Analyzing-Keno-ProgressivesGuide-to-Essential-Slot-Gaming-Terms-and-Vocabulary-h
Brandon James takes a have a look at a relatively new and extremely complicated Video Keno recreation for benefit play functions. The recreation comes with a Free Games characteristic, Extra Draws function in addition to ten totally different Progressives. James will present his full mathematical breakdown of this recreation (using math anyone can do) and shows you the means to go about doing it your self for comparable video games.
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Math Lesson
Greetings!
I have lately been offered with a new Video Keno recreation to research that I’m going to share with everybody here. I even have been requested not to truly title this Keno game as a result of it’s only available in sure markets that may mostly not apply to any readers of LCB.
So, why am I sharing the analysis?
Quite frankly, this is one of the most tough Video Keno games that I’ve ever analyzed, so I imagine that sharing my methodology might be useful for anybody on the market who needs to investigate keno progressives for themselves to seek out advantageous play points. In addition to land casinos, (and some comfort playing locations) I ought to think about that some on-line casinos even have Keno games with tough options or progressives to analyze.
Even if you’re simply someone who enjoys Keno and aren't essentially in search of advantageous conditions; it’s always good to know the method to decide the return-to-player on keno video games that include some type of gimmick--and this game has gimmicks galore.
Many folks would program a simulation and analyze this game that way--by merely creating the principles and having a computer program simulate hundreds of thousands of spins. There are a few things about that I should mention:
1.) I don’t know how to program. In fact, should you requested me to complete the easy task of creating a new E-Mail tackle, it will probably take me twice as long to do it as it will you.
2.) In this explicit case, and given the amount of time it would take for the simulations to be done---I consider that simply doing the math could be FASTER for this one. I assume I fully analyzed three totally different versions of this game in about ten hours.
The Game
As I’ve mentioned, I won’t be disclosing the title of this Keno recreation, however I’ve been giving a blessing to share all the other details of my analysis.
This game can primarily be broken down into three components, though these completely different parts are extremely intertwined and all come together to end in an general return-to-player and advantageous progressive points.
One other factor that I ought to point out is that this game has different paytables for anyplace from three to 10 different picks the participant can make. However, most benefit players are NOT going to want a ridiculous amount of Variance when going after a Video Keno play---particularly not a play on a sport that has several completely different options and is top-heavy anyway. For that reason, I decided to solely analyze the Pick three, Pick four and Pick 5 versions of this sport.
The very first thing that we are going to do is have a glance at all the totally different parts of the sport:
1.) Base Pays
-The first thing that the complete evaluation will consider is the base paytable of the game. Essentially, if the game had no other features in addition to selecting 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 three: 0.499513145082765
Choose four: zero.486358578130730
Choose 5: zero.421164278759215
THREE-SPOT:
2/3 PAYS: 2
3/3 PAYS: sixteen
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 recreation is that it's going to randomly draw 5 SEPARATE numbers from the player’s selections that will award either additional draws, free games, or both when the player hits either 3, four or all 5 out of five for the Free Games.
This will get extremely situational as a result of the way that this stuff are awarded is as follows:
three of 5: Five Extra Draws
4 of 5: Ten Extra Draws
5 of 5: Fifteen Free Games (No Additional Draws)
By itself, this may already be sophisticated sufficient, however, the additional draws on the three out of 5 hit may find yourself in Free Games AFTER the extra numbers are drawn (three out of 5 turns into five out of five) or can end result in another set of extra draws (three out of five turns into four out of five). In different phrases, there are a number of potential serieses of occasions attainable:
When you begin with three out of 5, the following issues can happen:
A.) You get zero further on this set of 5 and solely end up together with your five further attracts.
B.) One of these additional five numbers matches on this set of five and awards ten extra attracts for a complete of fifteen additional draws. Based on these ten further attracts AFTER the original 5, two things are possible:
-The player 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 further draws, the player might hit each of the opposite numbers in that set of five numbers, thereby getting Free Games but NOT getting the ten extra attracts.
The player can even instantly start with 4 out of 5, which will award ten further attracts from which:
A.) The player can hit the fifth quantity from that set of 5 and go into Free Games.
B.) The player does NOT hit the fifth quantity and does NOT go to Free Games.
Finally, the participant might simply hit 5/5 proper off the bat and go immediately into Free Games.
In different phrases, there could be both five, ten or fifteen further attracts (that will all come with their own corresponding chances that affect this---and other---aspects of the game) and the player can even hit Free Games or not hit Free Games.
***And, if that’s not already difficult sufficient, we must account for the fact that IF the participant hits any outcomes on this particular set of five numbers, then these hits CANNOT also double as hits on the player’s base Picks BECAUSE these numbers will all the time be separate from these ones as they aren't chosen by the participant.
***For instance, if a participant hits four out of five of this set of numbers on the initial draw, then that automatically signifies that four numbers DO NOT match the player’s base picks, nevertheless, the bottom pays are affected by these numbers as well as the additional draws. What that means is that we ALSO have to determine the possibilities for the numbers of picks that the player could have hit on the base alternatives given the decreased variety of possible matches.
***In effect, if we assume that the player matches three of those 5 numbers (which we even have to determine the possibilities for), then that can leave seventeen different attracts that both did or did not hit the player’s base numbers. We should work out the probabilities for each potential occasion and the way they relate to the extra draws.
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3.) Free Games:
-Okay, so the Free Games must be pretty straightforward, right? You have fifteen Free Games and it simply goes by the bottom paytable, yes?
Haha. No.
The Free Games ALSO give the likelihood for getting extra attracts as well as to retrigger extra Free Games. Here are the rules for a way the Free Games work:
A.) There are at all times fifteen Free Games initially.
B.) Free Games will include TWO Wild spots that depend in path of the Base Paytable in addition to the Jackpots. However, the WILDS do NOT go towards the lucky picks that award further draws and more 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 maximum variety of Free Games is 1,000, however we’re not going to go that far.
Okay, so the first thing that we've is identical additional draw mechanism as earlier than EXCEPT that the result of 4 out of five now offers ten extra draws AND extra Free Games while the 5 out of five for those alternatives awards further free video games solely. Therefore, the next occasions are potential.
a.) zero, 1 or 2 out of five---nothing occurs.
b.) Three out of 5, and then:
aa.) The player hits two extra numbers for a total of 5/5 after the 5 further attracts and is awarded an extra ten Free Games.
bb.) The participant hits zero additional numbers on this set of 5 and does NOT get any extra Free Games or additional draws---aside from the original additional 5.
cc.) The participant hits one of many set of 5 numbers (two remaining) and gets a total of 4 out of five which leads to ten more draws (fifteen total) and a minimal of 5 further Free Games, but:
aaa.) The ten further draws end in hitting the final one wanted for that set of five and getting ten additional Free Games---which would imply fifteen complete further Free Games.
