Analyzing Keno ProgressivesGuide to Essential Slot Gaming Terms and Vocabulary
Brandon James takes a look at a comparatively new and very difficult Video Keno sport for benefit play purposes. The game comes with a Free Games feature, Extra Draws characteristic in addition to ten totally different Progressives. James will present his full mathematical breakdown of this game (using math anyone can do) and shows you how to go about doing it your self for related video games.
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Math Lesson
Greetings!
I truly have lately been introduced with a brand new Video Keno game to analyze that I’m going to share with everyone right here. I even have been asked not to truly title this Keno sport as a result of it’s solely obtainable in sure markets that will largely not apply to any readers of LCB.
So, why am I sharing the analysis?
Quite frankly, this is one of the most difficult Video Keno games that I’ve ever analyzed, so I imagine that sharing my methodology will be helpful for anybody on the market who wants to research keno progressives for themselves to search out advantageous play points. In addition to land casinos, (and some comfort gambling locations) I should imagine that some on-line casinos even have Keno video games with troublesome features or progressives to research.
Even if you’re just someone who enjoys Keno and usually are not essentially on the lookout for advantageous conditions; it’s at all times nice to know the means to decide the return-to-player on keno games that include some sort of gimmick--and this recreation has gimmicks galore.
Many folks would program a simulation and analyze this game that way--by simply creating the foundations and having a pc program simulate millions of spins. There are a couple of issues about that I ought to mention:
1.) I don’t know how to program. In truth, if you requested me to complete the straightforward activity of creating a new E-Mail address, it will in all probability take me twice as lengthy to do it as it might you.
2.) In this specific case, and given the amount of time it will take for the simulations to be done---I consider that simply doing the maths might be FASTER for this one. I assume I totally analyzed three completely different variations of this recreation in about ten hours.
The Game
As I’ve talked about, I won’t be disclosing the identify of this Keno game, however I’ve been giving a blessing to share all the different particulars of my analysis.
This recreation can essentially be broken down into three elements, although these completely different components are extremely intertwined and all come collectively to result in an general return-to-player and advantageous progressive points.
One different factor that I ought to point out is that this game has different paytables for anywhere from 3 to 10 totally different picks the participant could make. However, most benefit players are NOT going to want a ridiculous quantity of Variance when going after a Video Keno play---particularly not a play on a sport that has several different features and is top-heavy anyway. For that cause, I determined to solely analyze the Pick 3, Pick four and Pick 5 variations of this recreation.
The first thing that we're going to do is take a glance at the entire different components of the sport:
1.) Base Pays
-The first thing that the full evaluation will consider is the base paytable of the game. Essentially, if the sport had no different options besides choosing your numbers and seeing what hits--how much would the return be? The good news is that the WoO Keno Calculator makes this an extremely trivial affair:
Based on WoO Keno Calculator:
Choose 3: zero.499513145082765
Choose 4: zero.486358578130730
Choose 5: zero.421164278759215
THREE-SPOT:
2/3 PAYS: 2
3/3 PAYS: 16
FOUR-SPOT:
2/4 PAYS: 1
3/4 PAYS: 3
4/4 PAYS: 47
FIVE-SPOT:
3/5 PAYS: 2
4/5 PAYS: thirteen
5/5 PAYS: 149
2.) Extra Draws and Free Games:
-The second component of this sport is that it is going to randomly draw 5 SEPARATE numbers from the player’s selections that will award either extra attracts, free games, or each when the participant hits both three, four or all 5 out of 5 for the Free Games.
This will get extremely situational as a result of the way in which that this stuff are awarded is as follows:
3 of 5: Five Extra Draws
4 of 5: Ten Extra Draws
5 of 5: Fifteen Free Games (No Additional Draws)
By itself, this would already be complicated enough, nevertheless, the extra attracts on the three out of 5 hit can lead to Free Games AFTER the extra numbers are drawn (three out of 5 turns into 5 out of five) or can lead to one other set of additional attracts (three out of five becomes 4 out of five). In different words, there are a few potential serieses of occasions potential:
When you begin with three out of 5, the next issues can happen:
A.) You get zero further on this set of five and solely find yourself with your five extra draws.
B.) One of those additional 5 numbers matches on this set of 5 and awards ten extra attracts for a total of fifteen extra attracts. Based on these ten additional draws AFTER the unique 5, two things are potential:
-The participant hits the fifth number and ALSO goes into Free Games.
-The player does not hit the fifth quantity and doesn't go into Free Games.
C.) On the five additional attracts, the player might hit each of the other numbers in that set of 5 numbers, thereby getting Free Games however NOT getting the ten extra draws.
The participant also can instantly begin with four out of 5, which is in a position to award ten further attracts from which:
A.) The participant can hit the fifth number from that set of five and go into Free Games.
B.) The participant does NOT hit the fifth quantity and does NOT go to Free Games.
Finally, the player might just hit 5/5 right off the bat and go immediately into Free Games.
In different words, there can be either 5, ten or fifteen further draws (that will all include their own corresponding possibilities that have an effect on this---and other---aspects of the game) and the player also can hit Free Games or not hit Free Games.
***And, if that’s not already sophisticated sufficient, we should account for the reality that IF the participant hits any outcomes on this explicit set of 5 numbers, then those hits CANNOT additionally double as hits on the player’s base Picks BECAUSE these numbers will always be separate from these ones as they aren't chosen by the player.
***For instance, if a player hits 4 out of five of this set of numbers on the initial draw, then that mechanically signifies that 4 numbers DO NOT match the player’s base picks, however, the base pays are affected by these numbers as nicely as the additional draws. What which means is that we ALSO have to find out the chances for the numbers of picks that the participant may have hit on the base choices given the lowered number of potential matches.
***In impact, if we assume that the player matches three of these 5 numbers (which we also have to determine out the probabilities for), then that can leave seventeen different attracts that both did or didn't hit the player’s base numbers. We should determine the probabilities for each potential event and how they relate to the additional draws.
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3.) Free Games:
-Okay, so the Free Games must be pretty simple, right? You have fifteen Free Games and it just goes by the bottom paytable, yes?
Haha. No.
The Free Games ALSO give the likelihood for getting extra draws in addition to to retrigger more Free Games. Here are the rules for the way the Free Games work:
A.) There are always fifteen Free Games initially.
B.) Free Games will embody TWO Wild spots that rely towards the Base Paytable in addition to the Jackpots. However, the WILDS do NOT go in the direction of the fortunate picks that award extra draws and extra Free Games.
C.) Lucky Picks throughout Free Games will award as follows:
3/5 Five Extra Draws
4/5 10 Extra Draws AND Five Additional Free Games
5/5 10 Additional Free Games
D.) The most variety of Free Games is 1,000, but we’re not going to go that far.
Okay, so the first thing that we now have is similar further draw mechanism as before EXCEPT that the results of 4 out of 5 now provides ten extra draws AND further Free Games whereas the five out of 5 for these choices awards extra free games solely. Therefore, the following events are possible.
a.) 0, 1 or 2 out of five---nothing occurs.
b.) Three out of five, after which:
aa.) The player hits two further numbers for a total of 5/5 after the 5 extra attracts and is awarded an additional ten Free Games.
bb.) The participant hits zero further numbers on this set of 5 and does NOT get any further Free Games or additional draws---aside from the unique additional five.
cc.) The player hits one of the set of 5 numbers (two remaining) and will get a complete of four out of five which leads to ten extra draws (fifteen total) and a minimal of 5 extra Free Games, however:
aaa.) The ten extra draws result in hitting the final one wanted for that set of 5 and getting ten additional Free Games---which would mean fifteen total extra Free Games.
Or:
bbb.) The ten further draws do NOT hit the ultimate quantity wanted on that set of five, so the player gets solely the 5 extra Free Games.
c.) Four out of five hits on an initial Free Games spin and then:
aa.) The ten extra draws result in hitting the final one needed for that set of 5 and getting ten additional Free Games---which would mean fifteen complete further Free Games.
Or:
bb.) The ten extra attracts do NOT hit the ultimate number needed on that set of 5, so the participant gets solely the 5 additional Free Games.
d.) Five out of 5 hits on an initial Free Games spin and simply awards ten further Free Games.
As a outcome, the next occasions are attainable:
NO extra attracts or Free Games awarded.
Five further attracts and no further games awarded.
Fifteen additional draws and 5 extra games awarded.
Fifteen further draws and fifteen extra video games awarded.
