Bar Cherry - - Yes
Single Bar Double Bar Single Bar Yes Yes Yes
Single Bar Double Bar Double Bar Yes Yes Yes
Single Bar Double Bar Triple Bar Yes Yes Yes
Single Bar Double Bar Seven Bar Yes Yes Yes
Single Bar Double Bar Seven Yes Yes -
Single Bar Double Bar Wild - Yes Yes
Single Bar Triple Bar Blank Yes Yes -
Single Bar Triple Bar Cherry - - Yes
Single Bar Triple Bar Single Bar Yes Yes Yes
Single Bar Triple Bar Double Bar Yes Yes Yes
Single Bar Triple Bar Triple Bar - Yes Yes
Single Bar Triple Bar Seven Bar Yes Yes Yes
Single Bar Triple Bar Seven Yes Yes -
Single Bar Triple Bar Wild - Yes Yes
Single Bar Seven Bar Blank Yes Yes -
Single Bar Seven Bar Cherry - - Yes
Single Bar Seven Bar Single Bar Yes Yes Yes
Single Bar Seven Bar Double Bar Yes Yes Yes
Single Bar Seven Bar Triple Bar Yes Yes Yes
Single Bar Seven Bar Seven Bar - Yes Yes
Single Bar Seven Bar Seven - Yes Yes
Single Bar Seven Bar Wild - Yes Yes
Single Bar Seven Blank - Yes -
Single Bar Seven Cherry - - Yes
Single Bar Seven Single Bar Yes - Yes
Single Bar Seven Double Bar Yes - Yes
Single Bar Seven Triple Bar Yes - Yes
Single Bar Seven Seven Bar - Yes Yes
Single Bar Seven Seven - Yes Yes
Single Bar Seven Wild - Yes Yes
Single Bar Wild Blank - Yes -
Single Bar Wild Cherry - Yes Yes
Single Bar Wild Single Bar Yes Yes Yes
Single Bar Wild Double Bar - Yes Yes
Single Bar Wild Triple Bar - Yes Yes
Single Bar Wild Seven Bar - Yes Yes
Single Bar Wild Seven - Yes Yes
Single Bar Wild Wild - Yes Yes
Double Bar Blank Blank Yes - -
Double Bar Blank Cherry - - Yes
Double Bar Blank Single Bar Yes - Yes
Double Bar Blank Double Bar Yes - Yes
Double Bar Blank Triple Bar Yes - Yes
Double Bar Blank Seven Bar Yes - Yes
Double Bar Blank Seven Yes - -
Double Bar Blank Wild Yes - Yes
Double Bar Cherry Blank - Yes -
Double Bar Cherry Cherry - Yes Yes
Double Bar Cherry Single Bar - Yes -
Double Bar Cherry Double Bar Yes - Yes
Double Bar Cherry Triple Bar - Yes -
Double Bar Cherry Seven Bar - Yes -
Double Bar Cherry Seven - Yes -
Double Bar Cherry Wild - Yes Yes
Double Bar Single Bar Blank Yes Yes -
Double Bar Single Bar Cherry - - Yes
Double Bar Single Bar Single Bar Yes Yes Yes
Double Bar Single Bar Double Bar Yes - Yes
Double Bar Single Bar Triple Bar Yes Yes Yes
Double Bar Single Bar Seven Bar Yes Yes Yes
Double Bar Single Bar Seven Yes Yes -
Double Bar Single Bar Wild Yes - Yes
Double Bar Double Bar Blank Yes Yes -
Double Bar Double Bar Cherry Yes Yes -
Double Bar Double Bar Single Bar Yes Yes Yes
Double Bar Double Bar Double Bar Yes Yes Yes
Double Bar Double Bar Triple Bar Yes Yes Yes
Double Bar Double Bar Seven Bar Yes Yes Yes
Double Bar Double Bar Seven Yes Yes -
Double Bar Double Bar Wild Yes Yes Yes
Double Bar Triple Bar Blank Yes Yes -
Double Bar Triple Bar Cherry - - Yes
Double Bar Triple Bar Single Bar Yes Yes Yes
Double Bar Triple Bar Double Bar Yes - Yes
Double Bar Triple Bar Triple Bar - Yes Yes
Double Bar Triple Bar Seven Bar Yes Yes Yes
Double Bar Triple Bar Seven Yes Yes -
Double Bar Triple Bar Wild - Yes Yes
Double Bar Seven Bar Blank Yes Yes -
Double Bar Seven Bar Cherry - - Yes
Double Bar Seven Bar Single Bar Yes Yes Yes
Double Bar Seven Bar Double Bar Yes - Yes
Double Bar Seven Bar Triple Bar Yes Yes Yes
Double Bar Seven Bar Seven Bar - Yes Yes
Double Bar Seven Bar Seven - Yes Yes
Double Bar Seven Bar Wild - Yes Yes
Double Bar Seven Blank Yes - -
Double Bar Seven Cherry - - Yes
Double Bar Seven Single Bar Yes - Yes
Double Bar Seven Double Bar Yes - Yes
Double Bar Seven Triple Bar Yes - Yes
Double Bar Seven Seven Bar - Yes Yes
Double Bar Seven Seven - Yes Yes
Double Bar Seven Wild - Yes Yes
Double Bar Wild Blank Yes Yes -
Double Bar Wild Cherry - Yes Yes
Double Bar Wild Single Bar Yes Yes Yes
Double Bar Wild Double Bar Yes Yes Yes
Double Bar Wild Triple Bar - Yes Yes
Double Bar Wild Seven Bar - Yes Yes
Double Bar Wild Seven - Yes Yes
Double Bar Wild Wild Yes Yes Yes
Triple Bar Blank Blank Yes - -
Triple Bar Blank Cherry - - Yes
Triple Bar Blank Single Bar Yes - Yes
Triple Bar Blank Double Bar Yes - Yes
Triple Bar Blank Triple Bar Yes - Yes
Triple Bar Blank Seven Bar Yes - Yes
Triple Bar Blank Seven Yes - -
Triple Bar Blank Wild Yes - Yes
Triple Bar Cherry Blank - Yes -
Triple Bar Cherry Cherry - Yes Yes
Triple Bar Cherry Single Bar - Yes -
Triple Bar Cherry Double Bar - Yes -
Triple Bar Cherry Triple Bar Yes - Yes
Triple Bar Cherry Seven Bar - Yes -
Triple Bar Cherry Seven - Yes -
Triple Bar Cherry Wild - Yes Yes
Triple Bar Single Bar Blank Yes Yes -
Triple Bar Single Bar Cherry - - Yes
Triple Bar Single Bar Single Bar Yes Yes Yes
Triple Bar Single Bar Double Bar Yes Yes Yes
Triple Bar Single Bar Triple Bar Yes - Yes
Triple Bar Single Bar Seven Bar Yes Yes Yes
Triple Bar Single Bar Seven Yes Yes -
Triple Bar Single Bar Wild Yes - Yes
Triple Bar Double Bar Blank Yes Yes -
Triple Bar Double Bar Cherry - - Yes
Triple Bar Double Bar Single Bar Yes Yes Yes
Triple Bar Double Bar Double Bar Yes Yes Yes
Triple Bar Double Bar Triple Bar Yes - Yes
Triple Bar Double Bar Seven Bar Yes Yes Yes
Triple Bar Double Bar Seven Yes Yes -
Triple Bar Double Bar Wild Yes - Yes
Triple Bar Triple Bar Blank Yes Yes -
Triple Bar Triple Bar Cherry Yes Yes -
Triple Bar Triple Bar Single Bar Yes Yes -
Triple Bar Triple Bar Double Bar Yes Yes -
Triple Bar Triple Bar Triple Bar Yes Yes Yes
Triple Bar Triple Bar Seven Bar Yes Yes -
Triple Bar Triple Bar Seven Yes Yes -
Triple Bar Triple Bar Wild Yes Yes Yes
Triple Bar Seven Bar Blank Yes Yes -
Triple Bar Seven Bar Cherry - - Yes
Triple Bar Seven Bar Single Bar Yes Yes Yes
Triple Bar Seven Bar Double Bar Yes Yes Yes
Triple Bar Seven Bar Triple Bar Yes - Yes
Triple Bar Seven Bar Seven Bar - Yes Yes
Triple Bar Seven Bar Seven - Yes Yes
Triple Bar Seven Bar Wild - Yes Yes
Triple Bar Seven Blank Yes - -
Triple Bar Seven Cherry - - Yes
Triple Bar Seven Single Bar Yes - Yes
Triple Bar Seven Double Bar Yes - Yes
Triple Bar Seven Triple Bar Yes - Yes
Triple Bar Seven Seven Bar - Yes Yes
Triple Bar Seven Seven - Yes Yes
Triple Bar Seven Wild - Yes Yes
Triple Bar Wild Blank Yes Yes -
Triple Bar Wild Cherry - Yes Yes
Triple Bar Wild Single Bar Yes Yes -
Triple Bar Wild Double Bar Yes Yes -
Triple Bar Wild Triple Bar Yes Yes Yes
Triple Bar Wild Seven Bar Yes Yes -
Triple Bar Wild Seven - Yes Yes
Triple Bar Wild Wild Yes Yes Yes
Seven Bar Blank Blank Yes - -
Seven Bar Blank Cherry - - Yes
Seven Bar Blank Single Bar Yes - Yes
Seven Bar Blank Double Bar Yes - Yes
Seven Bar Blank Triple Bar Yes - Yes
Seven Bar Blank Seven Bar Yes - Yes
Seven Bar Blank Seven Yes - Yes
Seven Bar Blank Wild Yes - Yes