Or:
bbb.) The ten additional draws do NOT hit the final number wanted on that set of five, so the participant gets solely the five extra Free Games.
c.) Four out of 5 hits on an initial Free Games spin and then:
aa.) The ten extra draws end in hitting the final one wanted for that set of 5 and getting ten further Free Games---which would imply fifteen whole further Free Games.
Or:
bb.) The ten further attracts do NOT hit the final quantity wanted on that set of five, so the player will get only the 5 extra Free Games.
d.) Five out of 5 hits on an initial Free Games spin and easily awards ten extra Free Games.
As a end result, the next occasions are attainable:
NO extra draws or Free Games awarded.
Five further attracts and no extra video games awarded.
Fifteen additional attracts and 5 further games awarded.
Fifteen extra draws and fifteen extra video games awarded.
Ten additional attracts and five further games awarded.
Ten extra draws and fifteen further games awarded.
Zero further attracts and ten further video games awarded.
Five extra draws and ten additional games awarded.
---And, we've to determine the probabilities of each one of these events.
The excellent news is that we DO NOT have to determine the likelihood for the entire variety of free games (between 15 and 1,000) that the player will obtain. One shortcut here is solely to discover out the AVERAGE quantity of Free Games that might be awarded per Free Games preliminary spin, then multiply by fifteen as a outcome of that is how many initial spins we begin with. After that, we'll multiply the average number of further spins awarded (per spin) by what number of extra Free Games we count on to get from the fifteen. We will then continue to repeat that process till we’re primarily not adding a significant amount of Free Games to the anticipated total anymore.*
*By using this technique, we are going to end up with an quantity of TOTAL Free Games per preliminary fifteen (18.3086771581 TOTAL, on average, was the result), but the outcome might be slightly brief. I mainly used this methodology till I reached the purpose that the subsequent step would have solely added lower than 1/10000th of a Free Game and known as that ok. If anyone does resolve to simulate this, they will find that my average result of 18.3086771581 Free Games TOTAL (per initial set of fifteen Free Games) is extraordinarily shut.
Not only that, but as quickly as again, we've to determine how these numbers being removed from the preliminary set of numbers that we could hit impacts the possibilities as relates our Base Picks and Progressives!!!
Oh, proper, 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 fairly as bad as it appears. The means the Progressives work is that five numbers on the Keno board are randomly chosen by the game (these numbers will NOT match the player’s numbers OR any numbers used for further draws/free games) and there might be completely different events that happen based on hitting three, 4 or five of those numbers.
If the player hits 5 out of 5 of these numbers, then the participant is awarded one of the PROGRESSIVES.
So, the place do the totally different progressives come into play?
Okay, the Progressives are represented by different symbols (ten of them) all of which have a special probability of appearance as the 5 further numbers. In other words, solely one of many Progressives can be awarded on anybody spin and all of them have completely different rates of look. In reality, simply figuring out how likely it is for a particular progressive to appear goes to be one of our tasks---which will solely happen after we take care of how probably hitting considered one of them is.
If you hit three or 4 out of 5 of those numbers, you won’t be paid something, however there might be an quantity added to the Progressive meter for whichever progressive is energetic on the time. Of course, how much is added varies based mostly upon which meter it occurs to be, so we should decide all of those probabilities (and adds to the meter) individually so as to work out how a lot the progressive meter additions contribute to the base return of the sport.
---If that’s not tough sufficient yet, don’t worry, it’ll get a little more durable:
A.) The progressive five-spot possibilities are ALSO conditional with the Extra Draw chances. For example, if you hit three/five of the additional draw set of five to be able to get five further draws, then that implies that three of your progressive set of 5 mechanically DID NOT hit, so you must determine the probabilities based on seventeen completely different numbers that could hit it.*
*Technically, this could also be checked out as true for the bottom game hits, but it doesn’t matter in the case of the base sport hits. The purpose why it doesn’t matter is as a outcome of nothing concerning the base recreation numbers directly influences the likelihood of hitting the Progressive numbers. In other phrases, base sport hits and progressive hits can be handled individually from one another, but not from the Extra Draw and Free Games set of numbers.
B.) There are two WILD numbers that are randomly chosen during Free Games that may substitute as Progressive Numbers or substitute for the bottom paytable. Actually, they technically substitute for both simultaneously when they're hit. These numbers will NOT match any of the numbers selected for the Base Game, any of the progressive numbers OR any of the numbers selected for Free Games/Extra Draws.
On To The Math!!!
Okay, if the best way the sport works was difficult to comply with, (I asked a number of clarifying questions and in addition asked for pictures of the entire rules screens) don’t worry---because the mathematical breakdown isn’t going to make it any easier.
I would say simply do your greatest to comply with along, but don’t be afraid to cease and come back to it. I know that I stopped in my evaluation of those three paytables--several times---and came again to it later.
The good news is that I will reexplain each particular person phase of the sport under and the maths shall be accomplished individually for every component. Okay, right here we go:
Base Game Return (Without Extra Draws)
Based on WoO Keno Calculator:
Choose three: zero.499513145082765
Choose four: zero.486358578130730
Choose 5: 0.421164278759215
It is also potential to hit fifteen (total) additional draws by first hitting five additional numbers with three out of 5, however then having a kind of numbers hit and unlocking 4/5 with ten extra attracts. By this mechanism, the Free Games may also be hit.
Therefore, we should first decide our total Extra Draw possibilities after which our complete chances for Free Games accordingly.
3/5 Hits: zero.083935052289483
In order to obtain another ten attracts, this must first occur after which be adopted up with hitting one of many two remaining spots. Hitting both spots will simply unlock fifteen Free Games. We could additionally unlock the ten additional attracts and unlock Free Games after that.
At this level, there might be sixty numbers remaining, of which 5 shall be drawn.