Ten extra draws and 5 extra video games awarded.
Ten additional attracts and fifteen further games awarded.
Zero extra attracts and ten extra games awarded.
Five extra draws and ten additional games awarded.
---And, we've to determine out the chances of every certainly one of these occasions.
The good news is that we DO NOT have to determine the likelihood for the whole variety of free video games (between 15 and 1,000) that the player will receive. One shortcut right here is solely to find out the AVERAGE quantity of Free Games that shall be awarded per Free Games preliminary spin, then multiply by fifteen as a end result of that is how many initial spins we begin with. After that, we are going to multiply the common variety of extra spins awarded (per spin) by how many additional Free Games we anticipate to get from the fifteen. We will then continue to repeat that course of until we’re basically not including a meaningful amount of Free Games to the anticipated whole anymore.*
*By using this technique, we will end up with an quantity of TOTAL Free Games per initial fifteen (18.3086771581 TOTAL, on average, was the result), but the result will be slightly short. I mainly used this methodology till I reached the point that the subsequent step would have only added less than 1/10000th of a Free Game and called that ok. If anybody does decide to simulate this, they will find that my average results of 18.3086771581 Free Games TOTAL (per preliminary set of fifteen Free Games) is extraordinarily close.
Not only that, however as quickly as once more, we've to discover out how these numbers being faraway from the preliminary set of numbers that we might hit impacts the possibilities as relates our Base Picks and Progressives!!!
Oh, right, the Progressives…
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four.) The Progressives:
That’s right, PROGRESSIVES, plural, isn’t that fun?
In truth, there are ten of them.
It’s not quite as dangerous because it appears. The method the Progressives work is that 5 numbers on the Keno board are randomly selected by the sport (these numbers will NOT match the player’s numbers OR any numbers used for further draws/free games) and there might be totally different occasions that occur primarily based on hitting three, four or 5 of these numbers.
If the player hits five out of five of those numbers, then the player is awarded one of the PROGRESSIVES.
So, where do the different progressives come into play?
Okay, the Progressives are represented by different symbols (ten of them) all of which have a unique likelihood of look because the 5 additional numbers. In other words, only one of the Progressives could be awarded on anybody spin and all of them have totally different rates of appearance. In fact, just determining how doubtless it's for a specific progressive to seem goes to be considered one of our tasks---which will solely occur after we take care of how probably hitting considered one of them is.
If you hit three or four out of 5 of these numbers, you won’t be paid anything, however there shall be an quantity added to the Progressive meter for whichever progressive is lively on the time. Of course, how much is added varies primarily based upon which meter it occurs to be, so we will have to decide all of those probabilities (and provides to the meter) individually so as to figure out how much the progressive meter additions contribute to the base return of the game.
---If that’s not difficult sufficient but, don’t fear, it’ll get slightly tougher:
A.) The progressive five-spot possibilities are ALSO conditional with the Extra Draw possibilities. For example, if you hit three/five of the additional draw set of five in order to get five further attracts, then that implies that three of your progressive set of 5 routinely DID NOT hit, so you want to determine the probabilities based mostly on seventeen completely different numbers that could hit it.*
*Technically, this is also looked at as true for the bottom game hits, but it doesn’t matter within the case of the base game hits. The reason why it doesn’t matter is as a outcome of nothing in regards to the base recreation numbers directly influences the chance of hitting the Progressive numbers. In other phrases, base sport hits and progressive hits could be handled separately from one another, however not from the Extra Draw and Free Games set of numbers.
B.) There are two WILD numbers that are randomly chosen throughout Free Games that may substitute as Progressive Numbers or substitute for the base paytable. Actually, they technically substitute for both simultaneously when they're hit. These numbers will NOT match any of the numbers chosen for the Base Game, any of the progressive numbers OR any of the numbers chosen for Free Games/Extra Draws.
On To The Math!!!
Okay, if the finest way the sport works was difficult to follow, (I asked several clarifying questions and likewise requested for photos of all of the rules screens) don’t worry---because the mathematical breakdown isn’t going to make it any simpler.
I would say simply do your greatest to observe alongside, however don’t be afraid to stop and come again to it. I know that I stopped in my evaluation of these three paytables--several times---and got here again to it later.
The good news is that I will reexplain each individual section of the game under and the mathematics might be accomplished individually for each element. Okay, right here we go:
Base Game Return (Without Extra Draws)
Based on WoO Keno Calculator:
Choose three: 0.499513145082765
Choose 4: zero.486358578130730
Choose 5: zero.421164278759215
It can be attainable to hit fifteen (total) additional draws by first hitting 5 further numbers with three out of five, but then having one of those numbers hit and unlocking 4/5 with ten additional attracts. By this mechanism, the Free Games may also be hit.
Therefore, we should first determine our total Extra Draw probabilities after which our total possibilities for Free Games accordingly.
3/5 Hits: zero.083935052289483
In order to obtain another ten draws, this should first occur after which be followed up with hitting one of many two remaining spots. Hitting both spots will simply unlock fifteen Free Games. We might also unlock the ten extra draws and unlock Free Games after that.
At this point, there shall be sixty numbers remaining, of which five shall be drawn.
nCr(2,1)*nCr(58,4)/nCr(60,5) = zero.1553672316384181
Therefore, that is the chance of this event and the overall chance of getting fifteen extra draws is:
zero.1553672316384181*0.083935052289483 = zero.0130407567116428407376253868423
The second factor that we'll have a look at is hitting both of our different five numbers which is ready to unlock Free Games:
nCr(2,2)*nCr(58,3)/nCr(60,5) = zero.0056497175141243
This shall be added to our running probability of Free Games, but first, we must factor in the likelihood of being on this state of affairs within the first place:
zero.0056497175141243*0.083935052289483 = zero.0004742093349688310301048447369
0.0004742093349688310301048447369+0.000644924695558 = zero.0011191340305268310301048447369
Now, we should take a glance at the chance of hitting ⅗, getting the 5 further attracts, turning that into 4/5 and then hitting Free Games. There might be fifty five remaining numbers of which just one is useful, ten numbers are drawn:
nCr(1,1)*nCr(54,9)/nCr(55,10) = zero.1818181818181818
Of course, so much has to happen in the first place. This is similar as our chance of getting fifteen further draws (at all) from above:
0.1818181818181818*0.0130407567116428407376253868423 = zero.0023710466748441526242817664868574411340838756
Therefore, our working likelihood of hitting the Free Games is:
zero.0023710466748441526242817664868574411340838756+0.0011191340305268310301048447369 = zero.0034901807053709836543866112237574411340838756
Finally, we could hit 4/5 of those numbers and be awarded ten extra draws. This will trigger sixty numbers to stay of which just one helps us of the ten chosen:
nCr(1,1)*nCr(59,9)/nCr(60,10) = 0.1666666666666667
Of course, we have to be in this state of affairs within the first place having obtained ⅘ on the initial draw, so:
zero.1666666666666667*0.012092338041705 = zero.0020153896736175004030779347235
0.0020153896736175004030779347235+0.0034901807053709836543866112237574411340838756 = zero.0055055703789884840574645459472574411340838756
Therefore, we all know the following chances:
Extra Draw Probabilities:
3/5 Hits: zero.083935052289483***
Zero Extra: nCr(2,0)*nCr(58,5)/nCr(60,5) = zero.8389830508474576
One Extra (Initial Draw) = 0.1553672316384181
Two Extra (Initial Draw) = 0.0056497175141243
PROOF: zero.0056497175141243+0.1553672316384181+0.8389830508474576 = 1
PROBABILITY 3/5 + 0 EXTRA: 0.083935052289483*0.8389830508474576 = zero.0704200862428713282322697684208
PROBABILITY 3/5 + 1 EXTRA: 0.1553672316384181*0.083935052289483 = 0.0130407567116428407376253868423***
PROBABILITY 3/5 + 2 EXTRA: zero.083935052289483*0.0056497175141243 = 0.0004742093349688310301048447369---FREE GAMES TRIGGERED
***This is the one potential way to get fifteen additional numbers.
***PROBABILITY OF 3/5 + 1 EXTRA, + 1 EXTRA AGAIN---FREE GAMES:
zero.0130407567116428407376253868423*0.1818181818181818 = zero.00237104667484415262428176648691198658862933014
***PROBABILITY OF 3/5 + 1 EXTRA + 0 EXTRA:
zero.0130407567116428407376253868423*.818181818181 = 0.0106697100367880181662021541830964883198563
4/5 Hits: zero.012092338041705***
What matters here is whether or not or not or not we get Free Games, which may solely be accomplished if an additional number results in Free Games.