Seven Bar Cherry Blank - Yes -
Seven Bar Cherry Cherry - Yes Yes
Seven Bar Cherry Single Bar - Yes -
Seven Bar Cherry Double Bar - Yes -
Seven Bar Cherry Triple Bar - Yes -
Seven Bar Cherry Seven Bar Yes - Yes
Seven Bar Cherry Seven Yes - Yes
Seven Bar Cherry Wild - Yes Yes
Seven Bar Single Bar Blank Yes - -
Seven Bar Single Bar Cherry - - Yes
Seven Bar Single Bar Single Bar Yes Yes Yes
Seven Bar Single Bar Double Bar Yes Yes Yes
Seven Bar Single Bar Triple Bar Yes Yes Yes
Seven Bar Single Bar Seven Bar Yes - Yes
Seven Bar Single Bar Seven Yes - Yes
Seven Bar Single Bar Wild Yes - Yes
Seven Bar Double Bar Blank Yes - -
Seven Bar Double Bar Cherry - - Yes
Seven Bar Double Bar Single Bar Yes Yes Yes
Seven Bar Double Bar Double Bar Yes Yes Yes
Seven Bar Double Bar Triple Bar Yes Yes Yes
Seven Bar Double Bar Seven Bar Yes - Yes
Seven Bar Double Bar Seven Yes - Yes
Seven Bar Double Bar Wild Yes - Yes
Seven Bar Triple Bar Blank Yes - -
Seven Bar Triple Bar Cherry - - Yes
Seven Bar Triple Bar Single Bar Yes Yes Yes
Seven Bar Triple Bar Double Bar Yes Yes Yes
Seven Bar Triple Bar Triple Bar - Yes Yes
Seven Bar Triple Bar Seven Bar Yes - Yes
Seven Bar Triple Bar Seven Yes - Yes
Seven Bar Triple Bar Wild Yes - Yes
Seven Bar Seven Bar Blank Yes Yes -
Seven Bar Seven Bar Cherry Yes Yes -
Seven Bar Seven Bar Single Bar Yes Yes -
Seven Bar Seven Bar Double Bar Yes Yes -
Seven Bar Seven Bar Triple Bar Yes Yes -
Seven Bar Seven Bar Seven Bar Yes Yes Yes
Seven Bar Seven Bar Seven Yes Yes Yes
Seven Bar Seven Bar Wild Yes Yes Yes
Seven Bar Seven Blank Yes Yes -
Seven Bar Seven Cherry Yes Yes -
Seven Bar Seven Single Bar Yes Yes -
Seven Bar Seven Double Bar Yes Yes -
Seven Bar Seven Triple Bar Yes Yes -
Seven Bar Seven Seven Bar Yes Yes Yes
Seven Bar Seven Seven Yes Yes Yes
Seven Bar Seven Wild Yes Yes Yes
Seven Bar Wild Blank Yes Yes -
Seven Bar Wild Cherry - Yes Yes
Seven Bar Wild Single Bar Yes Yes -
Seven Bar Wild Double Bar Yes Yes -
Seven Bar Wild Triple Bar Yes Yes -
Seven Bar Wild Seven Bar Yes Yes Yes
Seven Bar Wild Seven Yes Yes Yes
Seven Bar Wild Wild Yes Yes Yes
Seven Blank Blank Yes - -
Seven Blank Cherry - - Yes
Seven Blank Single Bar Yes - -
Seven Blank Double Bar Yes - -
Seven Blank Triple Bar Yes - -
Seven Blank Seven Bar Yes - Yes
Seven Blank Seven Yes - Yes
Seven Blank Wild Yes - Yes
Seven Cherry Blank - Yes -
Seven Cherry Cherry - Yes Yes
Seven Cherry Single Bar - Yes -
Seven Cherry Double Bar - Yes -
Seven Cherry Triple Bar - Yes -
Seven Cherry Seven Bar Yes - Yes
Seven Cherry Seven Yes - Yes
Seven Cherry Wild - Yes Yes
Seven Single Bar Blank Yes - -
Seven Single Bar Cherry - - Yes
Seven Single Bar Single Bar - Yes Yes
Seven Single Bar Double Bar - Yes Yes
Seven Single Bar Triple Bar - Yes Yes
Seven Single Bar Seven Bar Yes - Yes
Seven Single Bar Seven Yes - Yes
Seven Single Bar Wild Yes - Yes
Seven Double Bar Blank Yes - -
Seven Double Bar Cherry - - Yes
Seven Double Bar Single Bar - Yes Yes
Seven Double Bar Double Bar - Yes Yes
Seven Double Bar Triple Bar - Yes Yes
Seven Double Bar Seven Bar Yes - Yes
Seven Double Bar Seven Yes - Yes
Seven Double Bar Wild Yes - Yes
Seven Triple Bar Blank Yes - -
Seven Triple Bar Cherry - - Yes
Seven Triple Bar Single Bar - Yes Yes
Seven Triple Bar Double Bar - Yes Yes
Seven Triple Bar Triple Bar - Yes Yes
Seven Triple Bar Seven Bar Yes - Yes
Seven Triple Bar Seven Yes - Yes
Seven Triple Bar Wild Yes - Yes
Seven Seven Bar Blank Yes Yes -
Seven Seven Bar Cherry Yes Yes -
Seven Seven Bar Single Bar Yes Yes -
Seven Seven Bar Double Bar Yes Yes -
Seven Seven Bar Triple Bar Yes Yes -
Seven Seven Bar Seven Bar Yes Yes Yes
Seven Seven Bar Seven Yes Yes Yes
Seven Seven Bar Wild Yes Yes Yes
Seven Seven Blank Yes Yes -
Seven Seven Cherry Yes Yes -
Seven Seven Single Bar Yes Yes -
Seven Seven Double Bar Yes Yes -
Seven Seven Triple Bar Yes Yes -
Seven Seven Seven Bar Yes Yes Yes
Seven Seven Seven Yes Yes Yes
Seven Seven Wild Yes Yes Yes
Seven Wild Blank Yes Yes -
Seven Wild Cherry - Yes Yes
Seven Wild Single Bar Yes Yes -
Seven Wild Double Bar Yes Yes -
Seven Wild Triple Bar Yes Yes -
Seven Wild Seven Bar Yes Yes Yes
Seven Wild Seven Yes Yes Yes
Seven Wild Wild Yes Yes Yes
Wild Blank Blank Yes - -
Wild Blank Cherry Yes - Yes
Wild Blank Single Bar Yes - -
Wild Blank Double Bar Yes - Yes
Wild Blank Triple Bar Yes - Yes
Wild Blank Seven Bar Yes - Yes
Wild Blank Seven Yes - Yes
Wild Blank Wild Yes - Yes
Wild Cherry Blank Yes Yes -
Wild Cherry Cherry Yes Yes Yes
Wild Cherry Single Bar Yes Yes -
Wild Cherry Double Bar Yes Yes -
Wild Cherry Triple Bar Yes Yes -
Wild Cherry Seven Bar Yes Yes -
Wild Cherry Seven Yes Yes -
Wild Cherry Wild Yes Yes Yes
Wild Single Bar Blank Yes - -
Wild Single Bar Cherry Yes - Yes
Wild Single Bar Single Bar Yes Yes Yes
Wild Single Bar Double Bar Yes - Yes
Wild Single Bar Triple Bar Yes - Yes
Wild Single Bar Seven Bar Yes - Yes
Wild Single Bar Seven Yes - Yes
Wild Single Bar Wild Yes - Yes
Wild Double Bar Blank Yes - -
Wild Double Bar Cherry Yes - Yes
Wild Double Bar Single Bar Yes Yes Yes
Wild Double Bar Double Bar Yes Yes Yes
Wild Double Bar Triple Bar Yes - Yes
Wild Double Bar Seven Bar Yes - Yes
Wild Double Bar Seven Yes - Yes
Wild Double Bar Wild Yes Yes Yes
Wild Triple Bar Blank Yes Yes -
Wild Triple Bar Cherry Yes - Yes
Wild Triple Bar Single Bar Yes Yes -
Wild Triple Bar Double Bar Yes Yes -
Wild Triple Bar Triple Bar Yes Yes Yes
Wild Triple Bar Seven Bar Yes - Yes
Wild Triple Bar Seven Yes - Yes
Wild Triple Bar Wild Yes Yes Yes
Wild Seven Bar Blank Yes Yes -
Wild Seven Bar Cherry Yes - Yes
Wild Seven Bar Single Bar Yes Yes -
Wild Seven Bar Double Bar Yes Yes -
Wild Seven Bar Triple Bar Yes Yes -
Wild Seven Bar Seven Bar Yes Yes Yes
Wild Seven Bar Seven Yes Yes Yes
Wild Seven Bar Wild Yes Yes Yes
Wild Seven Blank Yes Yes -
Wild Seven Cherry Yes - Yes
Wild Seven Single Bar Yes Yes -
Wild Seven Double Bar Yes Yes -
Wild Seven Triple Bar Yes Yes -
Wild Seven Seven Bar Yes Yes Yes
Wild Seven Seven Yes Yes Yes
Wild Seven Wild Yes Yes Yes
Wild Wild Blank Yes Yes -
Wild Wild Cherry Yes Yes Yes
Wild Wild Single Bar Yes Yes -
Wild Wild Double Bar Yes Yes Yes
Wild Wild Triple Bar Yes Yes Yes
Wild Wild Seven Bar Yes Yes Yes
Wild Wild Seven Yes Yes Yes
Wild Wild Wild Yes Yes Yes
Last edited by: JB on Dec 2, 2009
dk
dk
dk
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December 1st, 2009 at 5:14:31 PM permalink
Removed due to incorrect assumptions.
Last edited by: dk on Dec 2, 2009
The ratio of people to cake is too big.
dk
dk
dk
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December 1st, 2009 at 7:35:49 PM permalink
Quote: dk