nCr(2,1)*nCr(58,4)/nCr(60,5) = zero.1553672316384181
Therefore, that's the chance of this event and the general probability of getting fifteen further attracts is:
0.1553672316384181*0.083935052289483 = 0.0130407567116428407376253868423
The second thing that we're going to look at is hitting both of our other 5 numbers which can unlock Free Games:
nCr(2,2)*nCr(58,3)/nCr(60,5) = 0.0056497175141243
This will be added to our working chance of Free Games, but first, we should factor in the likelihood of being in this situation in the first place:
0.0056497175141243*0.083935052289483 = zero.0004742093349688310301048447369
0.0004742093349688310301048447369+0.000644924695558 = 0.0011191340305268310301048447369
Now, we should look at the probability of hitting ⅗, getting the 5 additional attracts, turning that into 4/5 and then hitting Free Games. There might be fifty five remaining numbers of which only one is useful, ten numbers are drawn:
nCr(1,1)*nCr(54,9)/nCr(55,10) = zero.1818181818181818
Of course, lots has to occur within the first place. This is identical as our likelihood of getting fifteen extra attracts (at all) from above:
zero.1818181818181818*0.0130407567116428407376253868423 = 0.0023710466748441526242817664868574411340838756
Therefore, our running chance of hitting the Free Games is:
zero.0023710466748441526242817664868574411340838756+0.0011191340305268310301048447369 = zero.0034901807053709836543866112237574411340838756
Finally, we might hit 4/5 of those numbers and be awarded ten additional draws. This will cause sixty numbers to remain of which just one helps us of the ten chosen:
nCr(1,1)*nCr(59,9)/nCr(60,10) = 0.1666666666666667
Of course, we should be in this state of affairs in the first place having acquired ⅘ on the preliminary draw, so:
zero.1666666666666667*0.012092338041705 = zero.0020153896736175004030779347235
zero.0020153896736175004030779347235+0.0034901807053709836543866112237574411340838756 = 0.0055055703789884840574645459472574411340838756
Therefore, we all know the following probabilities:
Extra Draw Probabilities:
3/5 Hits: 0.083935052289483***
Zero Extra: nCr(2,0)*nCr(58,5)/nCr(60,5) = zero.8389830508474576
One Extra (Initial Draw) = 0.1553672316384181
Two Extra (Initial Draw) = zero.0056497175141243
PROOF: 0.0056497175141243+0.1553672316384181+0.8389830508474576 = 1
PROBABILITY 3/5 + 0 EXTRA: zero.083935052289483*0.8389830508474576 = zero.0704200862428713282322697684208
PROBABILITY 3/5 + 1 EXTRA: 0.1553672316384181*0.083935052289483 = 0.0130407567116428407376253868423***
PROBABILITY 3/5 + 2 EXTRA: 0.083935052289483*0.0056497175141243 = zero.0004742093349688310301048447369---FREE GAMES TRIGGERED
***This is the one potential approach to get fifteen additional numbers.
***PROBABILITY OF 3/5 + 1 EXTRA, + 1 EXTRA AGAIN---FREE GAMES:
zero.0130407567116428407376253868423*0.1818181818181818 = 0.00237104667484415262428176648691198658862933014
***PROBABILITY OF 3/5 + 1 EXTRA + 0 EXTRA:
0.0130407567116428407376253868423*.818181818181 = 0.0106697100367880181662021541830964883198563
4/5 Hits: 0.012092338041705***
What matters here is whether or not or not we get Free Games, which may only be carried out if an additional number leads to Free Games.
From Above: nCr(1,1)*nCr(59,9)/nCr(60,10) = zero.1666666666666667
Therefore, the general likelihood of hitting 4/5 adopted by Free Games is:
zero.1666666666666667*0.012092338041705 = 0.0020153896736175004030779347235
The chance that this will not happen is:
0.012092338041705*.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: 0.0704201 or 1 in 14.2 (Five Extra Draws)
3/5 Hit, One Extra, No Free Games After: 0.0106697 or 1 in ninety three.72 (Fifteen Extra Draws Total)
3/5 Hit, One Extra, One Extra Again, Free Games: 0.0023711 or 1 in 421.seventy five (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: zero.0100769 or 1 in ninety nine.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.sixty three (Initial)
5/5 Hit: zero.0055056 or 1 in 181.sixty three (Total)
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FINAL EXTRA DRAW PROBABILITIES:
FIVE EXTRA: 0.0704201 + zero.0004742 = zero.0708943
TEN EXTRA: zero.0100769 + 0.0020154 = zero.0120923
FIFTEEN EXTRA: zero.0106697 + 0.0023711 = zero.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: zero.0055056 or 1 in 181.sixty three
BASE RETURN BEFORE FREE GAMES INCLUDING EXTRA DRAWS:
Based on WoO Keno Calculator:
Choose three: 0.499513145082765
Choose four: zero.486358578130730
Choose 5: 0.421164278759215
THREE-SPOT:
2/3 PAYS: 2
3/3 PAYS: 16
FOUR-SPOT:
2/4 PAYS: 1
3/4 PAYS: 3
4/4 PAYS: forty seven
FIVE-SPOT:
3/5 PAYS: 2
4/5 PAYS: 13
5/5 PAYS: 149
RETURNS BASED ON TOTAL NUMBER DRAWN:
DRAW TWENTY TOTAL:
Choose three: 0.499513145082765 * zero.9039726 = zero.451546196494644292239***
Choose four: 0.486358578130730 * 0.9039726 = 0.439654828405139137998***
Choose 5: 0.421164278759215 * 0.9039726 = zero.380720968097092357509***
***The cause for this calculation is because there’s lower than a 100% probability of drawing ONLY twenty numbers. Therefore, we must decide the different possibilities based mostly on the whole variety of balls which might be drawn by the time the sport ends. In this case, there might be only a ninety.39726% chance that only twenty numbers will be drawn with the opposite possibilities being greater than that.
THREE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):***
***Remember right here that the additional draws have already happened, so they MUST be assumed as having not hit other numbers as a result of they couldn’t have. That means, with five extra draws (three of the set of 5 that controls additional attracts hit) that there are only 22 numbers obtainable to be doing different issues. This may even apply differently in phrases of having ten or fifteen additional draws.
THREE-SPOT:
nCr(3,2)*nCr(74,20)/nCr(77,22) * 2 = 0.3473684210526316
nCr(3,3)*nCr(74,19)/nCr(77,22) * 16 = zero.3368421052631579
ADDED VALUE: (0.3473684210526316+0.3368421052631579) * zero.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) * zero.0120923 = 0.0127459378378378371842
DRAW 35 TOTAL (OUT OF 31 BECAUSE FOUR UNLOCKED EXTRA DRAWS)
THREE-SPOT:
nCr(3,2)*nCr(73,29)/nCr(76,31) * 2 = zero.5953058321479374
nCr(3,3)*nCr(73,28)/nCr(76,31) * 16 = 1.0230440967283073
ADDED VALUE: (1.0230440967283073+0.5953058321479374) * zero.0130408 = 0.02110457775248933188376
THREE SPOT TOTAL RETURNS:
zero.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 right here that the extra attracts have already happened, in order that they MUST be assumed as having not hit other numbers because they couldn’t have. That means, with five extra draws (three of the set of 5 that controls additional attracts hit) that there are only 22 numbers obtainable to be doing different issues. This will also apply in a different way in relation to having ten or fifteen further attracts.
nCr(4,2)*nCr(73,20)/nCr(77,22) * 1 = 0.2534850640113798
nCr(4,3)*nCr(73,19)/nCr(77,22) * three = 0.1877667140825036
nCr(4,4)*nCr(73,18)/nCr(77,22) * forty seven = 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 = 0.3103139188214891
nCr(4,3)*nCr(72,23)/nCr(76,26) * three = 0.3039809817026832
nCr(4,4)*nCr(72,22)/nCr(76,26) * forty seven = 0.5476724020343343
ADDED VALUE: (0.5476724020343343+0.3039809817026832+0.3103139188214891)* 0.0120923 = 0.01405085721272822935918
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) * three = 0.4729827159531558
nCr(4,4)*nCr(72,27)/nCr(76,31) * 47 = 1.1526763966562092
ADDED VALUE: (1.1526763966562092+0.4729827159531558+0.3588144741713595)*0.0130408 = 0.0258791231504900720596
FOUR SPOT TOTAL RETURNS:
0.439654828405139137998+0.04929322025604552550625+0.01405085721272822935918+0.0258791231504900720596 = zero.52887802902440296492303
FIVE SPOT ADDED RETURNS:
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):***
***Remember here that the extra draws have already occurred, in order that they MUST be assumed as having not hit other numbers as a outcome 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 obtainable to be doing different issues. This may even apply in a different way in relation to having ten or fifteen further attracts.
