Analyzing Keno Progressives

Brandon James takes a look at a relatively new and extremely complicated Video Keno game for advantage play purposes. The game comes with a Free Games feature, Extra Draws feature as well as ten 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 yourself for similar games.

Math Lesson

Greetings!

I have recently been presented with a new Video Keno game to analyze that I’m going to share with everyone here. I have been asked not to actually name this Keno game because it’s only available in certain markets that will mostly 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 believe that sharing my methodology will be helpful for anyone out there who wants to analyze keno progressives for themselves to find advantageous play points. In addition to land casinos, (and some convenience gambling locations) I should imagine that some online casinos also have Keno games with difficult features or progressives to analyze.

Even if you’re just someone who enjoys Keno and are not necessarily looking for advantageous situations; it’s always nice to know how to determine the return-to-player on keno games that come with some sort of gimmick--and this game has gimmicks galore.

Many people would program a simulation and analyze this game that way--by simply creating the rules and having a computer program simulate millions of spins. There are a couple of things about that I should mention:

1.) I don’t know how to program. In fact, if you asked me to complete the simple task of creating a new E-Mail address, it would probably take me twice as long to do it as it would you.

2.) In this particular case, and given the amount of time it would take for the simulations to be done---I believe that just doing the math might be FASTER for this one. I think I fully analyzed three different versions of this game in about ten hours.

The Game

As I’ve mentioned, I won’t be disclosing the name of this Keno game, but I’ve been giving a blessing to share all of the other details of my analysis.

This game can essentially be broken down into three parts, though these different parts are highly intertwined and all come together to result in an overall return-to-player and advantageous progressive points.

One other thing that I should mention is that this game has different paytables for anywhere from 3 to 10 different picks the player can make. However, most advantage players are NOT going to want a ridiculous amount of Variance when going after a Video Keno play---particularly not a play on a game that has several different features and is top-heavy anyway. For that reason, I decided to only analyze the Pick 3, Pick 4 and Pick 5 versions of this game.

The first thing that we are going to do is look at all of the different components of the game:

1.) Base Pays

-The first thing that the full analysis will consider is the base paytable of the game. Essentially, if the game had no other features besides picking 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: 0.499513145082765

Choose 4: 0.486358578130730

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: 47

FIVE-SPOT:

3/5 PAYS: 2

4/5 PAYS: 13

5/5 PAYS: 149

2.) Extra Draws and Free Games:

-The second component of this game is that it will randomly draw five SEPARATE numbers from the player’s selections that will award either extra draws, free games, or both when the player hits either 3, 4 or all 5 out of five for the Free Games.

This gets extremely situational because the way that these things 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, however, the extra draws on the 3 out of 5 hit can result in Free Games AFTER the extra numbers are drawn (three out of five becomes five out of five) or can result in another set of extra draws (three out of five becomes four out of five). In other words, there are a few potential serieses of events possible:

When you start with three out of five, the following things can happen:

A.) You get zero additional on this set of five and only end up with your five extra draws.

B.) One of these extra five numbers matches on this set of five and awards ten extra draws for a total of fifteen extra draws. Based on these ten additional draws AFTER the original five, two things are possible:

-The player hits the fifth number and ALSO goes into Free Games.

-The player does not hit the fifth number and does not go into Free Games.

C.) On the five extra draws, the player could hit both of the other numbers in that set of five numbers, thereby getting Free Games but NOT getting the ten extra draws.

The player can also immediately start with four out of five, which will award ten extra draws from which:

A.) The player can hit the fifth number from that set of five and go into Free Games.

B.) The player does NOT hit the fifth number and does NOT go to Free Games.

Finally, the player could just hit 5/5 right off the bat and go directly into Free Games.

In other words, there can be either five, ten or fifteen extra draws (that will all come with their own corresponding probabilities that affect this---and other---aspects of the game) and the player can also hit Free Games or not hit Free Games.

***And, if that’s not already complicated enough, we will have to account for the fact that IF the player hits any results on this particular set of five numbers, then those hits CANNOT also double as hits on the player’s base Picks BECAUSE these numbers will always be separate from those ones as they are not chosen by the player.

***For example, if a player hits four out of five of this set of numbers on the initial draw, then that automatically means that four numbers DO NOT match the player’s base picks, however, the base pays are affected by these numbers as well as the extra draws. What that means is that we ALSO have to determine the probabilities for the numbers of picks that the player could have hit on the base selections given the reduced number of possible matches.