Is 5x, Cherry, Blank worth 0 or 10?


JB, are you valuing the above hand at 10? If so, I think that makes most of the difference between our calculations. Otherwise, I don't see much value in holding a Wild and a Cherry. I wasn't sure if it should be 10 or 0, so I went with 0 because it looked like that's the way the OP treated it.
The ratio of people to cake is too big.
JB
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JB
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December 2nd, 2009 at 12:34:45 AM permalink
Note: Figures revised 12/2/2009 4:18 PM

Quote: dk

Quote: dk

Is 5x, Cherry, Blank worth 0 or 10?


JB, are you valuing the above hand at 10? If so, I think that makes most of the difference between our calculations. Otherwise, I don't see much value in holding a Wild and a Cherry. I wasn't sure if it should be 10 or 0, so I went with 0 because it looked like that's the way the OP treated it.



I found the source of our differences; we each had a mistake.

To answer this question: that pays 25, since the wild takes the place of a second cherry, 2 cherries pays 5, and the wild multiplies that by 5x.

My mistake was that I neglected to score 3 natural mixed bars correctly. Recomputing everything, I get a base return of 89.2434%, which is more realistic. I edited my earlier post to reflect the revised figures and strategy.

Here are the total combinations for each prize, on the deal and after the draw:

Prize Deal Draw
0
2028001
3812114901657
2
44474
407597896170
5
13677
92521181125
10
4311
36210518751
20
616
6899392808
25
4970
30789733274
30
105
1248475233
40
30
345233070
50
497
5266871337
60
9
110811393
100
221
2438729525
125
35
203809611
150
50
581956770
200
41
465426849
250
54
566900230
300
15
178902999
500
26
282484810
1000
12
132875948
1500
7
79776911
5000
1
10632633
Totals
2097152
4398046511104
JB
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December 2nd, 2009 at 12:39:44 AM permalink
Note: Figures/strategy revised 12/2/2009 4:15 PM

Here is the strategy for the break-even point when the jackpot exceeds $53,432.50:

Reel 1 Reel 2 Reel 3 Hold Reel 1? Hold Reel 2? Hold Reel 3?
Blank Blank Blank - - -
Blank Blank Cherry - - Yes
Blank Blank Single Bar - - Yes
Blank Blank Double Bar - - Yes
Blank Blank Triple Bar - - Yes
Blank Blank Seven Bar - - Yes
Blank Blank Seven - - Yes
Blank Blank Wild - - Yes
Blank Cherry Blank - Yes -
Blank Cherry Cherry - Yes Yes
Blank Cherry Single Bar - Yes -
Blank Cherry Double Bar - Yes -
Blank Cherry Triple Bar - Yes -
Blank Cherry Seven Bar - Yes -
Blank Cherry Seven - Yes -
Blank Cherry Wild - Yes Yes
Blank Single Bar Blank - Yes -
Blank Single Bar Cherry - - Yes
Blank Single Bar Single Bar - Yes Yes
Blank Single Bar Double Bar - Yes Yes
Blank Single Bar Triple Bar - Yes Yes
Blank Single Bar Seven Bar - Yes Yes
Blank Single Bar Seven - Yes -
Blank Single Bar Wild - - Yes
Blank Double Bar Blank - Yes -
Blank Double Bar Cherry - - Yes
Blank Double Bar Single Bar - Yes Yes
Blank Double Bar Double Bar - Yes Yes
Blank Double Bar Triple Bar - Yes Yes
Blank Double Bar Seven Bar - Yes Yes
Blank Double Bar Seven - Yes -
Blank Double Bar Wild - Yes Yes
Blank Triple Bar Blank - Yes -
Blank Triple Bar Cherry - - Yes
Blank Triple Bar Single Bar - Yes Yes
Blank Triple Bar Double Bar - Yes Yes
Blank Triple Bar Triple Bar - Yes Yes
Blank Triple Bar Seven Bar - Yes Yes
Blank Triple Bar Seven - Yes -
Blank Triple Bar Wild - Yes Yes
Blank Seven Bar Blank - Yes -
Blank Seven Bar Cherry - - Yes
Blank Seven Bar Single Bar - Yes Yes
Blank Seven Bar Double Bar - Yes Yes
Blank Seven Bar Triple Bar - Yes Yes
Blank Seven Bar Seven Bar - Yes Yes
Blank Seven Bar Seven - Yes Yes
Blank Seven Bar Wild - Yes Yes
Blank Seven Blank - Yes -
Blank Seven Cherry - - Yes
Blank Seven Single Bar - Yes -
Blank Seven Double Bar - - Yes
Blank Seven Triple Bar - - Yes
Blank Seven Seven Bar - Yes Yes
Blank Seven Seven - Yes Yes
Blank Seven Wild - Yes Yes
Blank Wild Blank - Yes -
Blank Wild Cherry - Yes Yes
Blank Wild Single Bar - Yes -
Blank Wild Double Bar - Yes Yes
Blank Wild Triple Bar - Yes Yes
Blank Wild Seven Bar - Yes Yes
Blank Wild Seven - Yes Yes
Blank Wild Wild - Yes Yes
Cherry Blank Blank Yes - -
Cherry Blank Cherry Yes - Yes
Cherry Blank Single Bar Yes - -
Cherry Blank Double Bar Yes - -
Cherry Blank Triple Bar Yes - -
Cherry Blank Seven Bar Yes - -
Cherry Blank Seven Yes - -
Cherry Blank Wild Yes - Yes
Cherry Cherry Blank Yes Yes -
Cherry Cherry Cherry Yes Yes Yes
Cherry Cherry Single Bar Yes Yes -
Cherry Cherry Double Bar Yes Yes -
Cherry Cherry Triple Bar Yes Yes -
Cherry Cherry Seven Bar Yes Yes -
Cherry Cherry Seven Yes Yes -
Cherry Cherry Wild Yes Yes Yes
Cherry Single Bar Blank Yes - -
Cherry Single Bar Cherry Yes - Yes
Cherry Single Bar Single Bar - Yes Yes
Cherry Single Bar Double Bar Yes - -
Cherry Single Bar Triple Bar Yes - -
Cherry Single Bar Seven Bar Yes - -
Cherry Single Bar Seven Yes - -
Cherry Single Bar Wild Yes - Yes
Cherry Double Bar Blank Yes - -
Cherry Double Bar Cherry Yes - Yes
Cherry Double Bar Single Bar Yes - -
Cherry Double Bar Double Bar - Yes Yes
Cherry Double Bar Triple Bar Yes - -
Cherry Double Bar Seven Bar Yes - -
Cherry Double Bar Seven Yes - -
Cherry Double Bar Wild Yes - Yes
Cherry Triple Bar Blank Yes - -
Cherry Triple Bar Cherry Yes - Yes
Cherry Triple Bar Single Bar Yes - -
Cherry Triple Bar Double Bar Yes - -
Cherry Triple Bar Triple Bar - Yes Yes
Cherry Triple Bar Seven Bar Yes - -
Cherry Triple Bar Seven Yes - -
Cherry Triple Bar Wild Yes - Yes
Cherry Seven Bar Blank Yes - -
Cherry Seven Bar Cherry Yes - Yes
Cherry Seven Bar Single Bar Yes - -
Cherry Seven Bar Double Bar Yes - -
Cherry Seven Bar Triple Bar Yes - -
Cherry Seven Bar Seven Bar - Yes Yes
Cherry Seven Bar Seven - Yes Yes
Cherry Seven Bar Wild Yes - Yes
Cherry Seven Blank Yes - -
Cherry Seven Cherry Yes - Yes
Cherry Seven Single Bar Yes - -
Cherry Seven Double Bar Yes - -
Cherry Seven Triple Bar Yes - -
Cherry Seven Seven Bar - Yes Yes
Cherry Seven Seven - Yes Yes
Cherry Seven Wild Yes - Yes
Cherry Wild Blank Yes Yes -
Cherry Wild Cherry Yes Yes Yes
Cherry Wild Single Bar Yes Yes -
Cherry Wild Double Bar Yes Yes -
Cherry Wild Triple Bar Yes Yes -
Cherry Wild Seven Bar Yes Yes -
Cherry Wild Seven Yes Yes -
Cherry Wild Wild - Yes Yes
Single Bar Blank Blank Yes - -
Single Bar Blank Cherry - - Yes
Single Bar Blank Single Bar Yes - Yes
Single Bar Blank Double Bar Yes - Yes
Single Bar Blank Triple Bar Yes - Yes
Single Bar Blank Seven Bar Yes - Yes
Single Bar Blank Seven - - Yes
Single Bar Blank Wild - - Yes
Single Bar Cherry Blank - Yes -
Single Bar Cherry Cherry - Yes Yes
Single Bar Cherry Single Bar - Yes -
Single Bar Cherry Double Bar - Yes -
Single Bar Cherry Triple Bar - Yes -
Single Bar Cherry Seven Bar - Yes -
Single Bar Cherry Seven - Yes -
Single Bar Cherry Wild - Yes Yes
Single Bar Single Bar Blank Yes Yes -
Single Bar Single Bar Cherry Yes Yes -
Single Bar Single Bar Single Bar Yes Yes Yes
Single Bar Single Bar Double Bar Yes Yes Yes
Single Bar Single Bar Triple Bar Yes Yes Yes
Single Bar Single Bar Seven Bar Yes Yes Yes
Single Bar Single Bar Seven Yes Yes -
Single Bar Single Bar Wild Yes Yes Yes
Single Bar Double Bar Blank Yes Yes -
Single Bar Double Bar Cherry - - Yes
Single Bar Double Bar Single Bar Yes Yes Yes
Single Bar Double Bar Double Bar Yes Yes Yes
Single Bar Double Bar Triple Bar Yes Yes Yes
Single Bar Double Bar Seven Bar Yes Yes Yes
Single Bar Double Bar Seven Yes Yes -
Single Bar Double Bar Wild - Yes Yes
Single Bar Triple Bar Blank Yes Yes -
Single Bar Triple Bar Cherry - - Yes
Single Bar Triple Bar Single Bar Yes Yes Yes
Single Bar Triple Bar Double Bar Yes Yes Yes
Single Bar Triple Bar Triple Bar - Yes Yes
Single Bar Triple Bar Seven Bar Yes Yes Yes
Single Bar Triple Bar Seven Yes Yes -
Single Bar Triple Bar Wild - Yes Yes
Single Bar Seven Bar Blank Yes Yes -
Single Bar Seven Bar Cherry - - Yes
Single Bar Seven Bar Single Bar Yes Yes Yes
Single Bar Seven Bar Double Bar Yes Yes Yes
Single Bar Seven Bar Triple Bar Yes Yes Yes
Single Bar Seven Bar Seven Bar - Yes Yes
Single Bar Seven Bar Seven - Yes Yes
Single Bar Seven Bar Wild - Yes Yes
Single Bar Seven Blank - Yes -
Single Bar Seven Cherry - - Yes
Single Bar Seven Single Bar Yes - Yes
Single Bar Seven Double Bar Yes - Yes
Single Bar Seven Triple Bar Yes - Yes
Single Bar Seven Seven Bar - Yes Yes
Single Bar Seven Seven - Yes Yes
Single Bar Seven Wild - Yes Yes
Single Bar Wild Blank - Yes -
Single Bar Wild Cherry - Yes Yes
Single Bar Wild Single Bar Yes Yes Yes
Single Bar Wild Double Bar - Yes Yes
Single Bar Wild Triple Bar - Yes Yes
Single Bar Wild Seven Bar - Yes Yes
Single Bar Wild Seven - Yes Yes
Single Bar Wild Wild - Yes Yes
Double Bar Blank Blank Yes - -
Double Bar Blank Cherry - - Yes
Double Bar Blank Single Bar Yes - Yes
Double Bar Blank Double Bar Yes - Yes
Double Bar Blank Triple Bar Yes - Yes
Double Bar Blank Seven Bar Yes - Yes
Double Bar Blank Seven Yes - -
Double Bar Blank Wild Yes - Yes
Double Bar Cherry Blank - Yes -
Double Bar Cherry Cherry - Yes Yes
Double Bar Cherry Single Bar - Yes -
Double Bar Cherry Double Bar Yes - Yes
Double Bar Cherry Triple Bar - Yes -
Double Bar Cherry Seven Bar - Yes -
Double Bar Cherry Seven - Yes -
Double Bar Cherry Wild - Yes Yes
Double Bar Single Bar Blank Yes Yes -
Double Bar Single Bar Cherry - - Yes
Double Bar Single Bar Single Bar Yes Yes Yes
Double Bar Single Bar Double Bar Yes - Yes
Double Bar Single Bar Triple Bar Yes Yes Yes
Double Bar Single Bar Seven Bar Yes Yes Yes
Double Bar Single Bar Seven Yes Yes -
Double Bar Single Bar Wild Yes - Yes
Double Bar Double Bar Blank Yes Yes -
Double Bar Double Bar Cherry Yes Yes -
Double Bar Double Bar Single Bar Yes Yes Yes
Double Bar Double Bar Double Bar Yes Yes Yes
Double Bar Double Bar Triple Bar Yes Yes Yes
Double Bar Double Bar Seven Bar Yes Yes Yes
Double Bar Double Bar Seven Yes Yes -
Double Bar Double Bar Wild Yes Yes Yes
Double Bar Triple Bar Blank Yes Yes -
Double Bar Triple Bar Cherry - - Yes
Double Bar Triple Bar Single Bar Yes Yes Yes
Double Bar Triple Bar Double Bar Yes - Yes
Double Bar Triple Bar Triple Bar - Yes Yes
Double Bar Triple Bar Seven Bar Yes Yes Yes
Double Bar Triple Bar Seven Yes Yes -
Double Bar Triple Bar Wild - Yes Yes
Double Bar Seven Bar Blank Yes Yes -
Double Bar Seven Bar Cherry - - Yes
Double Bar Seven Bar Single Bar Yes Yes Yes
Double Bar Seven Bar Double Bar Yes - Yes
Double Bar Seven Bar Triple Bar Yes Yes Yes
Double Bar Seven Bar Seven Bar - Yes Yes
Double Bar Seven Bar Seven - Yes Yes
Double Bar Seven Bar Wild - Yes Yes
Double Bar Seven Blank Yes - -
Double Bar Seven Cherry - - Yes
Double Bar Seven Single Bar Yes - Yes
Double Bar Seven Double Bar Yes - Yes
Double Bar Seven Triple Bar Yes - Yes
Double Bar Seven Seven Bar - Yes Yes
Double Bar Seven Seven - Yes Yes
Double Bar Seven Wild - Yes Yes
Double Bar Wild Blank - Yes -
Double Bar Wild Cherry - Yes Yes
Double Bar Wild Single Bar Yes Yes Yes
Double Bar Wild Double Bar Yes Yes Yes
Double Bar Wild Triple Bar - Yes Yes
Double Bar Wild Seven Bar - Yes Yes
Double Bar Wild Seven - Yes Yes
Double Bar Wild Wild - Yes Yes