FIVE-SPOT:
nCr(5,3)*nCr(72,19)/nCr(77,22) * 2 = 0.2314932091428126
nCr(5,4)*nCr(72,18)/nCr(77,22) * 13 = zero.2647167715660866
nCr(5,5)*nCr(72,17)/nCr(77,22) * 149 = zero.1985931136616068
ADDED VALUE: (0.1985931136616068+0.2647167715660866+0.2314932091428126) * 0.0708943 = zero.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 = zero.4817416551374735
nCr(5,4)*nCr(71,27)/nCr(76,31) * 13 = zero.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 + 0.03714550630214930212464 + zero.01694489201833412307246 + zero.0492575790132309635158 = zero.4840689454308067462219 or forty eight.4069%
BASE GAME FINAL RETURNS:
THREE-SPOT: zero.533903
FOUR-SPOT: 0.528878
FIVE-SPOT: zero.484069
FIGURING OUT THE FREE GAMES:
The subsequent factor that we've to do is determine how the Free Games will impression our returns primarily based on the variety of spots that we now have selected.
The following are the summarized guidelines 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 in addition to the Jackpots. However, the WILDS do NOT go towards the fortunate picks that award further draws and extra Free Games.
3.) 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
4.) The maximum variety of Free Games is 1,000, but we’re not going to go that far.
Average Free Games
The first thing that we’re going to do is approximate the average variety of Free Games that a player receives per preliminary occurrence. The good news is that the chances for Free Games retrigger are simply going to be a variation of the above probabilities for the 3/5 and 4/5 potentialities from above.
3/5 Hit, No Extra Numbers: 0.0704201 or 1 in 14.2 (Five Extra Draws, no Retrigger)
3/5 Hit, One Extra, No Free Games After: 0.0106697 or 1 in 93.seventy two (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: zero.0100769 or 1 in 99.24 (Ten Extra Draws Total, Five Free Games Retrigger)
4/5 Hit, One Extra, Free Games: zero.0020154 or 1 in 496.18 (Ten Extra Draws Total, Fifteen Free Games Retriggered)
5/5 Hit: 0.0006449 or 1 in 1550.sixty three (Initial, Ten Free Games Retriggered)
5/5 Hit: zero.0055056 or 1 in 181.63 (Total)
FINAL EXTRA DRAW PROBABILITIES:
FIVE EXTRA: zero.0704201 + 0.0004742 = 0.0708943
TEN EXTRA: zero.0100769 + zero.0020154 = zero.0120923
FIFTEEN EXTRA: zero.0106697 + 0.0023711 = zero.0130408
ZERO EXTRA: 1 - (0.0708943+0.0120923+0.0130408) = zero.9039726
FINAL FREE GAMES RETRIGGER PROBABILITIES:
Five Free Games: zero.0106697 + 0.0100769 = zero.0207466
Ten Free Games: zero.0004742 + 0.0006449 = zero.0011191
Fifteen Free Games: zero.0023711 + 0.0020154 = zero.0043865
FREE GAMES EXPECTED BASED ON INITIAL SPINS:
The subsequent thing that we are going to do is determine how many additional free games we are expected to receive based mostly upon the fifteen preliminary Free Games. The first step to doing this is to find out what quantity of extra are expected on a per spin basis, like so:
(5 * zero.0207466) + (10 * 0.0011191) + (15 * 0.0043865) = 0.1807215
Therefore, we anticipate to obtain 0.1807215 retriggered Free Games per every of the initial fifteen spins. We can multiply this number by fifteen to get the whole number of anticipated extra Free Games per preliminary spins:
0.1807215*15 = 2.7108225
Okay, so now we are going to do the same thing primarily based on our anticipated 2.7108225 retriggered games:
2.7108225 * .1807215 = zero.48990390843
And then once more with that:
zero.48990390843 * .1807215 = 0.08853616918
And again:
zero.08853616918 * .1807215 = zero.01600038929
One last time:
zero.01600038929 * .1807215 = 0.00289161435
We will now add all these anticipated extra Free Games to our preliminary fifteen, which is going to get us shut sufficient:
15 + 2.7108225 + zero.48990390843 + 0.08853616918 + 0.01600038929 + zero.00289161435 = 18.3081545812
Therefore, for each initial set of fifteen Free Games, we are expected to get one thing alongside the lines of 18.308155 Free Games. If we had gone one more retrigger occurrence, it will have added about .00052257688 expected 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 name it 18.308677 Free Games expected per preliminary set.
While that will appear low, remember that the likelihood 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 in any respect. Granted, they can certainly add up rapidly when they are hitting, but we’ve principally accounted for that by occurring a foundation of expected further Free Games per preliminary set of spins.
Base Return Of Free Games
The excellent news here is that the Extra Draw possibilities are the identical, with the significant change being that there at the moment are two WILD spots during the Free Games. Another thing that is the same are the base return tables, so basically, you have 5 spots to hit both 2 or three spots on a three-spot card. You can also hit 4 and five (and be awarded as if you hit three).