***In effect, if we assume that the player matches three of these five numbers (which we also have to figure out the probabilities for), then that will leave seventeen other draws that either did or did not hit the player’s base numbers. We must figure out the probabilities for every possible event and how they relate to the extra draws.

3.) Free Games:

-Okay, so the Free Games should be pretty easy, right? You have fifteen Free Games and it just goes by the base paytable, yes?

Haha. No.

The Free Games ALSO give the possibility for getting extra draws as well as to retrigger more Free Games. Here are the rules for how the Free Games work:

A.) There are always fifteen Free Games initially.

B.) Free Games will include TWO Wild spots that count towards the Base Paytable as well as the Jackpots. However, the WILDS do NOT go towards the lucky picks that award extra draws and more Free Games.

C.) 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

D.) The maximum number of Free Games is 1,000, but we’re not going to go that far.

Okay, so the first thing that we have is the same extra draw mechanism as before EXCEPT that the result of four out of five now gives ten extra draws AND extra Free Games while the five out of five for those selections awards extra free games only. Therefore, the following events are possible.

a.) 0, 1 or 2 out of five---nothing happens.

b.) Three out of five, and then:

aa.) The player hits two extra numbers for a total of 5/5 after the five extra draws and is awarded an additional ten Free Games.

bb.) The player hits zero extra numbers on this set of five and does NOT get any extra Free Games or additional draws---aside from the original extra five.

cc.) The player hits one of the set of five numbers (two remaining) and gets a total of four out of five which results in ten more draws (fifteen total) and at least five additional Free Games, but:

aaa.) The ten extra draws result in hitting the last one needed for that set of five and getting ten additional Free Games---which would mean fifteen total additional Free Games.

Or:

bbb.) The ten extra draws do NOT hit the final number needed on that set of five, so the player gets only the five 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 last one needed for that set of five and getting ten additional Free Games---which would mean fifteen total additional Free Games.

Or:

bb.) The ten extra draws do NOT hit the final number needed on that set of five, so the player gets only the five extra Free Games.

d.) Five out of five hits on an initial Free Games spin and simply awards ten extra Free Games.

As a result, the following events are possible:

• NO extra draws or Free Games awarded.
• Five extra draws and no extra games awarded. 
• Fifteen extra draws and five extra games awarded. 
• Fifteen extra draws and fifteen extra games awarded. 
• Ten extra draws and five extra games awarded. 
• Ten extra draws and fifteen extra games awarded. 
• Zero extra draws and ten extra games awarded.
• Five extra draws and ten extra games awarded.

---And, we have to figure out the probabilities of each one of these events.

The good news is that we DO NOT have to determine the probability for the total number of free games (between 15 and 1,000) that the player will receive. One shortcut here is simply to determine the AVERAGE amount of Free Games that will be awarded per Free Games initial spin, then multiply by fifteen because that is how many initial spins we start with. After that, we will multiply the average number of extra spins awarded (per spin) by how many extra Free Games we expect to get from the fifteen. We will then continue to repeat that process until we’re essentially not adding a meaningful amount of Free Games to the expected total anymore.*

*By using this method, we will end up with an amount of TOTAL Free Games per initial fifteen (18.3086771581 TOTAL, on average, was the result), but the result will be slightly short. I basically used this method until I reached the point that the next step would have only added less than 1/10000th of a Free Game and called that good enough. If anyone does decide to simulate this, they will find that my average result of 18.3086771581 Free Games TOTAL (per initial set of fifteen Free Games) is extremely close.

Not only that, but once again, we have to determine how these numbers being removed from the initial set of numbers that we could hit impacts the probabilities as relates our Base Picks and Progressives!!!

Oh, right, the Progressives…

4.) The Progressives:

That’s right, PROGRESSIVES, plural, isn’t that fun?

In fact, there are ten of them.

It’s not quite as bad as it seems. The way the Progressives work is that five numbers on the Keno board are randomly selected by the game (these numbers will NOT match the player’s numbers OR any numbers used for extra draws/free games) and there will be different events that occur based on hitting three, four or five of these numbers.

If the player hits five out of five of these 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 different probability of appearance as the five extra numbers. In other words, only one of the Progressives can be awarded on any one spin and all of them have different rates of appearance. In fact, just figuring out how likely it is for a particular progressive to appear is going to be one of our tasks---which will only occur after we deal with how likely hitting one of them is.