Triple Bar Blank Blank Yes - -
Triple Bar Blank Cherry - - Yes
Triple Bar Blank Single Bar Yes - Yes
Triple Bar Blank Double Bar Yes - Yes
Triple Bar Blank Triple Bar Yes - Yes
Triple Bar Blank Seven Bar Yes - Yes
Triple Bar Blank Seven Yes - -
Triple Bar Blank Wild Yes - Yes
Triple Bar Cherry Blank - Yes -
Triple Bar Cherry Cherry - Yes Yes
Triple Bar Cherry Single Bar - Yes -
Triple Bar Cherry Double Bar - Yes -
Triple Bar Cherry Triple Bar Yes - Yes
Triple Bar Cherry Seven Bar - Yes -
Triple Bar Cherry Seven - Yes -
Triple Bar Cherry Wild - Yes Yes
Triple Bar Single Bar Blank Yes Yes -
Triple Bar Single Bar Cherry - - Yes
Triple Bar Single Bar Single Bar Yes Yes Yes
Triple Bar Single Bar Double Bar Yes Yes Yes
Triple Bar Single Bar Triple Bar Yes - Yes
Triple Bar Single Bar Seven Bar Yes Yes Yes
Triple Bar Single Bar Seven Yes Yes -
Triple Bar Single Bar Wild Yes - Yes
Triple Bar Double Bar Blank Yes Yes -
Triple Bar Double Bar Cherry - - Yes
Triple Bar Double Bar Single Bar Yes Yes Yes
Triple Bar Double Bar Double Bar Yes Yes Yes
Triple Bar Double Bar Triple Bar Yes - Yes
Triple Bar Double Bar Seven Bar Yes Yes Yes
Triple Bar Double Bar Seven Yes Yes -
Triple Bar Double Bar Wild Yes - Yes
Triple Bar Triple Bar Blank Yes Yes -
Triple Bar Triple Bar Cherry Yes Yes -
Triple Bar Triple Bar Single Bar Yes Yes -
Triple Bar Triple Bar Double Bar Yes Yes -
Triple Bar Triple Bar Triple Bar Yes Yes Yes
Triple Bar Triple Bar Seven Bar Yes Yes -
Triple Bar Triple Bar Seven Yes Yes -
Triple Bar Triple Bar Wild Yes Yes Yes
Triple Bar Seven Bar Blank Yes Yes -
Triple Bar Seven Bar Cherry - - Yes
Triple Bar Seven Bar Single Bar Yes Yes Yes
Triple Bar Seven Bar Double Bar Yes Yes Yes
Triple Bar Seven Bar Triple Bar Yes - Yes
Triple Bar Seven Bar Seven Bar - Yes Yes
Triple Bar Seven Bar Seven - Yes Yes
Triple Bar Seven Bar Wild - Yes Yes
Triple Bar Seven Blank Yes - -
Triple Bar Seven Cherry - - Yes
Triple Bar Seven Single Bar Yes - Yes
Triple Bar Seven Double Bar Yes - Yes
Triple Bar Seven Triple Bar Yes - Yes
Triple Bar Seven Seven Bar - Yes Yes
Triple Bar Seven Seven - Yes Yes
Triple Bar Seven Wild - Yes Yes
Triple Bar Wild Blank Yes Yes -
Triple Bar Wild Cherry - Yes Yes
Triple Bar Wild Single Bar Yes Yes -
Triple Bar Wild Double Bar Yes Yes -
Triple Bar Wild Triple Bar Yes Yes Yes
Triple Bar Wild Seven Bar Yes Yes -
Triple Bar Wild Seven - Yes Yes
Triple Bar Wild Wild Yes Yes Yes
Seven Bar Blank Blank Yes - -
Seven Bar Blank Cherry - - Yes
Seven Bar Blank Single Bar Yes - Yes
Seven Bar Blank Double Bar Yes - Yes
Seven Bar Blank Triple Bar Yes - Yes
Seven Bar Blank Seven Bar Yes - Yes
Seven Bar Blank Seven Yes - Yes
Seven Bar Blank Wild Yes - Yes
Seven Bar Cherry Blank - Yes -
Seven Bar Cherry Cherry - Yes Yes
Seven Bar Cherry Single Bar - Yes -
Seven Bar Cherry Double Bar - Yes -
Seven Bar Cherry Triple Bar - Yes -
Seven Bar Cherry Seven Bar Yes - Yes
Seven Bar Cherry Seven Yes - Yes
Seven Bar Cherry Wild - Yes Yes
Seven Bar Single Bar Blank Yes - -
Seven Bar Single Bar Cherry - - Yes
Seven Bar Single Bar Single Bar Yes Yes Yes
Seven Bar Single Bar Double Bar Yes Yes Yes
Seven Bar Single Bar Triple Bar Yes Yes Yes
Seven Bar Single Bar Seven Bar Yes - Yes
Seven Bar Single Bar Seven Yes - Yes
Seven Bar Single Bar Wild Yes - Yes
Seven Bar Double Bar Blank Yes - -
Seven Bar Double Bar Cherry - - Yes
Seven Bar Double Bar Single Bar Yes Yes Yes
Seven Bar Double Bar Double Bar Yes Yes Yes
Seven Bar Double Bar Triple Bar Yes Yes Yes
Seven Bar Double Bar Seven Bar Yes - Yes
Seven Bar Double Bar Seven Yes - Yes
Seven Bar Double Bar Wild Yes - Yes
Seven Bar Triple Bar Blank Yes - -
Seven Bar Triple Bar Cherry - - Yes
Seven Bar Triple Bar Single Bar Yes Yes Yes
Seven Bar Triple Bar Double Bar Yes Yes Yes
Seven Bar Triple Bar Triple Bar - Yes Yes
Seven Bar Triple Bar Seven Bar Yes - Yes
Seven Bar Triple Bar Seven Yes - Yes
Seven Bar Triple Bar Wild Yes - Yes
Seven Bar Seven Bar Blank Yes Yes -
Seven Bar Seven Bar Cherry Yes Yes -
Seven Bar Seven Bar Single Bar Yes Yes -
Seven Bar Seven Bar Double Bar Yes Yes -
Seven Bar Seven Bar Triple Bar Yes Yes -
Seven Bar Seven Bar Seven Bar Yes Yes Yes
Seven Bar Seven Bar Seven Yes Yes Yes
Seven Bar Seven Bar Wild Yes Yes Yes
Seven Bar Seven Blank Yes Yes -
Seven Bar Seven Cherry Yes Yes -
Seven Bar Seven Single Bar Yes Yes -
Seven Bar Seven Double Bar Yes Yes -
Seven Bar Seven Triple Bar Yes Yes -
Seven Bar Seven Seven Bar Yes Yes Yes
Seven Bar Seven Seven Yes Yes Yes
Seven Bar Seven Wild Yes Yes Yes
Seven Bar Wild Blank Yes Yes -
Seven Bar Wild Cherry - Yes Yes
Seven Bar Wild Single Bar Yes Yes -
Seven Bar Wild Double Bar Yes Yes -
Seven Bar Wild Triple Bar Yes Yes -
Seven Bar Wild Seven Bar Yes Yes Yes
Seven Bar Wild Seven Yes Yes Yes
Seven Bar Wild Wild Yes Yes Yes
Seven Blank Blank Yes - -
Seven Blank Cherry - - Yes
Seven Blank Single Bar Yes - -
Seven Blank Double Bar Yes - -
Seven Blank Triple Bar Yes - -
Seven Blank Seven Bar Yes - Yes
Seven Blank Seven Yes - Yes
Seven Blank Wild Yes - Yes
Seven Cherry Blank - Yes -
Seven Cherry Cherry - Yes Yes
Seven Cherry Single Bar - Yes -
Seven Cherry Double Bar - Yes -
Seven Cherry Triple Bar - Yes -
Seven Cherry Seven Bar Yes - Yes
Seven Cherry Seven Yes - Yes
Seven Cherry Wild - Yes Yes
Seven Single Bar Blank Yes - -
Seven Single Bar Cherry - - Yes
Seven Single Bar Single Bar - Yes Yes
Seven Single Bar Double Bar - Yes Yes
Seven Single Bar Triple Bar - Yes Yes
Seven Single Bar Seven Bar Yes - Yes
Seven Single Bar Seven Yes - Yes
Seven Single Bar Wild Yes - Yes
Seven Double Bar Blank Yes - -
Seven Double Bar Cherry - - Yes
Seven Double Bar Single Bar - Yes Yes
Seven Double Bar Double Bar - Yes Yes
Seven Double Bar Triple Bar - Yes Yes
Seven Double Bar Seven Bar Yes - Yes
Seven Double Bar Seven Yes - Yes
Seven Double Bar Wild Yes - Yes
Seven Triple Bar Blank Yes - -
Seven Triple Bar Cherry - - Yes
Seven Triple Bar Single Bar - Yes Yes
Seven Triple Bar Double Bar - Yes Yes
Seven Triple Bar Triple Bar - Yes Yes
Seven Triple Bar Seven Bar Yes - Yes
Seven Triple Bar Seven Yes - Yes
Seven Triple Bar Wild Yes - Yes
Seven Seven Bar Blank Yes Yes -
Seven Seven Bar Cherry Yes Yes -
Seven Seven Bar Single Bar Yes Yes -
Seven Seven Bar Double Bar Yes Yes -
Seven Seven Bar Triple Bar Yes Yes -
Seven Seven Bar Seven Bar Yes Yes Yes
Seven Seven Bar Seven Yes Yes Yes
Seven Seven Bar Wild Yes Yes Yes
Seven Seven Blank Yes Yes -
Seven Seven Cherry Yes Yes -
Seven Seven Single Bar Yes Yes -
Seven Seven Double Bar Yes Yes -
Seven Seven Triple Bar Yes Yes -
Seven Seven Seven Bar Yes Yes Yes
Seven Seven Seven Yes Yes Yes
Seven Seven Wild Yes Yes Yes
Seven Wild Blank Yes Yes -
Seven Wild Cherry - Yes Yes
Seven Wild Single Bar Yes Yes -
Seven Wild Double Bar Yes Yes -
Seven Wild Triple Bar Yes Yes -
Seven Wild Seven Bar Yes Yes Yes
Seven Wild Seven Yes Yes Yes
Seven Wild Wild Yes Yes Yes
Wild Blank Blank Yes - -
Wild Blank Cherry Yes - Yes
Wild Blank Single Bar Yes - -
Wild Blank Double Bar Yes - Yes
Wild Blank Triple Bar Yes - Yes
Wild Blank Seven Bar Yes - Yes
Wild Blank Seven Yes - Yes
Wild Blank Wild Yes - Yes
Wild Cherry Blank Yes Yes -
Wild Cherry Cherry Yes Yes Yes
Wild Cherry Single Bar Yes Yes -
Wild Cherry Double Bar Yes Yes -
Wild Cherry Triple Bar Yes Yes -
Wild Cherry Seven Bar Yes Yes -
Wild Cherry Seven Yes Yes -
Wild Cherry Wild Yes - Yes
Wild Single Bar Blank Yes - -
Wild Single Bar Cherry Yes - Yes
Wild Single Bar Single Bar Yes Yes Yes
Wild Single Bar Double Bar Yes - Yes
Wild Single Bar Triple Bar Yes - Yes
Wild Single Bar Seven Bar Yes - Yes
Wild Single Bar Seven Yes - Yes
Wild Single Bar Wild Yes - Yes
Wild Double Bar Blank Yes - -
Wild Double Bar Cherry Yes - Yes
Wild Double Bar Single Bar Yes - -
Wild Double Bar Double Bar Yes Yes Yes
Wild Double Bar Triple Bar Yes - Yes
Wild Double Bar Seven Bar Yes - Yes
Wild Double Bar Seven Yes - Yes
Wild Double Bar Wild Yes - Yes
Wild Triple Bar Blank Yes Yes -
Wild Triple Bar Cherry Yes - Yes
Wild Triple Bar Single Bar Yes Yes -
Wild Triple Bar Double Bar Yes Yes -
Wild Triple Bar Triple Bar Yes Yes Yes
Wild Triple Bar Seven Bar Yes - Yes
Wild Triple Bar Seven Yes - Yes
Wild Triple Bar Wild Yes - Yes
Wild Seven Bar Blank Yes Yes -
Wild Seven Bar Cherry Yes - Yes
Wild Seven Bar Single Bar Yes Yes -
Wild Seven Bar Double Bar Yes Yes -
Wild Seven Bar Triple Bar Yes Yes -
Wild Seven Bar Seven Bar Yes Yes Yes
Wild Seven Bar Seven Yes Yes Yes
Wild Seven Bar Wild Yes Yes Yes
Wild Seven Blank Yes Yes -
Wild Seven Cherry Yes - Yes
Wild Seven Single Bar Yes Yes -
Wild Seven Double Bar Yes Yes -
Wild Seven Triple Bar Yes Yes -
Wild Seven Seven Bar Yes Yes Yes
Wild Seven Seven Yes Yes Yes
Wild Seven Wild Yes Yes Yes
Wild Wild Blank Yes Yes -
Wild Wild Cherry Yes Yes -
Wild Wild Single Bar Yes Yes -
Wild Wild Double Bar Yes Yes -
Wild Wild Triple Bar Yes Yes -
Wild Wild Seven Bar Yes Yes Yes
Wild Wild Seven Yes Yes Yes
Wild Wild Wild Yes Yes Yes
miplet
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December 2nd, 2009 at 4:34:32 AM permalink
Quote: JB