BASE RETURN FREE GAMES BEFORE INCLUDING EXTRA DRAWS:
Based on WoO Keno Calculator:
Choose three: 2.087671821849037
Choose four: 2.190619507075203
Choose 5: 2.428321678321678
THREE-SPOT (BECOMES FIVE-SPOT):
2/3 PAYS: 2
3/3, 4/3, 5/3 PAYS: 16
FOUR-SPOT (BECOMES SIX-SPOT):
2/4 PAYS: 1
3/4 PAYS: 3
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 4: 2.190619507075203 * 0.9039726 = 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) * sixteen = 0.3258052573121066
nCr(5,5)*nCr(72,17)/nCr(77,22) * sixteen = zero.0213254350240652
ADDED VALUE: (1.8519456731425008+0.6134570042284534+0.3258052573121066+0.0213254350240652)*0.0708943 = 0.1993925844720279027818
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) * 16 = 0.6473669054779365
nCr(5,5)*nCr(71,21)/nCr(76,26) * 16 = 0.0569682876820584
ADDED VALUE: (0.0569682876820584+0.6473669054779365+2.7583459450799033+0.6895864862699758)*0.0120923 = 0.0502104657958607493702
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) * 16 = 3.8539332410997876
nCr(5,4)*nCr(71,27)/nCr(76,31) * sixteen = 1.2262514858044779
nCr(5,5)*nCr(71,26)/nCr(76,31) * 16 = zero.1471501782965373
ADDED VALUE: (0.1471501782965373+1.2262514858044779+3.8539332410997876+0.7143065921003917)*0.0130408 = 0.0774837584380054172356
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 ultimate few steps will have us multiplying the expected return (in credits) by the whole expected variety of Free Games after which factoring that in opposition to the likelihood of hitting Free Games in the first place:
(2.2142849334458940693876*18.3086771581) * 0.0055056 = 0.2232005
THREE SPOT BASE RETURN WITHOUT PROGRESSIVES:
0.2232005 + 0.533903 = 0.7571035 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) * 47 = 2.1533691225472047
nCr(6,5)*nCr(71,17)/nCr(77,22) * 47 = 0.287115883006294
nCr(6,6)*nCr(71,16)/nCr(77,22) * forty seven = 0.0147908182154757
ADDED VALUE: (0.0147908182154757+0.287115883006294+2.1533691225472047+0.5112141701903778+0.3322892106237456) * zero.0708943 = 0.23386464256347551036254
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
nCr(6,2)*nCr(70,24)/nCr(76,26) * 1 = 0.3423651217044598
nCr(6,3)*nCr(70,23)/nCr(76,26) * 3 = 0.6992989719920882
nCr(6,4)*nCr(70,22)/nCr(76,26) * forty seven = three.937198899601288
nCr(6,5)*nCr(70,21)/nCr(76,26) * 47 = zero.7070887819692109
nCr(6,6)*nCr(70,20)/nCr(76,26) * 47 = 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 = 0.3169106711431315
nCr(6,3)*nCr(70,28)/nCr(76,31) * 3 = zero.8752770917286489
nCr(6,4)*nCr(70,27)/nCr(76,31) * 47 = 6.696887515784314
nCr(6,5)*nCr(70,26)/nCr(76,31) * 47 = 1.6437814811470589
nCr(6,6)*nCr(70,25)/nCr(76,31) * forty seven = zero.158290068554902
ADDED VALUE: (0.158290068554902+1.6437814811470589+6.696887515784314+0.8752770917286489+0.3169106711431315) * zero.0130408 = zero.12638030755925172755624
FOUR SPOT TOTAL EXPECTED RETURN PER FREE GAME:
Add them up: 2.190619507075203 + 0.12638030755925172755624 + 0.06935475772953615190391 + zero.23386464256347551036254 = 2.62021921492746638982269
FOUR-SPOT ADDED RETURN OF FREE GAMES:
The final few steps could have us multiplying the anticipated return (in credits) by the whole expected variety of Free Games after which factoring that towards the likelihood of hitting Free Games in the first place:
(2.62021921492746638982269*18.3086771581) * zero.0055056 = 0.2641187596796262218413489029829228343184
FOUR SPOT BASE RETURN WITHOUT PROGRESSIVES:
0.2641187596796262218413489029829228343184 + 0.528878 = 0.7929967596796262218413489029829228343184 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) * 13 = 1.0374287561727268
nCr(7,5)*nCr(70,17)/nCr(77,22) * 149 = 2.4229758409769986
nCr(7,6)*nCr(70,16)/nCr(77,22) * 149 = 0.254262896892648
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 = 0.29506536408535234892582
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
FIVE-SPOT:
nCr(7,3)*nCr(70,23)/nCr(76,26) * 2 = 0.8158488006574362
nCr(7,4)*nCr(70,22)/nCr(76,26) * thirteen = 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 = 0.0615347183641124
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 = 4.3221047087685998
nCr(7,5)*nCr(70,26)/nCr(76,31) * 149 = 18.238979625918962
nCr(7,6)*nCr(70,25)/nCr(76,31) * 149 = three.5126923723992075
nCr(7,7)*nCr(70,24)/nCr(76,31) * 149 = 0.2727245630744726
ADDED VALUE: (1.0211566070167571+4.3221047087685998+18.238979625918962+3.5126923723992075+0.2727245630744726)*0.0130408 = zero.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 anticipated return (in credits) by the whole anticipated variety of Free Games after which factoring that towards the probability of hitting Free Games within the first place:
(2.99658864230849032317022*18.3086771581) * zero.0055056 = zero.3020568931666592841333559378737508509792
FIVE SPOT BASE RETURN WITHOUT PROGRESSIVES:
0.3020568931666592841333559378737508509792 + 0.484069 = zero.7861258931666592841333559378737508509792 OR seventy eight.6126%.
FINAL RETURNS BEFORE PROGRESSIVES AND METER CONTRIBUTIONS:
THREE-SPOT: 0.7571035
FOUR-SPOT: 0.792997
FIVE-SPOT: zero.786126
THE PROGRESSIVE METERS:
This is an uncommon recreation in that there are ten totally different progressive meters all primarily based on a five-spot that's separate from the Bonus Games and from the participant spots chosen.
Each of these ten progressives has a certain possibility of being chosen, which work as follows:
5/5: Awards Progressive
3/5 & 4/5 : Adds to Progressive
The amounts added to the Progressive range as do the probabilities of a particular progressive even coming up. In order to figure within the progressives, we have to take a look at the possibilities of hitting three out of 5, 4 out of 5 and five out of 5 each within the base recreation and through Free Games. Fortunately, we've already carried out these probabilities:
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 = zero.0005829942538474
DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):
(nCr(5,3)*nCr(72,19)/nCr(77,22)) *.0708943 = 0.0082057745084666
(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 = 0.0004892596769444
(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)) * 0.0130408 = 0.0031411482881584
(nCr(5,4)*nCr(71,27)/nCr(76,31)) * zero.0130408 = zero.0009994562735049
(nCr(5,5)*nCr(71,26)/nCr(76,31)) * zero.0130408 = 0.0001199347528206
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) = 0.0138634666884298
PROGRESSIVE HIT: 0.0005829942538474+0.0000944907367642+0.0000430548515711+0.0001199347528206 = 0.0008404745950033
FREE GAMES AND PROGRESSIVE METERS (HITS)
The next thing that we have to do is account for the chances of transferring the Progressive Meters or hitting a progressive during the Free Games. We must remember that the WILD spots additionally depend in course of the Progressive five-spot, so we're essentially drawing from seven as an alternative.