If you hit three or four out of five of these numbers, you won’t be paid anything, but there will be an amount added to the Progressive meter for whichever progressive is active at the time. Of course, how much is added varies based upon which meter it happens to be, so we will have to determine all of these probabilities (and adds to the meter) individually in order to figure out how much the progressive meter additions contribute to the base return of the game.

---If that’s not difficult enough yet, don’t worry, it’ll get a little tougher:

A.) The progressive five-spot probabilities are ALSO conditional with the Extra Draw probabilities. For example, if you hit three/five of the extra draw set of five in order to get five extra draws, then that means that three of your progressive set of five automatically DID NOT hit, so you have to figure out the probabilities based on seventeen different numbers that could hit it.*

*Technically, this could also be looked at as true for the base game hits, but it doesn’t matter in the case of the base game hits. The reason why it doesn’t matter is because nothing about the base game numbers directly influences the probability of hitting the Progressive numbers. In other words, base game hits and progressive hits can be treated separately from each other, but not from the Extra Draw and Free Games set of numbers.

B.) There are two WILD numbers that are randomly selected during Free Games that can substitute as Progressive Numbers or substitute for the base paytable. Actually, they technically substitute for both simultaneously when they are 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 way the game works was difficult to follow, (I asked several clarifying questions and also asked for pictures of all of the rules screens) don’t worry---because the mathematical breakdown isn’t going to make it any easier.

I would say just do your best to follow along, but don’t be afraid to stop and come back to it. I know that I stopped in my analysis of these three paytables--several times---and came back to it later.

The good news is that I will reexplain each individual segment of the game below and the math will be done individually for each component. Okay, here we go:

Base Game Return (Without Extra Draws)

Based on WoO Keno Calculator:

Choose 3: 0.499513145082765

Choose 4: 0.486358578130730

Choose 5: 0.421164278759215

It is also possible to hit fifteen (total) extra draws by first hitting five extra numbers with three out of five, but then having one of those numbers hit and unlocking 4/5 with ten additional draws. By this mechanism, the Free Games can also be hit.

Therefore, we should first determine our total Extra Draw probabilities and then our total probabilities for Free Games accordingly.

 3/5 Hits: 0.083935052289483 

In order to receive another ten draws, this must first happen and then be followed up with hitting one of the two remaining spots. Hitting both spots will just unlock fifteen Free Games. We could also unlock the ten extra draws and unlock Free Games after that.

At this point, there will be sixty numbers remaining, of which five will be drawn.

nCr(2,1)*nCr(58,4)/nCr(60,5) = 0.1553672316384181

Therefore, that is the probability of this event and the overall probability of getting fifteen extra draws is:

0.1553672316384181*0.083935052289483 = 0.0130407567116428407376253868423

The second thing that we will look at is hitting both of our other five numbers which will unlock Free Games:

nCr(2,2)*nCr(58,3)/nCr(60,5) = 0.0056497175141243

This will be added to our running probability of Free Games, but first, we must factor in the probability of being in this situation in the first place:

0.0056497175141243*0.083935052289483 = 0.0004742093349688310301048447369

0.0004742093349688310301048447369+0.000644924695558 = 0.0011191340305268310301048447369

Now, we must look at the probability of hitting ⅗, getting the five extra draws, turning that into 4/5 and then hitting Free Games. There will be 55 remaining numbers of which only one is helpful, ten numbers are drawn:

nCr(1,1)*nCr(54,9)/nCr(55,10) = 0.1818181818181818

Of course, a lot has to happen in the first place. This is the same as our probability of getting fifteen extra draws (at all) from above:

0.1818181818181818*0.0130407567116428407376253868423 = 0.0023710466748441526242817664868574411340838756

Therefore, our running probability of hitting the Free Games is:

0.0023710466748441526242817664868574411340838756+0.0011191340305268310301048447369 = 0.0034901807053709836543866112237574411340838756

Finally, we could hit 4/5 of these numbers and be awarded ten extra draws. This will cause sixty numbers to remain of which only one helps us of the ten chosen:

nCr(1,1)*nCr(59,9)/nCr(60,10) = 0.1666666666666667

Of course, we must be in this situation in the first place having received ⅘ on the initial draw, so:

0.1666666666666667*0.012092338041705 = 0.0020153896736175004030779347235

0.0020153896736175004030779347235+0.0034901807053709836543866112237574411340838756 = 0.0055055703789884840574645459472574411340838756

Therefore, we know the following probabilities:

Extra Draw Probabilities:

3/5 Hits: 0.083935052289483***

Zero Extra: nCr(2,0)*nCr(58,5)/nCr(60,5) = 0.8389830508474576

One Extra (Initial Draw) = 0.1553672316384181

Two Extra (Initial Draw) = 0.0056497175141243

PROOF: 0.0056497175141243+0.1553672316384181+0.8389830508474576 = 1

PROBABILITY 3/5 + 0 EXTRA: 0.083935052289483*0.8389830508474576 = 0.0704200862428713282322697684208

PROBABILITY 3/5 + 1 EXTRA: 0.1553672316384181*0.083935052289483 = 0.0130407567116428407376253868423***

PROBABILITY 3/5 + 2 EXTRA: 0.083935052289483*0.0056497175141243 = 0.0004742093349688310301048447369---FREE GAMES TRIGGERED

***This is the only possible way to get fifteen extra numbers.

***PROBABILITY OF 3/5 + 1 EXTRA, + 1 EXTRA AGAIN---FREE GAMES:

0.0130407567116428407376253868423*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 we get Free Games, which can only be done if an extra number results in Free Games.

From Above: nCr(1,1)*nCr(59,9)/nCr(60,10) = 0.1666666666666667

Therefore, the overall probability of hitting 4/5 followed by Free Games is:

0.1666666666666667*0.012092338041705 = 0.0020153896736175004030779347235

The probability that this will not happen is:

0.012092338041705*.833333333 = 0.010076948364056720652765

5/5 Hits: 0.000644924695558

5/5 Hits Total Probability: 0.000644924695558+0.0020153896736175004030779347235+0.00237104667484415262428176648691198658862933014+0.0004742093349688310301048447369 = 0.00550557037898848405746454594731198658862933014 (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 93.72 (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: 0.0004742 or 1 in 2108.81 (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: 0.0020154 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)

FINAL EXTRA DRAW PROBABILITIES:

FIVE EXTRA: 0.0704201 + 0.0004742 = 0.0708943

TEN EXTRA: 0.0100769 + 0.0020154 = 0.0120923

FIFTEEN EXTRA: 0.0106697 + 0.0023711 = 0.0130408

ZERO EXTRA: 1 - (0.0708943+0.0120923+0.0130408) = 0.9039726

FINAL FREE GAMES PROBABILITY (REGARDLESS OF NUMBER OF EXTRA DRAWS)

5/5 Hit: 0.0055056 or 1 in 181.63

BASE RETURN BEFORE FREE GAMES INCLUDING EXTRA DRAWS:

Based on WoO Keno Calculator:

Choose 3: 0.499513145082765

Choose 4: 0.486358578130730

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: 47

FIVE-SPOT:

3/5 PAYS: 2

4/5 PAYS: 13

5/5 PAYS: 149

RETURNS BASED ON TOTAL NUMBER DRAWN:

DRAW TWENTY TOTAL:

Choose 3: 0.499513145082765 * 0.9039726 = 0.451546196494644292239***

Choose 4: 0.486358578130730 * 0.9039726 = 0.439654828405139137998***

Choose 5: 0.421164278759215 * 0.9039726 = 0.380720968097092357509***

***The reason for this calculation is because there’s less than a 100% chance of drawing ONLY twenty numbers. Therefore, we must determine the different probabilities based on the total number of balls that are drawn by the time the game ends. In this case, there is only a 90.39726% chance that only twenty numbers will be drawn with the other possibilities being more 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, so 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 five that controls extra draws hit) that there are only 22 numbers available to be doing other things. This will 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) * 16 = 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) * 16 = 0.5917496443812233

ADDED VALUE: (0.5917496443812233+0.4623044096728307) * 0.0120923 = 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 = 0.5953058321479374

nCr(3,3)*nCr(73,28)/nCr(76,31) * 16 = 1.0230440967283073

ADDED VALUE: (1.0230440967283073+0.5953058321479374) * 0.0130408 = 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, so 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 five that controls extra draws hit) that there are only 22 numbers available to be doing other things. This will also apply in a different way when it comes to having ten or fifteen extra draws.