From Above: nCr(1,1)*nCr(59,9)/nCr(60,10) = zero.1666666666666667
Therefore, the general probability of hitting 4/5 adopted by Free Games is:
zero.1666666666666667*0.012092338041705 = zero.0020153896736175004030779347235
The likelihood that this won't happen is:
zero.012092338041705*.833333333 = zero.010076948364056720652765
5/5 Hits: 0.000644924695558
5/5 Hits Total Probability: 0.000644924695558+0.0020153896736175004030779347235+0.00237104667484415262428176648691198658862933014+0.0004742093349688310301048447369 = zero.00550557037898848405746454594731198658862933014 (Agrees with Running Total From Above)
FINAL PROBABILITIES+FREQUENCIES ON FIVE NUMBER EXTRA DRAW + FREE GAME DRAWS:
3/5 Hit, No Extra Numbers: zero.0704201 or 1 in 14.2 (Five Extra Draws)
3/5 Hit, One Extra, No Free Games After: 0.0106697 or 1 in ninety three.seventy two (Fifteen Extra Draws Total)
3/5 Hit, One Extra, One Extra Again, Free Games: 0.0023711 or 1 in 421.75 (Fifteen Extra Total)
3/5 Hit, Two Extra, Free Games: zero.0004742 or 1 in 2108.eighty one (Five Extra Draws)
4/5 Hit, No Extra: 0.0100769 or 1 in 99.24 (Ten Extra Draws Total)
4/5 Hit, One Extra, Free Games: zero.0020154 or 1 in 496.18 (Ten Extra Draws Total)
5/5 Hit: 0.0006449 or 1 in 1550.63 (Initial)
5/5 Hit: 0.0055056 or 1 in 181.63 (Total)
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FINAL EXTRA DRAW PROBABILITIES:
FIVE EXTRA: 0.0704201 + zero.0004742 = 0.0708943
TEN EXTRA: 0.0100769 + zero.0020154 = 0.0120923
FIFTEEN EXTRA: 0.0106697 + 0.0023711 = 0.0130408
ZERO EXTRA: 1 - (0.0708943+0.0120923+0.0130408) = zero.9039726
FINAL FREE GAMES PROBABILITY (REGARDLESS OF NUMBER OF EXTRA DRAWS)
5/5 Hit: 0.0055056 or 1 in 181.sixty three
BASE RETURN BEFORE FREE GAMES INCLUDING EXTRA DRAWS:
Based on WoO Keno Calculator:
Choose 3: 0.499513145082765
Choose four: zero.486358578130730
Choose 5: 0.421164278759215
THREE-SPOT:
2/3 PAYS: 2
3/3 PAYS: sixteen
FOUR-SPOT:
2/4 PAYS: 1
3/4 PAYS: three
4/4 PAYS: forty seven
FIVE-SPOT:
3/5 PAYS: 2
4/5 PAYS: thirteen
5/5 PAYS: 149
RETURNS BASED ON TOTAL NUMBER DRAWN:
DRAW TWENTY TOTAL:
Choose 3: 0.499513145082765 * 0.9039726 = zero.451546196494644292239***
Choose 4: zero.486358578130730 * zero.9039726 = zero.439654828405139137998***
Choose 5: zero.421164278759215 * 0.9039726 = zero.380720968097092357509***
***The reason for this calculation is because there’s less than a 100 percent chance of drawing ONLY twenty numbers. Therefore, we should determine the completely different probabilities based mostly on the whole variety of balls that are drawn by the point the sport ends. In this case, there's only a ninety.39726% likelihood that only twenty numbers might be drawn with the other prospects being greater than that.
THREE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):***
***Remember here that the extra draws have already happened, in order that they MUST be assumed as having not hit different numbers as a end result of they couldn’t have. That means, with 5 additional attracts (three of the set of five that controls extra attracts hit) that there are solely 22 numbers available to be doing other issues. This may also apply in a different way when it comes to having ten or fifteen extra draws.
THREE-SPOT:
nCr(3,2)*nCr(74,20)/nCr(77,22) * 2 = 0.3473684210526316
nCr(3,3)*nCr(74,19)/nCr(77,22) * sixteen = 0.3368421052631579
ADDED VALUE: (0.3473684210526316+0.3368421052631579) * 0.0708943 = 0.04850662631578947554985
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
THREE-SPOT:
nCr(3,2)*nCr(73,24)/nCr(76,26) * 2 = 0.4623044096728307
nCr(3,3)*nCr(73,23)/nCr(76,26) * sixteen = 0.5917496443812233
ADDED VALUE: (0.5917496443812233+0.4623044096728307) * 0.0120923 = zero.0127459378378378371842
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
THREE-SPOT:
nCr(3,2)*nCr(73,29)/nCr(76,31) * 2 = 0.5953058321479374
nCr(3,3)*nCr(73,28)/nCr(76,31) * sixteen = 1.0230440967283073
ADDED VALUE: (1.0230440967283073+0.5953058321479374) * zero.0130408 = 0.02110457775248933188376
THREE SPOT TOTAL RETURNS:
0.451546196494644292239+0.04850662631578947554985+0.0127459378378378371842+0.02110457775248933188376 = 0.533903
FOUR-SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS)***
***Remember here that the extra draws have already happened, in order that they MUST be assumed as having not hit different numbers because they couldn’t have. That means, with five extra attracts (three of the set of five that controls extra attracts hit) that there are only 22 numbers available to be doing other issues. This may even apply another way when it comes to having ten or fifteen additional draws.
nCr(4,2)*nCr(73,20)/nCr(77,22) * 1 = 0.2534850640113798
nCr(4,3)*nCr(73,19)/nCr(77,22) * 3 = zero.1877667140825036
nCr(4,4)*nCr(73,18)/nCr(77,22) * 47 = 0.2540540540540541
ADDED VALUE: (0.2534850640113798+0.1877667140825036+0.2540540540540541)*0.0708943 = zero.04929322025604552550625
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
FOUR-SPOT:
nCr(4,2)*nCr(72,24)/nCr(76,26) * 1 = zero.3103139188214891
nCr(4,3)*nCr(72,23)/nCr(76,26) * three = zero.3039809817026832
nCr(4,4)*nCr(72,22)/nCr(76,26) * forty seven = zero.5476724020343343
ADDED VALUE: (0.5476724020343343+0.3039809817026832+0.3103139188214891)* 0.0120923 = zero.01405085721272822935918
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
FOUR-SPOT:
nCr(4,2)*nCr(72,29)/nCr(76,31) * 1 = 0.3588144741713595
nCr(4,3)*nCr(72,28)/nCr(76,31) * 3 = zero.4729827159531558
nCr(4,4)*nCr(72,27)/nCr(76,31) * 47 = 1.1526763966562092
ADDED VALUE: (1.1526763966562092+0.4729827159531558+0.3588144741713595)*0.0130408 = zero.0258791231504900720596
FOUR SPOT TOTAL RETURNS:
0.439654828405139137998+0.04929322025604552550625+0.01405085721272822935918+0.0258791231504900720596 = 0.52887802902440296492303
FIVE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):***
***Remember here that the additional draws have already occurred, so they MUST be assumed as having not hit other numbers because they couldn’t have. That means, with 5 extra attracts (three of the set of five that controls additional draws hit) that there are solely 22 numbers available to be doing different things. This may even apply in a unique way in relation to having ten or fifteen additional draws.