PROBABILITY OF HITTING FREE GAMES: 0.0055056
AVERAGE NUMBER OF FREE GAMES: 18.3086771581
NORMAL PROBABILITIES * 0.9039726 (PROBABILITY OF NO EXTRA DRAWS)
(nCr(7,3)*nCr(73,17)/nCr(80,20)) * .9039726 = zero.1581890954550239
(nCr(7,4)*nCr(73,16)/nCr(80,20)) * .9039726 = 0.0471792039076387
(nCr(7,5)*nCr(73,15)/nCr(80,20)) * .9039726 = zero.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 = zero.0154837443092076
(nCr(7,4)*nCr(70,18)/nCr(77,22)) * .0708943 = zero.0056575219591335
(nCr(7,5)*nCr(70,17)/nCr(77,22)) * .0708943 = 0.0011528535312951
(nCr(7,6)*nCr(70,16)/nCr(77,22)) * .0708943 = 0.0001209784569878
(nCr(7,7)*nCr(70,15)/nCr(77,22)) * .0708943 = zero.0000050276761346
DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):
(nCr(7,3)*nCr(69,23)/nCr(76,26)) * .0120923 = zero.0033119854089495
(nCr(7,4)*nCr(69,22)/nCr(76,26)) * .0120923 = 0.0016207588171455
(nCr(7,5)*nCr(69,21)/nCr(76,26)) * .0120923 = zero.000445708674715
(nCr(7,6)*nCr(69,20)/nCr(76,26)) * .0120923 = 0.0000636726678164
(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 = zero.0039950097242352
(nCr(7,4)*nCr(69,27)/nCr(76,31)) * zero.0130408 = 0.0026633398161568
(nCr(7,5)*nCr(69,26)/nCr(76,31)) * zero.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)) * 0.0130408 = zero.0000156856316566
ADDED PROBABILITIES PER FREE GAME:
3/5 : 0.1581890954550239+0.0154837443092076+0.0033119854089495+0.0039950097242352 = 0.1809798348974162
4/5 : zero.0471792039076387+0.0056575219591335+0.0016207588171455+0.0026633398161568 = zero.0571208245000745
PROGRESSIVE HIT: 0.0078089716812643+0.0006617772611241+0.0000220592420375+0.0011528535312951+0.0001209784569878+0.0000050276761346+0.000445708674715+0.0000636726678164+0.0000036384381609+0.0010033977912033+0.0001976389588734+0.0000156856316566 = zero.011501410011269
TOTAL ADDED PROBABILITIES DURING FREE GAMES * FREE GAME PROBABILITY:
3/5 (0.1809798348974162*18.3086771581) * 0.0055056 = zero.018242813138614555793199790556832
4/5 (0.0571208245000745*18.3086771581) * 0.0055056 = 0.00575779355898468825039996183432
PROGRESSIVE HIT: (0.011501410011269*18.3086771581) * zero.0055056 = 0.00115934503855140059648226887184
TOTAL PROBABILITIES OF METER HITS OR PROGRESSIVE AWARDS:
3/5 0.0893065819128652 + 0.018242813138614555793199790556832 = 0.107549395051479755793199790556832
4/5 0.0138634666884298 + 0.00575779355898468825039996183432 = zero.01962126024741448825039996183432
PROGRESSIVE: zero.0008404745950033 + 0.00115934503855140059648226887184 = 0.00199981963355470059648226887184 OR 1 in 500.05
Simplified:
3/5 : .1075494
4/5 : .0196213
5/5 : .0019998
We additionally know that there may be multiple progressives hit or meters moved throughout one preliminary spin, (because of Free Games) but by taking the average number of Free Spins per prevalence and the possibilities of each outcome on every initial Free Game, we fully account for all expectations.
Progressive Meter Added Values
Okay, since I’m maintaining the actual identify of this game secret, we’re simply going to call these Progressives 1-10. The method it really works is each individual progressive has its own probability of displaying up on an preliminary spin and the value added to the meter (based on one credit bet) can be totally different.
The very first thing we’re going to do is list the chances that a progressive comes up on a spin (or a Free Game)---one will ALWAYS come up---and then issue within the chances of concurrently hitting 3/5 or 4/5 in addition to corresponding meter move amount.
Progresive 1:
3 Hits: 4
4 Hits: 8
Probability: .005
Progressive 2:
three Hits: 2
4 Hits: 6
Probability: .016
Progressive three:
3 Hits: 2
four Hits: four
Probability: .026
Progressive 4:
three Hits: 2
4 Hits: 3
Probability: .031
Progressive 5:
three Hits: 2
4 Hits: 3
Probability: .041
Progressive 6:
3 Hits: 2
4 Hits: three
Probability: .052
Progressives 7-10:
three Hits: .5
four Hits: 1
Probability: .828 (Combined)
Armed with this information, we must now take the meter enhance probabilities & values and multiply them by the probabilities of having that exact meter.
Progressive 1:
((.1075494*4)+(.0196213*8)) * .005 = zero.00293584
Progressive 2:
((.1075494*2)+(.0196213*6)) * .016 = 0.0053252256
Progressive 3:
((.1075494*2)+(.0196213*4)) * .026 = zero.007633184
Progressive 4:
((.1075494*2)+(.0196213*3)) * .031 = 0.0084928437
Progressive 5:
((.1075494*2)+(.0196213*3)) * .041 = zero.0112324707
Progressive 6:
((.1075494*2)+(.0196213*3)) * .052 = zero.0142460604
Progressives 7-10:
((.1075494*.5)+(.0196213*1)) * .828 = 0.060771888
TOTAL METER CONTRIBUTIONS:
0.00293584+0.0053252256+0.007633184+0.0084928437+0.0112324707+0.0142460604+0.060771888 = 0.1106375124 or .1106375
RETURNS WITH METER MOVE ADDED:
THREE-SPOT: zero.7571035 + .1106375 = 0.867741
FOUR-SPOT: zero.792997 + .1106375 = zero.9036345
FIVE-SPOT: zero.786126 + .1106375 = zero.8967635
HITTING THE JACKPOTS:
The last factor we want to take a look at is the precise returns of the base jackpots together with the probability of hitting them. The jackpots have different starting values, so we will have to look at them all individually.
Also, the chance for every jackpot to be out there on a given recreation is completely different, however total, it will comply with the identical methodology because the earlier section.