My mistake was that I neglected to score 3 natural mixed bars correctly. Recomputing everything, I get a base return of 89.2434%, which is more realistic. I edited my earlier post to reflect the revised figures and strategy.

Here are the total combinations for each prize, on the deal and after the draw:

Prize Deal Draw
0
2028106
3811472040826
2
44474
409412415978
5
13677
92530209605
10
4311
36277860575
20
616
6912481576
25
4970
30791948442
40
30
339112110
50
497
5268518057
60
9
109259649
100
221
2442012341
125
35
204151755
150
50
574844706
200
41
462953121
250
54
566953350
300
15
176692695
500
26
282686666
1000
12
132557228
1500
7
79179791
5000
1
10632633
Totals
2097152
4398046511104



You are missing a line for a payout for any 3 sevens of 30. My first spin results are the same exept for the following.
Prize Deal
0
2027823
10
4489
30
105

Edit to add
Ok I found a typo in my program that accounted for my extra 10 payouts, so now they will be the same exept for the any 3 sevens

Prize Deal
0
2028001
30
105
“Man Babes” #AxelFabulous
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December 2nd, 2009 at 9:29:29 AM permalink
JB,

Based on my revised calculations using 5x, Cherry as payout of 25 instead of 0 (no idea were I came up with 10), I get a return of 88.6777%. Compared to your revised optimal strategy, I come up with 13 first spin outcomes where we differ in strategy.

With 2 of one type of seven and one of the other type of seven, you always hold the two of a kind, whereas I always hold all 3. This accounts for 6 differences.

With r1=Seven Bar, r2=any other bar, r3=Blank, you hold r1 & r2, whereas I hold only r1. This accounts for 3 differences.

With r1=Seven, r2=Triple Bar, r3=Blank; r1=Seven, r2=Blank, r3=Triple or Double Bar; and r1=Blank, r2=Seven, r3=Single, you hold the bar whereas I hold the Seven. This accounts for 4 differences.

In all cases my calculations show that my play has a higher expectation; however you came up with a higher payback than I did. Any guess as to what I'm missing?
The ratio of people to cake is too big.
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December 2nd, 2009 at 1:31:30 PM permalink
JB missed a the part of the paytable that says you get 30 for any three 7s.
With r1=Seven Bar, r2=any other bar, r3=Blank I get a return of .608520507812 for holding r1 and 0.59375 for holding r1 and r2, so I agree with dk. Also agree with dk for the other differances. I'll post my chart tomarrow if you want it. I'll have to mess with some coding so it will post here.
“Man Babes” #AxelFabulous
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December 2nd, 2009 at 1:42:36 PM permalink
Quote: miplet

JB missed a the part of the paytable that says you get 30 for any three 7s.


I thought of that, but if he just missed part of the pay table, his return should be lower than mine, not higher.

miplet, I got the same returns for the plays you listed. What overall return did you come up with?
The ratio of people to cake is too big.
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December 2nd, 2009 at 1:53:37 PM permalink
0.900619121371
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December 2nd, 2009 at 4:20:36 PM permalink
Quote: miplet

JB missed a the part of the paytable that says you get 30 for any three 7s.
With r1=Seven Bar, r2=any other bar, r3=Blank I get a return of .608520507812 for holding r1 and 0.59375 for holding r1 and r2, so I agree with dk. Also agree with dk for the other differances. I'll post my chart tomarrow if you want it. I'll have to mess with some coding so it will post here.



Thanks, you're right. I missed both the natural mixed bars and natural mixed 7's in my original version (I did have them with the wild, but missed the naturals). Your 90.0619% figure exactly matches what I calculated after adding in the natural mixed 7's.

My previous posts have been edited to reflect the corrected results.
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December 2nd, 2009 at 4:26:02 PM permalink
Okay, I finally got around to this.

For only the original spin I get:

Hit Frequency = 0.032937
Expected return = 0.226268

Here is a frequency of each win.


Win Hits
5000 1
1500 7
1000 12
500 26
300 15
250 54
200 41
150 80
125 35
100 221
60 9
50 497
40 30
30 294
25 4890
20 616
10 4311
5 13461
2 44474
0 2028078


For the respin my results don't seem reasonable, but here they are:

Hit Frequency = 0.102392
Expected return = 0.57471
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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December 2nd, 2009 at 4:39:24 PM permalink
Craig, your probability of "One 5x + Two Red7" reflects that it happened 21 times. I would argue it can happen 15 ways only as follows:

R7 R7 5X = 1*3*1 = 3
R7 5X R7 = 1*1*3 = 3
5X R7 R7 = 1*3*3 = 9

3+3+9=15

Here is my table for the draw. This is probably wrong somewhere.

Win Hits
5000 2097152
1500 29192576
1000 40442624
500 95449472
300 109463040
250 181163648
200 232694272
150 339635968
125 143913344
100 1236706944
60 86299776
50 2315254144
40 218760960
30 1493475072
25 22682131328
20 4491891712
10 25257941760
5 67061415552
2 324305831168
0 3947722750592
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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December 3rd, 2009 at 8:38:21 PM permalink
Hey everyone,

THANK YOU for all the time put into this. It doesn't get any better than learning by example. I have really enjoyed following the process and seeing what it takes to get to the correct answer.

Again, thanks for the work. I find it extremely interesting to see the math behind slot machines because it is something that doesn't find the light of day very often.

-Craig
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December 4th, 2009 at 3:15:26 AM permalink
You're welcome. I hope this thread doesn't die out. I'd like to write this up on my Odds site. If anybody disagrees with my results, please challenge me.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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December 4th, 2009 at 5:30:22 AM permalink
Quote: Wizard

You're welcome. I hope this thread doesn't die out. I'd like to write this up on my Odds site. If anybody disagrees with my results, please challenge me.


I hereby challenge you... :)

I disagree with your counts on the initial spin for prizes of 0, 2, 5, 25, 30, and 150.

I didn't do all of them, but here's what I come up with for a prize of 30, which can only occur from three mixed sevens:

reel 1 reel 2 reel 3 weight total
red seven red seven seven bar 1*3*6 18
red seven seven bar seven bar 1*5*6 30
red seven seven bar red seven 1*5*3 15
seven bar red seven red seven 1*3*3 9
seven bar red seven seven bar 1*3*6 18
seven bar seven bar red seven 1*5*3 15
Total105


Your count for a prize of 30 is 294, which is 189 combinations too many.

I come up with a return of 22.2887% for the initial spin, and after the second spin I agree with miplet's result of 90.0619%, who I give credit to for being the first one to post what I believe is the correct result.
miplet
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December 4th, 2009 at 9:13:27 AM permalink
Quote: JB


I come up with a return of 22.2887% for the initial spin, and after the second spin I agree with miplet's result of 90.0619%, who I give credit to for being the first one to post what I believe is the correct result.


:+) All in 200 lines of php (including comments, blank lines, lots of echoing to see results ect. ). Of course I'm not much of a programmer, but it sure was fun.
“Man Babes” #AxelFabulous
Wizard
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December 4th, 2009 at 3:36:14 PM permalink
My total wins of 30 were wrong. Here are my new figures for the initial spin.

Win Hits
5000 1
1500 7
1000 12
500 26
300 15
250 54
200 41
150 50
125 35
100 221
60 9
50 497
40 30
30 105
25 4970
20 616
10 4311
5 13677
2 44474
0 2028001


Hit Frequency = 0.03295
Expected return = 0.224079

Where am I wrong now?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
miplet
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December 4th, 2009 at 6:39:14 PM permalink
Quote: Wizard

My total wins of 30 were wrong. Here are my new figures for the initial spin.


Hit Frequency = 0.03295
Expected return = 0.224079

Where am I wrong now?


Your chart is right , but your Expected return is wrong. 467428/(128^3)=.222887039185
“Man Babes” #AxelFabulous
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December 8th, 2009 at 3:21:31 PM permalink
Thanks, you're right. Following is my table for the draw. Any thoughts where this may be wrong?

Win Hits
5000 2097152
1500 28530048
1000 39848192
500 95449472
300 106993152
250 181163648
200 228144640
150 309765376
125 143913344
100 1236706944
60 85147776
50 2315254144
40 214068480
30 854620800
25 23009999488
20 4491891712
10 25224490752
5 67988566656
2 323654613248
0 3947835246080


Hit Frequency = 0.10236619
Expected return = 0.57108399
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
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December 8th, 2009 at 4:59:22 PM permalink
It seems that you aren't determining the best hold strategy correctly.

Looking at your draw combinations for 3 wilds, it looks like you have only one way to end up with 3 wilds, which is probably when all 3 were dealt, weighted to equal 128^3.

But that isn't the only way to end up with 3 wilds. For example, with wild-wild-blank, you should "discard" the blank and go for the third wild. With one or more wilds, the strategy should be:

With 3 dealt wilds, hold all 3.
With 2 dealt wilds, hold only the 2 wilds when the third symbol is a blank or single bar.
With 1 dealt wild, hold only the wild when the other symbols are blank-blank or blank-single bar.

Assuming a 5000x payout for 3 wilds, you should come up with 10,632,633 combinations for 3 wilds on the draw.
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December 9th, 2009 at 4:34:04 PM permalink
Thanks. I agree with miplet's figure now. Thanks to all who helped.