FIVE-SPOT:
nCr(5,3)*nCr(72,19)/nCr(77,22) * 2 = 0.2314932091428126
nCr(5,4)*nCr(72,18)/nCr(77,22) * thirteen = 0.2647167715660866
nCr(5,5)*nCr(72,17)/nCr(77,22) * 149 = zero.1985931136616068
ADDED VALUE: (0.1985931136616068+0.2647167715660866+0.2314932091428126) * zero.0708943 = 0.0492575790132309635158
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
FIVE-SPOT:
nCr(5,3)*nCr(71,23)/nCr(76,26) * 2 = zero.3447932431349879
nCr(5,4)*nCr(71,22)/nCr(76,26) * thirteen = 0.5259856107008234
nCr(5,5)*nCr(71,21)/nCr(76,26) * 149 = zero.5305171790391689
ADDED VALUE: (0.5305171790391689+0.5259856107008234+0.3447932431349879)*0.0120923 = zero.01694489201833412307246
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
FIVE-SPOT:
nCr(5,3)*nCr(71,28)/nCr(76,31) * 2 = 0.4817416551374735
nCr(5,4)*nCr(71,27)/nCr(76,31) * 13 = 0.9963293322161383
nCr(5,5)*nCr(71,26)/nCr(76,31) * 149 = 1.370336035386504
ADDED VALUE: (1.370336035386504+0.9963293322161383+0.4817416551374735)*0.0130408 = 0.03714550630214930212464
FIVE-SPOT TOTAL RETURNS: zero.380720968097092357509 + zero.03714550630214930212464 + zero.01694489201833412307246 + 0.0492575790132309635158 = zero.4840689454308067462219 or 48.4069%
BASE GAME FINAL RETURNS:
THREE-SPOT: zero.533903
FOUR-SPOT: zero.528878
FIVE-SPOT: zero.484069
FIGURING OUT THE FREE GAMES:
The subsequent factor that we now have to do is determine how the Free Games will impact our returns based mostly on the variety of spots that we've selected.
The following are the summarized rules for the Free Games:
1.) There are always fifteen Free Games initially.
2.) Free Games will embody TWO Wild spots that depend towards the Base Paytable as properly as the Jackpots. However, the WILDS do NOT go in the direction of the lucky picks that award further draws and extra Free Games.
three.) Lucky Picks during Free Games will award as follows:
3/5 Five Extra Draws
4/5 10 Extra Draws AND Five Additional Free Games
5/5 10 Additional Free Games
four.) The maximum variety of Free Games is 1,000, however we’re not going to go that far.
Average Free Games
The first thing that we’re going to do is approximate the average number of Free Games that a participant receives per initial occurrence. The good news is that the possibilities for Free Games retrigger are simply going to be a variation of the above possibilities for the 3/5 and 4/5 prospects from above.
3/5 Hit, No Extra Numbers: zero.0704201 or 1 in 14.2 (Five Extra Draws, no Retrigger)
3/5 Hit, One Extra, No Free Games After: zero.0106697 or 1 in 93.72 (Fifteen Extra Draws Total, Five Free Games Retrigger)
3/5 Hit, One Extra, One Extra Again, Free Games: zero.0023711 or 1 in 421.75 (Fifteen Extra Total, Fifteen Free Games Retrigger)
3/5 Hit, Two Extra, Free Games: 0.0004742 or 1 in 2108.81 (Five Extra Draws, 10 Free Games Retrigger)
4/5 Hit, No Extra: 0.0100769 or 1 in ninety nine.24 (Ten Extra Draws Total, Five Free Games Retrigger)
4/5 Hit, One Extra, Free Games: 0.0020154 or 1 in 496.18 (Ten Extra Draws Total, Fifteen Free Games Retriggered)
5/5 Hit: zero.0006449 or 1 in 1550.63 (Initial, Ten Free Games Retriggered)
5/5 Hit: zero.0055056 or 1 in 181.63 (Total)
FINAL EXTRA DRAW PROBABILITIES:
FIVE EXTRA: 0.0704201 + 0.0004742 = zero.0708943
TEN EXTRA: zero.0100769 + zero.0020154 = zero.0120923
FIFTEEN EXTRA: 0.0106697 + zero.0023711 = zero.0130408
ZERO EXTRA: 1 - (0.0708943+0.0120923+0.0130408) = 0.9039726
FINAL FREE GAMES RETRIGGER PROBABILITIES:
Five Free Games: 0.0106697 + zero.0100769 = 0.0207466
Ten Free Games: zero.0004742 + zero.0006449 = zero.0011191
Fifteen Free Games: zero.0023711 + 0.0020154 = 0.0043865
FREE GAMES EXPECTED BASED ON INITIAL SPINS:
The subsequent factor that we are going to do is determine what quantity of further free video games we are anticipated to obtain based mostly upon the fifteen initial Free Games. The first step to doing that is to discover out what number of extra are anticipated on a per spin foundation, like so:
(5 * 0.0207466) + (10 * 0.0011191) + (15 * zero.0043865) = zero.1807215
Therefore, we expect to obtain 0.1807215 retriggered Free Games per each of the preliminary fifteen spins. We can multiply this number by fifteen to get the entire number of expected further Free Games per preliminary spins:
0.1807215*15 = 2.7108225
Okay, so now we're going to do the identical thing primarily based on our expected 2.7108225 retriggered video games:
2.7108225 * .1807215 = zero.48990390843
And then once more with that:
zero.48990390843 * .1807215 = 0.08853616918
And again:
0.08853616918 * .1807215 = zero.01600038929
One last time:
0.01600038929 * .1807215 = 0.00289161435
We will now add all these anticipated extra Free Games to our preliminary fifteen, which is going to get us close sufficient:
15 + 2.7108225 + zero.48990390843 + 0.08853616918 + 0.01600038929 + zero.00289161435 = 18.3081545812
Therefore, for every initial set of fifteen Free Games, we are expected to get something along the traces of 18.308155 Free Games. If we had gone one more retrigger prevalence, it might have added about .00052257688 anticipated Free Games, so we are in a position to go ahead and try this.
18.3081545812+.00052257688 = 18.3086771581
So, we’ll break that off and call it 18.308677 Free Games anticipated per preliminary set.
While that will seem low, remember that the chance of ANY retrigger (per spin) is:
(0.0207466) + (0.0011191) + (0.0043865) = zero.0262522 or 2.62522%
So, the chance of going fifteen Free Games with no retrigger is:
(1-.0262522)^15 = 0.67096110017
Therefore, about 67.1% of the time you get Free Games, you won’t get any extras at all. Granted, they can actually add up quickly when they're hitting, but we’ve principally accounted for that by occurring a basis of expected extra Free Games per initial set of spins.
Base Return Of Free Games
The good news right here is that the Extra Draw possibilities are the identical, with the numerous change being that there are actually two WILD spots in the course of the Free Games. Another thing that's the similar are the bottom return tables, so essentially, you've five spots to hit either 2 or three spots on a three-spot card. You can also hit 4 and five (and be awarded as should you hit three).