Progressive 1:
(.0019998 * 200) * .005 = zero.0019998
Progressive 2:
(.0019998 * 125) * .016 = zero.0039996
Progressive 3:
(.0019998 * 100) * .026 = 0.00519948
Progressive four:
(.0019998 * 50) * .031 = zero.00309969
Progressive 5:
(.0019998 * 20) * .041 = zero.001639836
Progressive 6:
(.0019998 * 10) * .052 = 0.001039896
Progressive 7:
(.0019998 * 5) * .207 = zero.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 + zero.00519948 + zero.00309969 + zero.001639836 + 0.001039896 + zero.002069793 + 0.0012418758 + zero.0008279172 + zero.0004139586 = 0.0215318466
RETURNS WITH METER MOVE ADDED AND JACKPOTS:
THREE-SPOT: 0.7571035 + .1106375 = zero.867741 + zero.021531 = 0.889272
FOUR-SPOT: 0.792997 + .1106375 = zero.9036345 + 0.021531 = zero.9251655
FIVE-SPOT: 0.786126 + .1106375 = zero.8967635 + 0.021531 = 0.9182945
Advantage Play
The best approach to advantage play this may be to repair in on Progressives 7-10 and play those if they make the game positive.
The greatest returning version of this sport returns .9251655 with no increases to the jackpot AND together with meter strikes. Of course, you’re not going to hit every meter that you simply move. The first thing that we are going to do is calculate the likelihood of each individual jackpot hitting after which determine what the jackpot value would have to be (both together with and never together with meter transfer within the return) for the person jackpots:
Progressive 1:
.0019998 * .005 = zero.000009999 or 1 in 100,010
Progressive 2:
.0019998 * .016 = zero.0000319968 or 1 in 31,253
Progressive three:
.0019998 * .026 = 0.0000519948 or 1 in 19,233
Progressive four:
.0019998 * .031 = 0.0000619938 or 1 in 16,131
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 8:
.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 includes some simple algebra and solving for, “X.” The four-spot has the best return, so we’re going to make use of it. What we're going to need is for the added progressive return for each meter (by itself) to convey us up to one hundred pc.
Progressive 1:
With Meter: (1 - zero.9251655) = x * 0.000009999--->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 - zero.9251655) = x * 0.0000319968--->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 - 0.9251655) = x * 0.0000519948--->1439.268927/4 = $359.eighty two
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0000519948--->3567.125943/4 = $891.78
Progressive 4:
With Meter: (1 - 0.9251655) = 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 - zero.9251655) = x * zero.0000819918---912.707124/4 = $228.18
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0000819918--->2262.079867/4 = $565.fifty two
Progressive 6:
With Meter: (1 - 0.9251655) = x * zero.0001039896--->719.634463/4 = $179.ninety one
Without Meter: (1 - (0.9251655 -0.1106375)) = x * zero.0001039896-->1783.562972/4 = $445.89
Progressive 7:
With Meter: (1 - 0.9251655) = x * 0.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0004139586-->448.044804/4 = $112.01
Progressive eight:
With Meter: (1 - zero.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 9:
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 10:
With Meter: (1 - 0.9251655) = x * 0.0004139586-->180.77774/4 = $45.19
Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0004139586-->448.044804/4 = $112.01
Progressive 7-10 (Combined):***
With Meter: (1 - 0.9251655) = x * (4 * zero.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 typical that we'd need Progressives 7-10 at since all of them have the identical likelihood, but are not prone to have one positive by itself. By utilizing the typical, you could even hit certainly one of them and have the play proceed, but it allows you to basically play positively more often by treating it as one progressive.
Advantage Play:
As you probably can see, we did take a look at Progressives 7-10 as a complete, but this is not the kind of play where I would take a look at the entire thing when determining if I want to play.
The cause why is as a outcome of, nearly all the time, you’re going to finish up hitting a progressive that forces you to quit enjoying because it causes the general return of the game to drop again beneath 100%. In the meantime, you’ve additionally fed the meters of the other possible jackpots, but you won’t be staying to take them down, so it’s lost money.
For that reason, I recommend disregarding the meter motion when figuring out whether or not something is a play, with exception ONLY to the progressive, or progressives, if you’re going after 7-10, mixed, that you’re actually targeting.
Ideally, you’ll be taking part in for one thing considerably doubtless (such as the 7-10 combination play) that might be above 100 percent without breaking your back making an attempt to squeeze each little tenth or hundredth of a share to justify taking half in it.
Or, maybe you’ll have an individual jackpot that puts it above 100% by itself (ignoring meter transfer on everything else) and then any will increase over the progressive base are simply gravy. 105% when you’re ignoring other contributions is at all times better than 105% after adding each little scrap of worth you presumably can justify.
Calculating Other Keno Games And Help
The very first thing that I wish to reiterate is that this is, by far, probably the most tough video keno recreation that I’ve ever analyzed. Not solely does this game have a quantity of moving parts, (ten totally different conditional progressives and three units of numbers with interrelated probabilities) however there are also Free Games that include two partially Wild numbers (they can be used for Progressives and Base Pays, however do not substitute for extra attracts or further Free Games).
The math used in this is going to be helpful to you for MOST Video Keno games and Progressives as some facet of this sport also covers (in full) what you would need to know how to do in analyzing other Video Keno games.
In most instances, any Video Keno Progressive is going to be related to your base picks indirectly, so then it turns into a comparatively easy matter of simply determining the base return of the game based 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, typically has a Progressive for occasions corresponding to hitting 6 out of 6, 7 out of 7, eight out of eight...and so forth. That simply has to do with those base occasions (or hitting it during Free Games) so that’s all that must be found out. Of course, Cleopatra Keno is an example of a sport that’s already been totally 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 credit.
But, new Keno games seem to be always coming out lately, so understanding tips on how to do the sort of analysis included on this page is going to provide the advantage of being one of many solely individuals to know the play points for semi-complicated to very complicated Video Keno variants.
If you encounter a recreation and need help with calculating it, then you must be in a position to figure it out primarily based on what is included above. I may identify ten completely different Keno games that the methodology above would cowl, and more than that, you would solely have to know a section of what I’ve accomplished above as a end result of, once more, most Keno video games wouldn't have so many transferring elements.
However, if you discover yourself with a query as to methodology, then feel free to depart a remark to this text, to PM me on WizardofVegas at Mission146, or to good friend me (Brandon James) on Facebook and send me a direct message. I’ll answer any methodology questions that you've at no cost.
If you want to me to investigate a Video Keno sport FOR YOU, then be happy to shoot me a message and we can talk about my rates. I recommend making an attempt to do it your self as a outcome of I can already inform you it’s not going to be low cost. These kinds of analyses often take a quantity of hours---even for a comparatively simple sport. Most of it includes just repeating a variant of the same formulation used for other parts---as you'll find a way to see above.
Conclusion
Anyway, I hope that this has helped you to learn a variety of the methodologies that can be utilized in analyzing Video Keno games.
If you may have reached this conclusion and don't understand what was carried out above, be at liberty to leave any questions within the feedback. I tried to explain what I was doing and why with as much detail as attainable, in addition to to clearly separate the totally different aspects of the game, but it’s still an insanely sophisticated Keno game. Don’t be afraid to learn it multiple occasions if you end up confused or to take a break and are available back to it.
I certainly did.