Here is a preview of what I plan to write about the game.
Last edited by: Wizard on Dec 10, 2009
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
DesM
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December 20th, 2009 at 10:03:16 PM permalink
Quote: Wizard

Here is a preview of what I plan to write about the game.


There is a line:
Quote:

"The win for three 5X symbols is progressive, starting at 5000 coins."


It should be 5000 * 5 coins or 5000 bets.

Also it should be noted that there is optimal strategy with 100% return for jackpot size of 42,746.

And here are the return tables with just slightly improved outlook for better readability and a bit extended:

Return on the Deal


HandWinCombinationsProbabilityReturn
5x 5x 5x500010.0000000.002384
5x 5x Red7150070.0000030.005007
5x 5x 7-Bar1000120.0000060.005722
5x 5x 3-Bar500260.0000120.006199
5x Red7 Red7300150.0000070.002146
5x 5x 2-Bar250360.0000170.004292
5x 5x Cherry250180.0000090.002146
5x 7-Bar 7-Bar200410.0000200.003910
5x any7 any7150500.0000240.003576
5x 5x 1-Bar125350.0000170.002086
5x 3-Bar 3-Bar1002210.0001050.010538
Red7 Red7 Red76090.0000040.000257
5x 2-Bar 2-Bar503930.0001870.009370
5x Cherry Cherry501040.0000500.002480
7-Bar 7-Bar 7-Bar40300.0000140.000572
any7 any7 any7301050.0000500.001502
5x 5x252470.0001180.002944
5x 1-Bar 1-Bar253590.0001710.004280
5x Cherry2543640.0020810.052023
3-Bar 3-Bar 3-Bar206160.0002940.005875
5x anyBar anyBar1029290.0013970.013967
2-Bar 2-Bar 2-Bar1011900.0005670.005674
Cherry Cherry Cherry101920.0000920.000916
1-Bar 1-Bar 1-Bar510450.0004980.002491
Cherry Cherry5126320.0060230.030117
anyBar anyBar anyBar2444740.0212070.042414
-Nothing-020280010.9670260.000000
Total.20971521.0000000.222887


Return on the Draw


HandWinCombinationsProbabilityReturn
5x 5x 5x5000106326330.0000020.012088
5x 5x Red71500797769110.0000180.027209
5x 5x 7-Bar10001328759480.0000300.030212
5x 5x 3-Bar5002824848100.0000640.032115
5x Red7 Red73001789029990.0000410.012203
5x 5x 2-Bar2503462976040.0000790.019685
5x 5x Cherry2502206026260.0000500.012540
5x 7-Bar 7-Bar2004654268490.0001060.021165
5x any7 any71505819567700.0001320.019848
5x 5x 1-Bar1252038096110.0000460.005793
5x 3-Bar 3-Bar10024387295250.0005550.055450
Red7 Red7 Red7601108113930.0000250.001512
5x 2-Bar 2-Bar5038502228490.0008750.043772
5x Cherry Cherry5013960046480.0003170.015871
7-Bar 7-Bar 7-Bar403452330700.0000780.003140
any7 any7 any73012484752330.0002840.008516
5x 5x2510710782710.0002440.006088
5x 1-Bar 1-Bar2528946801750.0006580.016454
5x Cherry25268239748280.0060990.152477
3-Bar 3-Bar 3-Bar2068993928080.0015690.031375
5x anyBar anyBar10214397862810.0048750.048748
2-Bar 2-Bar 2-Bar10117101247900.0026630.026626
Cherry Cherry Cherry1027096624000.0006160.006161
1-Bar 1-Bar 1-Bar5100814170050.0022920.011461
Cherry Cherry5824397641200.0187450.093723
anyBar anyBar anyBar24098687185700.0931930.186387
-Nothing-038102156683770.8663430.000000
Total.43980465111041.0000000.900619
DesM
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December 20th, 2009 at 10:05:07 PM permalink
Full strategy for jackpot size up to 9300 bets (aka "5k" strategy).


  1. Do not respin if you have:

    *2\*15XRed-77-Bar3-Bar2-Bar1-BarCherry
    5X500015001000500250.250
    Red-73006030....
    7-Bar2003040....
    3-Bar100..20...
    2-Bar50.22102.
    1-Bar25.2225.
    Cherry50.....10
    any_7150......
    any_bar{1}.{2}{2}{2}{2}.


    {1}ValueReel1Reel2Reel3
    105X2-Bar1-Bar
    102-Bar5X1-Bar


    {2} All bars are differ in a kind

  2. Need to respin 1 reel:

    5XRed-77-Bar3-Bar2-Bar1-BarCherry
    5X -1-15516.014.512.810.37.1927.5
    -2-21125.822.713.411.2.27.9
    -3-21627.324.415.8..28.3
    Red-716.03.051.64....
    25.84.923.05....
    27.35.163.28....
    7-Bar14.51.642.73.719.719.719.
    22.73.054.30.625.625.625.
    24.43.284.56....
    3-Bar12.8..7192.41.719.719.
    13.4..6252.45.625.625.
    15.8..5942.84.594.594.
    2-Bar10.3..719.7191.91.719.
    11.2..625.6252.00.625.
    7.66..594.5941.22.594.
    1-Bar7.19..719.719.7191.28.
    ...625.625.625.859.
    ...594.594.594.969.
    Cherry27.5.....5.51
    27.9.....5.59
    28.3.....5.66


  3. Need to respin 2 reels:

    5XRed-77-Bar3-Bar2-Bar1-BarCherry
    1st8.00.486.609.450.329.251.967
    2nd6.74.295.453.452.334.312.886
    3rd6.53.277.443.419.451.291.806


  4. Need to respin all 3 reels:

    .223Blank Blank Blank


DesM
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December 20th, 2009 at 10:06:58 PM permalink
JP size dependence (change in hand values):



JP\comb 5 5 5 * 5 5 5 * 5 5 5 * 5 * * * 5 * * * 5
5k 5000 155 211 216 8.00 6.74 6.53
30k 30000 351 406 412 9.53 8.27 8.06
40k 40000 429 484 490 10.14 8.88 8.67
45k 45000 468 523 529 10.45 9.18 8.97
55k 55000 546 601 607 11.06 9.80 9.58
 ----- ------------------- ---------------------
per 1k +1000 +(1000/128)=7.81 +(1000/128**2)=.061



Difference in strategy from the base level of JP=5k.


Should lock only '5x' in case of:

hand \ JP_size16..17k18..19k20..37k38..41k42..49k50..56k
5x 5x + 2-Bar{1}+++++(any order)
5x 5x + Cherry{1}+++++(any order)
5x + 1-Bar++++++(any order)
2-Bar 5x (Any)--++++
5x 2-Bar 1-Bar---+++
5x 5x + 3-Bar----{1}+(any order)


{1} respin only for "5x * 5x" and "5x 5x *"
DesM
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December 20th, 2009 at 10:08:52 PM permalink
There are total of 113 optimal strategies, each starts at a shown JP size.

'Small return' is for return of small prizes only, i.e. without a jackpot.

'5k return' is return if "5k strategy" is used instead of optimal.

(I printed the whole table because it'll be helpful to compute the return of a play, described in the next message)

JP"5x 5x 5x" combsSmall Ret.Return"5k" Returndelta
082405690.8886940.8886940.888531.000163
179103213370.8886090.8890290.888964.000065
1101106326330.8885310.8911930.8911930
(5000)..0.9006190.9006190
9301108456250.8880810.9110170.911017.000000048
10051112224570.8872200.9128670.912830.000037
12276113365050.8869010.9185440.918209.000335
15776113883450.8867150.9275660.926671.000895
17101116832570.8855690.9309970.929874.001123
19956118319930.8848940.9385810.936776.001805
37696118559290.8846890.9863070.979664.006643
41301120361530.8829960.9960250.988380.007645
42051121672250.8817430.9980780.990193.007885
(42746)..1.0000003380.991873.008127
49101122819130.8804631.0175821.007237.010345
56896123382330.8797341.0393501.026082.013268
58356123579450.8794731.0434461.029612.013834
63476124219450.8785491.0578321.041990.015842
63653125528360.8766551.0583321.042417.015915
66976127704360.8733411.0678171.050451.017366
81056129281320.8704351.1087001.084491.024209
94016130759720.8672741.1467961.115822.030974
104436132364840.8634631.177776
105301133347880.8611091.180380
106051134167080.8591341.182654
107936135334440.8562691.188404
113101135498280.8558481.204298
118176136439080.8533201.219933
118693136688080.8526481.221537
131316137563600.8500341.260768
132416138577360.8469821.264209
134816139351760.8446081.271771
148133140091290.8421171.313966
153076141032090.8388421.329711
156916141496730.8371851.342025
157093141772460.8362001.342594
160416142237100.8345051.353306
169301142728620.8326131.382041
170051143220140.8307131.384475
177101143383980.8300531.407433
191397143732280.8285371.454040
227493144766460.8231881.572005
238501145950680.8167661.608240
245056146818520.8119301.629993
269216146963160.8110451.710645
273216147667160.8066721.724011
283456147737560.8062181.758393
293876147878360.8052771.793395
296256148312280.8023541.801398
304116148579800.8005041.827903
307616148701400.7996541.839728
320416148846040.7986001.883005
321856149268440.7955091.887879
331456150507480.7861711.920461
337856151421400.7791501.942363
342516151562200.7780541.958407
345716152821720.7681531.969434
349216153766360.7606521.981596
355616154385880.7556432.003972
358516154995160.7506762.014152
415397155761420.7434392.214611
466853156164800.7391572.396848
481445156681890.7334972.448661
484005157030190.7296642.457781
486309157587470.7235022.466007
487845157923620.7197732.471511
557733158039820.7182992.722462
693056158210060.7156173.208733
713716158340620.7134983.283053
717216158675980.7080293.295654
782757159539440.6926613.532118
813733159633230.6909263.644483
848293160477480.6746423.769924
1044901161209870.6572424.487314
1227173162368110.6249245.155430
1339813162547850.6194485.571277
1396133163416170.5918845.779431
1512256163528810.5880116.210903
1532916163627370.5845766.287722
1536416163699050.5820726.300743
1565093164263010.5620036.407482
1569189164352310.5588166.422780
2093477164576030.5481678.382012
2224549164623330.5457758.872487
2331456164665570.5435369.272651
2352116164677090.5429209.350003
2355616164687330.5423719.363108
2978213164728580.53957811.694458
3150656164766980.53682712.340342
3171316164774660.53627312.417742
3174816164781060.53581112.430855
3535269164854160.52993513.781359
3731877165285120.49336714.518315
4583845165390720.48236117.720137
4682149165482350.47260618.089814
4789056165493870.47135118.492066
4809716165497710.47093218.569807
4813216165501550.47051118.582978
5271973165539050.46601620.309312
5501349165604010.45789121.172666
5927333165625970.45493122.776666
6418853165635410.45355324.627680
8548773165642550.45216532.649200
9105829165648450.45094434.747226
9859493165652020.45014337.585834
10023333165662470.44776238.202936
10351013165666010.44692939.437217
11088293165725050.43204442.214410
11252133165763770.42213742.831784
11415973165792330.41472443.449302
20509093165806150.40827977.727465
41480613165812310.402470156.789999
62452133165813360.400979235.855469
83423653165813660.400410314.921441
125366693165813750.400153473.053669
DesM
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December 20th, 2009 at 10:11:24 PM permalink
Is "optimal" strategy really optimal?