BASE RETURN FREE GAMES BEFORE INCLUDING EXTRA DRAWS:
Based on WoO Keno Calculator:
Choose 3: 2.087671821849037
Choose 4: 2.190619507075203
Choose 5: 2.428321678321678
THREE-SPOT (BECOMES FIVE-SPOT):
2/3 PAYS: 2
3/3, 4/3, 5/3 PAYS: sixteen
FOUR-SPOT (BECOMES SIX-SPOT):
2/4 PAYS: 1
3/4 PAYS: three
4/4, 5/4, 6/4 PAYS: forty seven
FIVE-SPOT (BECOMES SEVEN SPOT):
3/5 PAYS: 2
4/5 PAYS: thirteen
5/5, 6/5, 7/5 PAYS: 149
RETURNS BASED ON TOTAL NUMBER DRAWN:
DRAW TWENTY TOTAL:
Choose 3: 2.087671821849037 * zero.9039726 = 1.88719812474
Choose four: 2.190619507075203 * zero.9039726 = 1.98026001142
Choose 5: 2.428321678321678 * zero.9039726 = 2.19513626119
FREE GAMES WITH EXTRA DRAWS ADDED:
THREE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):
THREE-SPOT:
nCr(5,2)*nCr(72,20)/nCr(77,22) * 2 = zero.6134570042284534
nCr(5,3)*nCr(72,19)/nCr(77,22) * sixteen = 1.8519456731425008
nCr(5,4)*nCr(72,18)/nCr(77,22) * 16 = 0.3258052573121066
nCr(5,5)*nCr(72,17)/nCr(77,22) * 16 = 0.0213254350240652
ADDED VALUE: (1.8519456731425008+0.6134570042284534+0.3258052573121066+0.0213254350240652)*0.0708943 = zero.1993925844720279027818
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
THREE-SPOT:
nCr(5,2)*nCr(71,24)/nCr(76,26) * 2 = 0.6895864862699758
nCr(5,3)*nCr(71,23)/nCr(76,26) * 16 = 2.7583459450799033
nCr(5,4)*nCr(71,22)/nCr(76,26) * sixteen = zero.6473669054779365
nCr(5,5)*nCr(71,21)/nCr(76,26) * sixteen = zero.0569682876820584
ADDED VALUE: (0.0569682876820584+0.6473669054779365+2.7583459450799033+0.6895864862699758)*0.0120923 = zero.0502104657958607493702
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
THREE-SPOT:
nCr(5,2)*nCr(71,29)/nCr(76,31) * 2 = 0.7143065921003917
nCr(5,3)*nCr(71,28)/nCr(76,31) * sixteen = 3.8539332410997876
nCr(5,4)*nCr(71,27)/nCr(76,31) * 16 = 1.2262514858044779
nCr(5,5)*nCr(71,26)/nCr(76,31) * sixteen = zero.1471501782965373
ADDED VALUE: (0.1471501782965373+1.2262514858044779+3.8539332410997876+0.7143065921003917)*0.0130408 = zero.0774837584380054172356
THREE SPOT TOTAL EXPECTED RETURN PER FREE GAME:
Add them up:
1.88719812474+0.1993925844720279027818+0.0502104657958607493702+0.0774837584380054172356 = 2.2142849334458940693876
THREE-SPOT ADDED RETURN OF FREE GAMES:
The last few steps could have us multiplying the expected return (in credits) by the whole anticipated number of Free Games and then factoring that in opposition to the probability of hitting Free Games in the first place:
(2.2142849334458940693876*18.3086771581) * 0.0055056 = zero.2232005
THREE SPOT BASE RETURN WITHOUT PROGRESSIVES:
zero.2232005 + zero.533903 = zero.7571035 or seventy five.71035% (Before Progressive)
FOUR-SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS)
nCr(6,2)*nCr(71,20)/nCr(77,22) * 1 = 0.3322892106237456
nCr(6,3)*nCr(71,19)/nCr(77,22) * 3 = 0.5112141701903778
nCr(6,4)*nCr(71,18)/nCr(77,22) * forty seven = 2.1533691225472047
nCr(6,5)*nCr(71,17)/nCr(77,22) * forty seven = zero.287115883006294
nCr(6,6)*nCr(71,16)/nCr(77,22) * forty seven = zero.0147908182154757
ADDED VALUE: (0.0147908182154757+0.287115883006294+2.1533691225472047+0.5112141701903778+0.3322892106237456) * zero.0708943 = zero.23386464256347551036254
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
nCr(6,2)*nCr(70,24)/nCr(76,26) * 1 = zero.3423651217044598
nCr(6,3)*nCr(70,23)/nCr(76,26) * three = 0.6992989719920882
nCr(6,4)*nCr(70,22)/nCr(76,26) * 47 = three.937198899601288
nCr(6,5)*nCr(70,21)/nCr(76,26) * forty seven = 0.7070887819692109
nCr(6,6)*nCr(70,20)/nCr(76,26) * forty seven = 0.0494962147378448
ADDED VALUE: (0.0494962147378448+0.7070887819692109+3.937198899601288+0.6992989719920882+0.3423651217044598) * .0120923 = zero.06935475772953615190391
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
FOUR-SPOT:
nCr(6,2)*nCr(70,29)/nCr(76,31) * 1 = zero.3169106711431315
nCr(6,3)*nCr(70,28)/nCr(76,31) * three = 0.8752770917286489
nCr(6,4)*nCr(70,27)/nCr(76,31) * 47 = 6.696887515784314
nCr(6,5)*nCr(70,26)/nCr(76,31) * forty seven = 1.6437814811470589
nCr(6,6)*nCr(70,25)/nCr(76,31) * 47 = 0.158290068554902
ADDED VALUE: (0.158290068554902+1.6437814811470589+6.696887515784314+0.8752770917286489+0.3169106711431315) * 0.0130408 = 0.12638030755925172755624
FOUR SPOT TOTAL EXPECTED RETURN PER FREE GAME:
Add them up: 2.190619507075203 + zero.12638030755925172755624 + 0.06935475772953615190391 + 0.23386464256347551036254 = 2.62021921492746638982269
FOUR-SPOT ADDED RETURN OF FREE GAMES:
The last few steps could have us multiplying the expected return (in credits) by the entire anticipated variety of Free Games after which factoring that towards the chance of hitting Free Games within the first place:
(2.62021921492746638982269*18.3086771581) * zero.0055056 = zero.2641187596796262218413489029829228343184
FOUR SPOT BASE RETURN WITHOUT PROGRESSIVES:
0.2641187596796262218413489029829228343184 + zero.528878 = zero.7929967596796262218413489029829228343184 OR 79.2997% (Before Progressive)
FIVE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):
FIVE-SPOT:
nCr(7,3)*nCr(70,19)/nCr(77,22) * 2 = 0.4368121078622008
nCr(7,4)*nCr(70,18)/nCr(77,22) * thirteen = 1.0374287561727268
nCr(7,5)*nCr(70,17)/nCr(77,22) * 149 = 2.4229758409769986
nCr(7,6)*nCr(70,16)/nCr(77,22) * 149 = zero.254262896892648
nCr(7,7)*nCr(70,15)/nCr(77,22) * 149 = zero.0105667697409932
ADDED VALUE: (0.0105667697409932+0.254262896892648+ 2.4229758409769986+1.0374287561727268+0.4368121078622008) * zero.0708943 = zero.29506536408535234892582
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
FIVE-SPOT:
nCr(7,3)*nCr(70,23)/nCr(76,26) * 2 = zero.8158488006574362
nCr(7,4)*nCr(70,22)/nCr(76,26) * 13 = 2.5410290770476398
nCr(7,5)*nCr(70,21)/nCr(76,26) * 149 = 7.8456765914243295
nCr(7,6)*nCr(70,20)/nCr(76,26) * 149 = 1.0983947227994061
nCr(7,7)*nCr(70,19)/nCr(76,26) * 149 = zero.0615347183641124
ADDED VALUE: (0.0615347183641124+1.0983947227994061+7.8456765914243295+2.5410290770476398+0.8158488006574362) *0.0120923 = zero.1494908641884351248852
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
FIVE-SPOT:
nCr(7,3)*nCr(70,28)/nCr(76,31) * 2 = 1.0211566070167571
nCr(7,4)*nCr(70,27)/nCr(76,31) * 13 = four.3221047087685998
nCr(7,5)*nCr(70,26)/nCr(76,31) * 149 = 18.238979625918962
nCr(7,6)*nCr(70,25)/nCr(76,31) * 149 = 3.5126923723992075
nCr(7,7)*nCr(70,24)/nCr(76,31) * 149 = zero.2727245630744726
ADDED VALUE: (1.0211566070167571+4.3221047087685998+18.238979625918962+3.5126923723992075+0.2727245630744726)*0.0130408 = 0.3568961528447028493592
FIVE SPOT TOTAL EXPECTED RETURN PER FREE GAME:
Add them up: 2.19513626119+0.3568961528447028493592+0.1494908641884351248852+0.29506536408535234892582 = 2.99658864230849032317022
FIVE-SPOT ADDED RETURN OF FREE GAMES:
The last few steps will have us multiplying the expected return (in credits) by the whole expected variety of Free Games and then factoring that in opposition to the chance of hitting Free Games within the first place:
(2.99658864230849032317022*18.3086771581) * 0.0055056 = 0.3020568931666592841333559378737508509792
FIVE SPOT BASE RETURN WITHOUT PROGRESSIVES:
zero.3020568931666592841333559378737508509792 + zero.484069 = zero.7861258931666592841333559378737508509792 OR seventy eight.6126%.
FINAL RETURNS BEFORE PROGRESSIVES AND METER CONTRIBUTIONS:
THREE-SPOT: zero.7571035
FOUR-SPOT: 0.792997
FIVE-SPOT: 0.786126
THE PROGRESSIVE METERS:
This is an uncommon recreation in that there are ten completely different progressive meters all primarily based on a five-spot that is separate from the Bonus Games and from the participant spots chosen.