I always say that 90% of gambling math is NOT figuring out the solutions to things---figuring out solutions is straightforward. 90% of gambling math, no much less than in my view, is determining what questions must be requested and then figuring out the way to ask them appropriately. I spent as a lot time figuring out HOW one might go about figuring out this sport as I spent truly doing the figuring.
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What does all of it mean?
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Guide to Slot Terms and Vocabulary
3D Video Slots – The most graphically advanced slots that supply completely different visible effects and animations.
3-Reel – A basic slot format that includes three spinning reels.
5-Reel – The most common layout for up to date slots, with five vertical spinning reels.
243 Ways to Win – Instead of payline-based wins, slots of this sort award wins whenever symbols line up on adjacent reels from left to right or proper to left.
All Ways – A slot whose paylines operating each left-to-right and right-to-left. Those are usually on-line games, and are considered highly profitable for gamers
Annuity Winner – A jackpot win that's paid in installments over a interval of years. Usually
Autospin – A function that allows slot players to set a pre-determined variety of spins at a certain bet stage. Beloved by on-line punters.
Bonus Feature – Any playing fan’s favorite a half of a slot sport. The Bonus is an extra round that has completely different rules from the bottom sport and normally offers greater payouts.
Branded Slots – Slots primarily based on popular franchises, similar to TV series, motion pictures, or books.
Brick and Mortar – Land-based on line casino, visited by real folks in bodily area. Usually refers to a casino or other business.
Cascading Reels – A comparatively current feature in online slots that has proven fairly popular. When you hit a winning combination, the winning symbols would ‘explode’, freeing space for new ones which may probably trigger an extra win. This is a re-spin of sorts.
Casino – The purveyor or supplier of slot motion. May be land-based or online.
Casino Bonus – This refers again to the new participant welcome bonus that almost all online casinos provide. These are normally more money that must be performed via slot video games.
Coins – In the past, slot machines have been performed with precise cash, but nowadays this is not overly practical – particularly online. However, most slot machines still permit you to measure your wager in cash or credit.
Coin Size – The denomination of the coins you're betting. Most slot machines let you set both the number and the denomination of the cash you will be betting per payline.
Cold Slots – Slang for a slot that doesn't pay usually, or that hasn’t paid out shortly.
Collect – This is a button in on-line slots that allows you to convert your credits back into foreign money. Oftentimes this is offered within the context of a chance or bonus characteristic.
Expanding Wilds – Feature (or Bonus Feature) – One or extra bonuses provided by a game. Oftentimes, they allow for giant wins with out When there are several obtainable, the characteristic refers to the high bonus round that offers the best payouts.
Free Spin – One of the most typical features available. These are extra spins usually triggered by scatter symbols, which are value you nothing and offer you wins primarily based in your last wager.
Fruit Machine – The widespread identify of slot games in the UK. Based on the fruit symbols that appeared on most slot machine fashions in the course of the Nineteen Twenties.
Gamble – A 50/50 likelihood offered by some slots to go ‘double or nothing’ on a win. Sometimes they are often played multiple instances in a row, providing the slot machines give you the option to double and even triple your winnings – if you're prepared to threat them. The Gamble characteristic is most video games is the equal of a coin toss. When you hit a profitable combination, you've the choice to wager it on a 50/50 likelihood to both double it, or lose every little thing
Gaming Symbols – The icons that appear on the reels, awarding wins when in the appropriate positions.
Hit and Run – A tactic typically used at land-based casinos. Players will strive a slot for a few spins, and immediately abandon it if it doesn’t pay out. A dubious tactic.
Hot Slots – Slang for a slot that pays out often, or that has lately paid out many occasions.
Icons – (see Gaming Symbols)
Instant Winner – This is a jackpot that's paid in one instalment to the fortunate winner.
Jackpot – The top prize obtainable. Jackpot might discuss 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 wager of £10 on a 10-payline slot yields a line wager of £1.
Linked Progressive Jackpot – When two or more slot machines whose progressive jackpots are all linked together. Though harder to set off, most of the largest wins in slot history have resulted from this type of jackpot.
Loose – A slot that pays out incessantly.
Max Bet – The most wager that could be made per spin, with all paylines energetic and the very best possible coin denomination. In most circumstances, slot technique dictates that you should play with all attainable paylines active.
Min Bet – The minimal quantity that may be wagered per spin. This can apply each to operating a single payline, or putting the minimum line wager on all paylines. The minimum wager allows you to stretch your payroll for a maximum number of spins.
Multiplier –
Nudge – A nudge slot machine allows the consumer to move or ‘nudge’ a key line up or down to help create a profitable combination.
One-Armed Bandit – An outdated nickname for a slot machine, going again to the beginnings of slot play. A disparaging time period, the “one-arm” refers to the massive lever – or arm – which was pulled to trigger the reels firstly of a brand new spin. The lever seems on just one aspect, giving the appearance of being ‘one-armed’. The “bandit” was intended to point out that the slot machine would leave its patrons poorer than it found them, in effect robbing them like a bandit.
Payline – The format of reel positions leading to payouts to players. Different video games offer various quantities and varieties of paylines, but extra of them normally means more frequent wins.
Pay Table – The list of player payouts offered for different image combos.
Pokies – An Australian time period for on-line slots (pokie machines).
Progressive Slot Machines –
Punter – A gambler or slot player.
Reels – The vertical strains of symbols, that spin throughout each play.
Re-spin – An further spin of the reels awarded without an additional guess. Essentially, two (or more) rounds for the worth of one.
RNG (Random Number Generator) – The most important software program present in modern slots, RNGs generate long strings of random numbers, which dictate the finish result of any particular person spin. The randomness of these numbers is carefully monitored to ensure fair play.
RTP (Return-to-player) – A mathematical expression of what proportion of complete bets placed on a given slot machine shall be returned to gamers. The greater the RTP, the better the chances for gamers. A crucial idea.
Rows – The horizontal lines of symbols.
Scatter – Special gaming symbols which pay regardless of payline place; typically, they open the best way to bonus options.
Spin – The spin button prompts the slot for a single spin – or round. It changed the lever in basic slot machines
Stacked Symbols – Matching symbols stacked on neighboring positions in a single reel. Stacked symbols tremendously enhance the odds of landing an enormous win.
Stacked Wilds – Wild symbols that appear vertically stacked on the identical reel, much like stacked symbols. An excellent turn of occasions for players.
Streaky – A recreation, player, or provider that seems to show hot or cold for lengthy durations of time. A perception seldom primarily based in mathematical fact.
Total Bet – All the cash that has been wagered on a single spin.
Volatility – (also known as variance) refers to the frequency of payouts vs. the scale of these wins. Low volatility / low variance slots are inclined to award frequent, small payouts, whereas excessive volatility / high variance slots award fewer wins of a bigger dimension.
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