At first, I have a question: does anybody know, what is the grown rate of a JackPot size? 1/25 of a bet?

It's strange, but I haven't seen on these sites the solution to the next problem. It's, as I understand, must be the well-demanded question to any progressive skill-based game, notably Video Poker.

So, assume there is only one player. And he plays continually until he wins a jackpot. So, the questions are: 1) at what jackpot size he must start to play to get zero profit on average; and 2) what strategy he should use in every single game?

To simplify computations, we can divide each bet by income into 3 pockets: a) small return (SR), which player gets back immediately with the probability of 1; 2) increase in jackpot size (dJP), which player gets after he wins; and 3) house income (1-SR-dJP). So, to find out 0-profit starting point, we can calculate total house income over the play. Obviously, they are equal.

We should compute:
SUM{n=1..inf}( HI(n) * Q(n-1) * P(n) ), where
HI(n)= SUM{k=1..n}( 1 - SR(k) - dJP ) // total house income after n'th game
Q(0)=1; Q(n-1)= PRODUCT{k=1..n-1}( 1 - P(k) )
P(n) is the probability of winning a jackpot using chosen strategy.

Luckily, for fixed strategies (which player does not change while JP grows) it's simple: player pays to the house (1-SR-dJP)/P on average whenever he starts. And here is the table for 'early' strategies:
(JP)0.100.080.060.040.020.00-- dJP
060341670827382380564873159405
17948541337621898304213894347465
110147441301721289295623783546108-- "5k" strategy
930148331294421054291643727445385
1005150081284620684285223636044198
1227650821284120600283593611843877
1577651311285420578283023602643749
1710154321296120490280193554843076
1995656151304920483279183535242786
3769656801309920518279373535642775
4130162131352120829281373544642754
4205165991382921058282873551742746
4910169961415821320284823564342805
5689672241435321482286113574042870


But for "optimal" strategies the sum of payments depends on starting point. So we need a sort of recursion. Hopefully, it doesn't need too much time to compute, so:

0.100.080.060.040.020.00-- dJP
70191410921163282693546842746-- optimal strategy
47441284120483279183535242746-- best fixed strategy


Look at the third row. These values are from the previous table. Seems like fixed strategy is always better than "optimal"! In other words, for calculating optimal long-run strategy, we must somehow include in the paytable dJP! I don't know how... yet. :)

To prove that fixed strategy can be better, assume dJP=.111306 and starting JP=0. Obviously, if we choose fixed "0"-strategy with SR=.888694, then at any JP we won't pay to a house even a penny on average. But at the "optimal" strategy our SR will be lower when we reach JP level of 179.

The things are even worse. We can try "intermediate" strategy, where we still play optimal strategies, but change them more rarely. Starting with the same strategy, we contribute to "virtual JP" (which points to optimal strategy) only the part of dJP. To be precise, k*dJP: k=0 if for strictly fixed strategy, k=1 is for fully "optimal" in terms of standard return table. Thus, for dJP=.08 we get:

k0.01.03.1.3.5.81
payment1284112841128431288213007132171370414109


So, at any k > 0 we lose more, and there is no better strategy, than fixed. I can't believe it's true. It must be a mistake somewhere. But where? I'm sure, that SR may be viewed as a single payment, especially on a long-term basis. I'm sure, it changes only variance, but not the return we are expecting.

PS. Maybe plain simulation without any assumptions could confirm or disprove such analysis, but right now I'm too sceptical doing so. The game's variance is huge, and each "play" consists of almost 400,000 single games on average.
DesM
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December 21st, 2009 at 10:35:42 PM permalink
Ok, I've done simulations. Not for "Lock & Roll" machine, but for the simplest one I could imagine.

It has 2 identical reels with 3 stops on each: 1 for "Seven" and 2 for "Bar". It pays 1 for a combination of "Bar Bar" and a jackpot payed on "Seven Seven".

With pencil and paper, I've found there are 4 strategies and corresponding numbers of combinations are:

(JP)"7 7""Bar Bar"Nothing
016416
.596012
2213624
5251640


Return equals to 1 if JP is (81-36*1)/21= 2.1429.

Here is the-same-way-calculated 0-profit table for fixed and optimal strategies. Again, fixed strategy is better than "optimal".

(JP).20.100-- dJP
00.88.917
.50.53331.43332.3333
21.37141.75712.1429
51.9522.2762.6
0.53331.43332.1429-- best fixed
0.94991.63552.1429-- optimal


To prove these calculations, here are simulation results of 1 mln plays each. Numbers are net profit of a player, including starting jackpot value (JP0). dJP is 0.10.


JP0 ----fixed---- ---optimal---
 sim. calc. sim. calc.
1.40 -0.035 -0.033 -0.195 -0.193
1.45 0.014 0.017 -0.143 -0.143
1.50 0.066 0.067 -0.112 -0.113
1.55 0.117 0.117 -0.061 -0.063
1.60 0.167 0.167 -0.034 -0.036
1.65 0.216 0.217 0.015 0.014
1.7 0.270 0.267 0.039 0.039
1.8 0.368 0.367 0.109 0.111
1.9 0.467 0.467 0.179 0.179
2.0 0.244 0.243 0.243 0.243
2.1 0.344 0.343 0.342 0.343
2.2 0.443 0.443 0.441 0.443


PS.
In the most primitive form the game may be reduced to 1 reel with 2 stops (a sort of "heads or tails"). Strategies and combinations are obvious:

(JP)HeadTail
013re-spin if 'head'
131re-spin if 'tail'


Zero-profit table:

dJP.25.20.15.10.050
fixed "0"-strategy0.2.4.6.81
"optimal".2813.425.5688.7125.85631


Simulation of 1 mln plays agrees with that (dJP=.25):

JP00.2.4.6.81.01.11.2
fixed sim..001.200.401.600.8001.000.433.533
fixed calc..000.200.400.600.8001.000.433.533
optimal sim.-.211-.010.119.225.300.500.433.533
optimal calc.-.211-.011.119.225.300.500.433.533


Any suggestions? Any suppositions? Any thoughts?
Last edited by: DesM on Dec 22, 2009
DesM
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December 22nd, 2009 at 9:42:07 PM permalink
The house edge for 'Lock & Roll'.


It's (1-SR-dJP) - JP0*P for fixed strategy:

JP0.10.08.06.04.02.00-- dJP
5k-.000619.020211.041655.061655.081655.104424
15k-.024795-.005565.014752.034752.054752.076759
25k-.048971-.031342-.012151.007849.027849.049094


And here is for optimal strategy:

JP0.10.08.06.04.02.00-- dJP
5k.005171.023582.042171.061008.080097.099381
15k-.020867-.002346.016351.035326.054707.074435
25k-.047551-.028964-.010232.008731.028027.047849
Last edited by: DesM on Dec 24, 2009
DesM
DesM
DesM
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Joined: Dec 17, 2009
December 23rd, 2009 at 9:55:33 PM permalink
Well, the question is over, at last. The key word for a fixed strategy to be useful was 'a single player game'. It lets you play longer and thus you'll harvest more, even though it returns less per every single bet.

So, if there are only a few players (or you are a member of a huge organized group), and your style of play can affect the longevity of it, then you should milk this cow gently and slowly using appropriate fixed strategy (or, since there are some other players, change it a bit over time).

As the number of players increases, you should not waste your time. So seek for a bigger single return and get as much milk as your force lets you, till you don't harm the cow.

PS. The lowest zero-profit point and thus suitable fixed strategy can be easily determined by the requirement: Return=1-dJP.
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