Each of these ten progressives has a certain risk of being chosen, which work as follows:
5/5: Awards Progressive
3/5 & 4/5 : Adds to Progressive
The quantities added to the Progressive vary as do the possibilities of a selected progressive even coming up. In order to figure in the progressives, we now have to take a look at the chances of hitting three out of 5, 4 out of five and 5 out of five both in the base sport and through Free Games. Fortunately, we now have already done these chances:
NORMAL PROBABILITIES * zero.9039726 (PROBABILITY OF NO EXTRA DRAWS)
(nCr(5,3)*nCr(75,17)/nCr(80,20)) * .9039726 = zero.0758749874492596
(nCr(5,4)*nCr(75,16)/nCr(80,20)) * .9039726 = zero.0109311422596391
(nCr(5,5)*nCr(75,15)/nCr(80,20)) * .9039726 = 0.0005829942538474
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):
(nCr(5,3)*nCr(72,19)/nCr(77,22)) *.0708943 = zero.0082057745084666
(nCr(5,4)*nCr(72,18)/nCr(77,22)) * .0708943 = zero.0014436084783414
(nCr(5,5)*nCr(72,17)/nCr(77,22)) * .0708943 = zero.0000944907367642
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
(nCr(5,3)*nCr(71,23)/nCr(76,26)) * .0120923 = zero.0020846716669806
(nCr(5,4)*nCr(71,22)/nCr(76,26)) * .0120923 = zero.0004892596769444
(nCr(5,5)*nCr(71,21)/nCr(76,26)) * .0120923 = 0.0000430548515711
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
(nCr(5,3)*nCr(71,28)/nCr(76,31)) * zero.0130408 = zero.0031411482881584
(nCr(5,4)*nCr(71,27)/nCr(76,31)) * 0.0130408 = 0.0009994562735049
(nCr(5,5)*nCr(71,26)/nCr(76,31)) * zero.0130408 = zero.0001199347528206
BASE PROGRESSIVE & METER MOVER PROBABILITIES:
3/5 : (0.0758749874492596+0.0082057745084666+0.0020846716669806+0.0031411482881584) = zero.0893065819128652
4/5 : (0.0109311422596391+0.0014436084783414+0.0004892596769444+0.0009994562735049) = zero.0138634666884298
PROGRESSIVE HIT: 0.0005829942538474+0.0000944907367642+0.0000430548515711+0.0001199347528206 = 0.0008404745950033
FREE GAMES AND PROGRESSIVE METERS (HITS)
The subsequent thing that we have to do is account for the possibilities of shifting the Progressive Meters or hitting a progressive in the course of the Free Games. We should do not forget that the WILD spots additionally count towards the Progressive five-spot, so we are basically drawing from seven as a substitute.
PROBABILITY OF HITTING FREE GAMES: 0.0055056
AVERAGE NUMBER OF FREE GAMES: 18.3086771581
NORMAL PROBABILITIES * zero.9039726 (PROBABILITY OF NO EXTRA DRAWS)
(nCr(7,3)*nCr(73,17)/nCr(80,20)) * .9039726 = 0.1581890954550239
(nCr(7,4)*nCr(73,16)/nCr(80,20)) * .9039726 = 0.0471792039076387
(nCr(7,5)*nCr(73,15)/nCr(80,20)) * .9039726 = 0.0078089716812643
(nCr(7,6)*nCr(73,14)/nCr(80,20)) * .9039726 = zero.0006617772611241
(nCr(7,7)*nCr(73,13)/nCr(80,20)) * .9039726 = 0.0000220592420375
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):
(nCr(7,3)*nCr(70,19)/nCr(77,22)) *.0708943 = 0.0154837443092076
(nCr(7,4)*nCr(70,18)/nCr(77,22)) * .0708943 = zero.0056575219591335
(nCr(7,5)*nCr(70,17)/nCr(77,22)) * .0708943 = zero.0011528535312951
(nCr(7,6)*nCr(70,16)/nCr(77,22)) * .0708943 = 0.0001209784569878
(nCr(7,7)*nCr(70,15)/nCr(77,22)) * .0708943 = 0.0000050276761346
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
(nCr(7,3)*nCr(69,23)/nCr(76,26)) * .0120923 = 0.0033119854089495
(nCr(7,4)*nCr(69,22)/nCr(76,26)) * .0120923 = zero.0016207588171455
(nCr(7,5)*nCr(69,21)/nCr(76,26)) * .0120923 = zero.000445708674715
(nCr(7,6)*nCr(69,20)/nCr(76,26)) * .0120923 = zero.0000636726678164
(nCr(7,7)*nCr(69,19)/nCr(76,26)) * .0120923 = 0.0000036384381609
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
(nCr(7,3)*nCr(69,28)/nCr(76,31)) * zero.0130408 = 0.0039950097242352
(nCr(7,4)*nCr(69,27)/nCr(76,31)) * zero.0130408 = 0.0026633398161568
(nCr(7,5)*nCr(69,26)/nCr(76,31)) * 0.0130408 = 0.0010033977912033
(nCr(7,6)*nCr(69,25)/nCr(76,31)) * zero.0130408 = zero.0001976389588734
(nCr(7,7)*nCr(69,24)/nCr(76,31)) * zero.0130408 = 0.0000156856316566
ADDED PROBABILITIES PER FREE GAME:
3/5 : 0.1581890954550239+0.0154837443092076+0.0033119854089495+0.0039950097242352 = zero.1809798348974162
4/5 : 0.0471792039076387+0.0056575219591335+0.0016207588171455+0.0026633398161568 = 0.0571208245000745
PROGRESSIVE HIT: zero.0078089716812643+0.0006617772611241+0.0000220592420375+0.0011528535312951+0.0001209784569878+0.0000050276761346+0.000445708674715+0.0000636726678164+0.0000036384381609+0.0010033977912033+0.0001976389588734+0.0000156856316566 = 0.011501410011269
TOTAL ADDED PROBABILITIES DURING FREE GAMES * FREE GAME PROBABILITY:
3/5 (0.1809798348974162*18.3086771581) * zero.0055056 = zero.018242813138614555793199790556832
4/5 (0.0571208245000745*18.3086771581) * zero.0055056 = zero.00575779355898468825039996183432
PROGRESSIVE HIT: (0.011501410011269*18.3086771581) * 0.0055056 = zero.00115934503855140059648226887184
TOTAL PROBABILITIES OF METER HITS OR PROGRESSIVE AWARDS:
3/5 zero.0893065819128652 + zero.018242813138614555793199790556832 = zero.107549395051479755793199790556832
4/5 zero.0138634666884298 + zero.00575779355898468825039996183432 = zero.01962126024741448825039996183432
PROGRESSIVE: 0.0008404745950033 + 0.00115934503855140059648226887184 = zero.00199981963355470059648226887184 OR 1 in 500.05
Simplified:
3/5 : .1075494
4/5 : .0196213
5/5 : .0019998
We also know that there could be multiple progressives hit or meters moved throughout one preliminary spin, (because of Free Games) however by taking the average variety of Free Spins per incidence and the possibilities of every result on each preliminary Free Game, we fully account for all expectations.
Progressive Meter Added Values
Okay, since I’m maintaining the actual name of this game secret, we’re just going to call these Progressives 1-10. The means it works is each individual progressive has its personal chance of displaying up on an preliminary spin and the worth added to the meter (based on one credit score bet) may be different.
The very first thing we’re going to do is list the probabilities that a progressive comes up on a spin (or a Free Game)---one will ALWAYS come up---and then issue in the chances of concurrently hitting 3/5 or 4/5 in addition to corresponding meter transfer quantity.
Progresive 1:
three Hits: four
4 Hits: 8
Probability: .005
Progressive 2:
3 Hits: 2
four Hits: 6
Probability: .016
Progressive three:
3 Hits: 2
4 Hits: four
Probability: .026
Progressive four:
3 Hits: 2
four Hits: three
Probability: .031
Progressive 5:
3 Hits: 2
four Hits: 3
Probability: .041
Progressive 6:
three Hits: 2
4 Hits: three
Probability: .052
Progressives 7-10:
3 Hits: .5
four Hits: 1
Probability: .828 (Combined)
Armed with this information, we must now take the meter increase chances & values and multiply them by the chances of having that exact meter.
Progressive 1:
((.1075494*4)+(.0196213*8)) * .005 = 0.00293584
Progressive 2:
((.1075494*2)+(.0196213*6)) * .016 = zero.0053252256
Progressive 3:
((.1075494*2)+(.0196213*4)) * .026 = zero.007633184
Progressive four:
((.1075494*2)+(.0196213*3)) * .031 = zero.0084928437
Progressive 5:
((.1075494*2)+(.0196213*3)) * .041 = 0.0112324707
Progressive 6:
((.1075494*2)+(.0196213*3)) * .052 = 0.0142460604
Progressives 7-10:
((.1075494*.5)+(.0196213*1)) * .828 = zero.060771888
TOTAL METER CONTRIBUTIONS:
zero.00293584+0.0053252256+0.007633184+0.0084928437+0.0112324707+0.0142460604+0.060771888 = zero.1106375124 or .1106375
RETURNS WITH METER MOVE ADDED:
THREE-SPOT: zero.7571035 + .1106375 = 0.867741
FOUR-SPOT: 0.792997 + .1106375 = zero.9036345
FIVE-SPOT: zero.786126 + .1106375 = 0.8967635
HITTING THE JACKPOTS:
The last factor we have to take a look at is the precise returns of the bottom jackpots along with the chance of hitting them. The jackpots have totally different starting values, so we must take a look at all of them individually.
Also, the likelihood for each jackpot to be obtainable on a given sport is completely different, but general, it will observe the same methodology because the previous section.
Progressive 1:
(.0019998 * 200) * .005 = zero.0019998
Progressive 2:
(.0019998 * 125) * .016 = 0.0039996
Progressive three:
(.0019998 * 100) * .026 = 0.00519948
Progressive 4:
(.0019998 * 50) * .031 = 0.00309969
Progressive 5:
(.0019998 * 20) * .041 = 0.001639836
Progressive 6:
(.0019998 * 10) * .052 = zero.001039896
Progressive 7:
(.0019998 * 5) * .207 = 0.002069793
Progressive 8:
(.0019998 * 3) * .207 = 0.0012418758
Progressive 9:
(.0019998 * 2) * .207 = 0.0008279172
Progressive 10:
(.0019998 * 1) * .207 = zero.0004139586
TOTAL BASE PROGRESSIVE CONTRIBUTION:
0.0019998 + 0.0039996 + 0.00519948 + 0.00309969 + zero.001639836 + 0.001039896 + zero.002069793 + zero.0012418758 + zero.0008279172 + 0.0004139586 = zero.0215318466
RETURNS WITH METER MOVE ADDED AND JACKPOTS:
THREE-SPOT: zero.7571035 + .1106375 = 0.867741 + 0.021531 = 0.889272
FOUR-SPOT: 0.792997 + .1106375 = zero.9036345 + zero.021531 = zero.9251655
FIVE-SPOT: zero.786126 + .1106375 = 0.8967635 + zero.021531 = 0.9182945
Advantage Play
The best method to advantage play this is in a position to be to repair in on Progressives 7-10 and play these in the occasion that they make the game positive.
The finest returning version of this sport returns .9251655 with no will increase to the jackpot AND together with meter moves. Of course, you’re not going to hit each meter that you simply transfer. The very first thing that we're going to do is calculate the probability of each particular person jackpot hitting and then decide what the jackpot value must be (both including and not together with meter transfer within the return) for the individual jackpots:
Progressive 1:
.0019998 * .005 = zero.000009999 or 1 in one hundred,010
Progressive 2:
.0019998 * .016 = zero.0000319968 or 1 in 31,253
Progressive three:
.0019998 * .026 = 0.0000519948 or 1 in 19,233
Progressive 4:
.0019998 * .031 = zero.0000619938 or 1 in sixteen,131
Progressive 5:
.0019998 * .041 = 0.0000819918 or 1 in 12,196
Progressive 6:
.0019998 * .052 = zero.0001039896 or 1 in 9,616
Progressive 7:
.0019998 * .207 = zero.0004139586 or 1 in 2,416
Progressive eight:
.0019998 * .207 = zero.0004139586 or 1 in 2,416
Progressive 9:
.0019998 * .207 = zero.0004139586 or 1 in 2,416
Progressive 10:
.0019998 * .207 = zero.0004139586 or 1 in 2,416
Positive Points
The subsequent step involves some easy algebra and solving for, “X.” The four-spot has the best return, so we’re going to use it. What we're going to need is for the added progressive return for each meter (by itself) to deliver us up to 100%.
Progressive 1:
With Meter: (1 - 0.9251655) = x * zero.000009999--->7484.19842/4 = $1,871.05
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.000009999--->18549.054905/4 = $4,637.26
Progressive 2:
With Meter: (1 - 0.9251655) = x * zero.0000319968--->2338.812006/4 = $584.70
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0000319968--->5796.579658/4 = $1,449.14
Progressive three:
With Meter: (1 - zero.9251655) = x * 0.0000519948--->1439.268927/4 = $359.eighty two
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0000519948--->3567.125943/4 = $891.seventy eight
Progressive 4:
With Meter: (1 - zero.9251655) = x * 0.0000619938--->1207.128777/4 = $301.78
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0000619938--->2991.783049/4 = $747.95
Progressive 5:
With Meter: (1 - 0.9251655) = x * 0.0000819918---912.707124/4 = $228.18
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0000819918--->2262.079867/4 = $565.52
Progressive 6:
With Meter: (1 - 0.9251655) = x * zero.0001039896--->719.634463/4 = $179.ninety one
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0001039896-->1783.562972/4 = $445.89
Progressive 7:
With Meter: (1 - zero.9251655) = x * 0.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0004139586-->448.044804/4 = $112.01
Progressive eight:
With Meter: (1 - zero.9251655) = x * 0.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0004139586-->448.044804/4 = $112.01
Progressive 9:
With Meter: (1 - 0.9251655) = x * zero.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0004139586-->448.044804/4 = $112.01
Progressive 10:
With Meter: (1 - 0.9251655) = x * zero.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0004139586-->448.044804/4 = $112.01
Progressive 7-10 (Combined):***
With Meter: (1 - zero.9251655) = x * (4 * 0.0004139586)-->45.194435/4 = $11.30
Without Meter: (1 - (0.9251655 -0.1106375)) = x * (4 * 0.0004139586)-->112.011201/4 = $28.00
***With this, we're determining the average that we'd want Progressives 7-10 at since all of them have the identical likelihood, but are not likely to have one positive by itself. By using the typical, you might even hit certainly one of them and have the play continue, however it lets you primarily play positively extra often by treating it as one progressive.
Advantage Play:
As you'll find a way to see, we did have a look at Progressives 7-10 as an entire, however this is not the kind of play the place I would have a look at the entire thing when determining if I need to play.
The reason why is as a end result of, virtually at all times, you’re going to end up hitting a progressive that forces you to stop enjoying as it causes the overall return of the game to drop back beneath 100%. In the meantime, you’ve additionally fed the meters of the opposite potential jackpots, however you won’t be staying to take them down, so it’s lost cash.
For that reason, I advocate disregarding the meter motion when figuring out whether or not one thing is a play, with exception ONLY to the progressive, or progressives, if you’re going after 7-10, mixed, that you’re really targeting.
Ideally, you’ll be playing for something considerably doubtless (such because the 7-10 combination play) that shall be above one hundred pc with out breaking your back attempting to squeeze every little tenth or hundredth of a share to justify taking part in it.
Or, perhaps you’ll have an individual jackpot that puts it above one hundred pc by itself (ignoring meter move on every thing else) and then any will increase over the progressive base are simply gravy. 105% when you’re ignoring different contributions is all the time better than 105% after adding each little scrap of value you can justify.
Calculating Other Keno Games And Help
The very first thing that I want to reiterate is that this is, by far, probably the most difficult video keno game that I’ve ever analyzed. Not solely does this recreation have several moving elements, (ten totally different conditional progressives and three units of numbers with interrelated probabilities) however there are additionally Free Games that include two partially Wild numbers (they can be used for Progressives and Base Pays, but do not substitute for extra attracts or extra Free Games).
The math used in that is going to be helpful to you for MOST Video Keno games and Progressives as some side of this recreation additionally covers (in full) what you would need to know tips on how to do in analyzing other Video Keno video games.
In most instances, any Video Keno Progressive is going to be associated to your base picks ultimately, so then it turns into a relatively easy matter of just figuring out the base return of the sport based mostly on the seed quantity, determining the contribution of the meter (that will add to the progressive as you play) and determining the return of the Free Games.
Cleopatra Keno, for one instance, usually has a Progressive for occasions similar to hitting 6 out of 6, 7 out of 7, eight out of 8...and so on. That just has to do with those base events (or hitting it throughout Free Games) so that’s all that must be found out. Of course, Cleopatra Keno is an example of a recreation that’s already been absolutely analyzed on Wizard of Odds.
With a handy calculator---making it a easy matter of simply inserting the paytable and changing the Progressive dollar amount to credits.
But, new Keno video games appear to be at all times coming out recently, so understanding how to do the kind of analysis included on this web page goes to provide the benefit of being one of many only people to know the play points for semi-complicated to very complicated Video Keno variants.
If you encounter a game and need assistance with calculating it, then you should be able to determine it out based on what is included above. I might name ten different Keno games that the methodology above would cowl, and greater than that, you would solely must know a segment of what I’ve accomplished above as a outcome of, once more, most Keno games don't have so many moving elements.
However, if you find yourself with a query as to methodology, then feel free to depart a remark to this article, to PM me on WizardofVegas at Mission146, or to friend me (Brandon James) on Facebook and send me a direct message. I’ll answer any methodology questions that you have got for free.
If you want to me to investigate a Video Keno recreation FOR YOU, then be happy to shoot me a message and we will focus on my charges. I recommend attempting to do it your self because I can already inform you it’s not going to be low-cost. These forms of analyses normally take a quantity of hours---even for a comparatively easy sport. Most of it includes just repeating a variant of the same formulation used for different parts---as you possibly can see above.
Conclusion
Anyway, I hope that this has helped you to study a number of the methodologies that can be utilized in analyzing Video Keno games.
If you could have reached this conclusion and do not perceive what was accomplished above, feel free to go away any questions within the comments. I tried to explain what I was doing and why with as much element as attainable, in addition to to obviously separate the totally different elements of the sport, however it’s nonetheless an insanely complicated Keno recreation. Don’t be afraid to read it a quantity of occasions if you end up confused or to take a break and are available back to it.
I definitely did.
I at all times say that 90% of gambling math is NOT figuring out the answers to things---figuring out solutions is straightforward. 90% of playing math, no much less than for my part, is figuring out what questions must be asked and then figuring out tips on how to ask them accurately. I spent as much time determining HOW one may go about determining this recreation as I spent actually doing the figuring.
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What does it all mean?
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Guide to Slot Terms and Vocabulary
3D Video Slots – The most graphically advanced slots that supply different visible effects and animations.
3-Reel – A basic slot format that includes three spinning reels.
5-Reel – The commonest structure for up to date slots, with 5 vertical spinning reels.
243 Ways to Win – Instead of payline-based wins, slots of this kind award wins each time symbols line up on adjacent reels from left to proper or right to left.
All Ways – A slot whose paylines operating both left-to-right and right-to-left. Those are often online video games, and are considered extremely worthwhile for players
Annuity Winner – A jackpot win that's paid in installments over a interval of years. Usually
Autospin – A function that permits slot gamers to set a pre-determined variety of spins at a sure guess stage. Beloved by online punters.
Bonus Feature – Any gambling fan’s favorite a part of a slot sport. The Bonus is an extra spherical that has totally different rules from the bottom recreation and often offers higher payouts.
Branded Slots – Slots based mostly on popular franchises, similar to TV collection, movies, or books.
Brick and Mortar – Land-based on line casino, visited by real people in bodily area. Usually refers to a casino or other business.
Cascading Reels – A relatively latest function in on-line slots that has confirmed fairly well-liked. When you hit a successful mixture, the winning symbols would ‘explode’, releasing area for brand spanking new ones that might potentially trigger a further win. This is a re-spin of kinds.
Casino – The purveyor or supplier of slot action. May be land-based or online.
Casino Bonus – This refers to the new participant welcome bonus that almost all on-line casinos provide. These are often more money that should be played by way of slot video games.
Coins – In the past, slot machines had been performed with actual cash, however nowadays this is not overly sensible – particularly on-line. However, most slot machines still let you measure your wager in cash or credit.
Coin Size – The denomination of the cash you are betting. Most slot machines permit you to set each the number and the denomination of the coins you'll be betting per payline.
Cold Slots – Slang for a slot that does not pay usually, or that hasn’t paid out shortly.
Collect – This is a button in online slots that permits you to convert your credit back into currency. Oftentimes that is provided within the context of a gamble or bonus function.
Expanding Wilds – Feature (or Bonus Feature) – One or more bonuses supplied by a sport. Oftentimes, they permit for large wins with out When there are several obtainable, the function refers again to the prime bonus spherical that gives the best payouts.
Free Spin – One of the most common features obtainable. These are extra spins usually triggered by scatter symbols, which are price you nothing and offer you wins based mostly in your last wager.
Fruit Machine – The widespread title of slot games in the UK. Based on the fruit symbols that appeared on most slot machine models in the course of the Twenties.
Gamble – A 50/50 chance offered by some slots to go ‘double or nothing’ on a win. Sometimes they are often played multiple occasions in a row, providing the slot machines provide the choice to double or even triple your winnings – if you are keen to risk them. The Gamble function is most video games is the equivalent of a coin toss. When you hit a profitable mixture, you've the choice to bet it on a 50/50 chance to either double it, or lose everything
Gaming Symbols – The icons that appear on the reels, awarding wins when in the applicable positions.
Hit and Run – A tactic generally used at land-based casinos. Players will attempt a slot for a couple of spins, and immediately abandon it if it doesn’t pay out. A doubtful tactic.
Hot Slots – Slang for a slot that pays out typically, or that has recently paid out many times.
Icons – (see Gaming Symbols)
Instant Winner – This is a jackpot that's paid in one instalment to the fortunate winner.
Jackpot – The prime prize obtainable. Jackpot might check with the highest-paying profitable combination, or to a separate jackpot that is triggered randomly.
Line Bet – The quantity bet per payline. For instance, a total guess of £10 on a 10-payline slot yields a line guess of £1.
Linked Progressive Jackpot – When two or more slot machines whose progressive jackpots are all linked collectively. Though harder to set off, a lot of the largest wins in slot historical past have resulted from this type of jackpot.
Loose – A slot that pays out regularly.
Max Bet – The maximum wager that may be made per spin, with all paylines energetic and the best possible coin denomination. In most cases, slot technique dictates that you must play with all potential paylines energetic.
Min Bet – The minimal amount that might be wagered per spin. This can apply each to working a single payline, or inserting the minimal line guess on all paylines. The minimal wager allows you to stretch your payroll for a most number of spins.
Multiplier –
Nudge – A nudge slot machine allows the user to maneuver or ‘nudge’ a key line up or down to assist create a winning mixture.
One-Armed Bandit – An previous nickname for a slot machine, going back to the beginnings of slot play. A disparaging term, the “one-arm” refers back to the massive lever – or arm – which was pulled to set off the reels at the start of a brand new spin. The lever appears on just one aspect, giving the looks of being ‘one-armed’. The “bandit” was intended to point out that the slot machine would go away its patrons poorer than it discovered them, in impact robbing them like a bandit.
Payline – The layout of reel positions leading to payouts to players. Different video games provide varying quantities and kinds of paylines, but extra of them often means more frequent wins.
Pay Table – The list of player payouts supplied for various symbol mixtures.
Pokies – An Australian time period for on-line slots (pokie machines).
Progressive Slot Machines –
Punter – A gambler or slot player.
Reels – The vertical lines of symbols, that spin throughout each play.
Re-spin – An extra spin of the reels awarded without a further wager. Essentially, two (or more) rounds for the price of one.
RNG (Random Number Generator) – The most important software present in trendy slots, RNGs generate long strings of random numbers, which dictate the end result of any individual spin. The randomness of these numbers is closely monitored to ensure honest play.
RTP (Return-to-player) – A mathematical expression of what share of whole bets positioned on a given slot machine might be returned to gamers. The higher the RTP, the better the odds for players. A crucial idea.
Rows – The horizontal strains of symbols.
Scatter – Special gaming symbols which pay regardless of payline place; often, they open the way to bonus options.
Spin – The spin button activates the slot for a single spin – or round. It replaced the lever in basic slot machines
Stacked Symbols – Matching symbols stacked on neighboring positions in a single reel. Stacked symbols greatly improve the chances of touchdown a giant win.
Stacked Wilds – Wild symbols that appear vertically stacked on the same reel, just like stacked symbols. An glorious turn of occasions for players.
Streaky – A recreation, participant, or provider that appears to show scorching or cold for lengthy durations of time. A notion seldom based mostly in mathematical truth.
Total Bet – All the money that has been wagered on a single spin.
Volatility – (also often recognized as variance) refers back to the frequency of payouts vs. the scale of those wins. Low volatility / low variance slots are inclined to award frequent, small payouts, while excessive volatility / high variance slots award fewer wins of a larger measurement.
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