nCr(4,2)*nCr(73,20)/nCr(77,22) * 1 = 0.2534850640113798

nCr(4,3)*nCr(73,19)/nCr(77,22) * 3 = 0.1877667140825036

nCr(4,4)*nCr(73,18)/nCr(77,22) * 47 = 0.2540540540540541

ADDED VALUE: (0.2534850640113798+0.1877667140825036+0.2540540540540541)*0.0708943 = 0.04929322025604552550625

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) * 3 = 0.3039809817026832

nCr(4,4)*nCr(72,22)/nCr(76,26) * 47 = 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) * 3 = 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 = 0.52887802902440296492303

FIVE SPOT ADDED RETURNS:

DRAW 25 TOTAL (OUT OF 22 BECAUSE THREE NUMBERS UNLOCKED EXTRA DRAWS):***

***Remember here that the extra draws have already happened, so 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 five that controls extra draws hit) that there are only 22 numbers available to be doing other things. This will also apply in a different way when it comes to having ten or fifteen extra draws.

FIVE-SPOT:

nCr(5,3)*nCr(72,19)/nCr(77,22) * 2 = 0.2314932091428126

nCr(5,4)*nCr(72,18)/nCr(77,22) * 13 = 0.2647167715660866

nCr(5,5)*nCr(72,17)/nCr(77,22) * 149 = 0.1985931136616068

ADDED VALUE: (0.1985931136616068+0.2647167715660866+0.2314932091428126) * 0.0708943 = 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 = 0.3447932431349879

nCr(5,4)*nCr(71,22)/nCr(76,26) * 13 = 0.5259856107008234

nCr(5,5)*nCr(71,21)/nCr(76,26) * 149 = 0.5305171790391689

ADDED VALUE: (0.5305171790391689+0.5259856107008234+0.3447932431349879)*0.0120923 = 0.01694489201833412307246

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: 0.380720968097092357509 + 0.03714550630214930212464 + 0.01694489201833412307246 + 0.0492575790132309635158 = 0.4840689454308067462219 or 48.4069%

BASE GAME FINAL RETURNS:

THREE-SPOT: 0.533903

FOUR-SPOT: 0.528878

FIVE-SPOT: 0.484069

FIGURING OUT THE FREE GAMES:

The next thing that we have to do is determine how the Free Games will impact our returns based on the number of spots that we have selected.

The following are the summarized rules for the Free Games:

1.) There are always fifteen Free Games initially.

2.) Free Games will include TWO Wild spots that count towards the Base Paytable as well as the Jackpots. However, the WILDS do NOT go towards the lucky picks that award extra draws and more Free Games.

3.) 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

4.) The maximum number 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 number of Free Games that a player receives per initial occurrence. The good news is that the probabilities for Free Games retrigger are just going to be a variation of the above probabilities for the 3/5 and 4/5 possibilities 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 93.72 (Fifteen Extra Draws Total, Five Free Games Retrigger)

3/5 Hit, One Extra, One Extra Again, Free Games: 0.0023711 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 99.24 (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: 0.0006449 or 1 in 1550.63 (Initial, Ten Free Games Retriggered)

5/5 Hit: 0.0055056 or 1 in 181.63 (Total)

FINAL EXTRA DRAW PROBABILITIES:

FIVE EXTRA: 0.0704201 + 0.0004742 = 0.0708943

TEN EXTRA: 0.0100769 + 0.0020154 = 0.0120923

FIFTEEN EXTRA: 0.0106697 + 0.0023711 = 0.0130408

ZERO EXTRA: 1 - (0.0708943+0.0120923+0.0130408) = 0.9039726

FINAL FREE GAMES RETRIGGER PROBABILITIES:

Five Free Games: 0.0106697 + 0.0100769 = 0.0207466

Ten Free Games: 0.0004742 + 0.0006449 = 0.0011191

Fifteen Free Games: 0.0023711 + 0.0020154 = 0.0043865

FREE GAMES EXPECTED BASED ON INITIAL SPINS:

The next thing that we are going to do is determine how many extra free games we are expected to receive based upon the fifteen initial Free Games. The first step to doing this is to determine how many extra are expected on a per spin basis, like so:

(5 * 0.0207466) + (10 * 0.0011191) + (15 * 0.0043865) = 0.1807215

Therefore, we expect to receive 0.1807215 retriggered Free Games per each of the initial fifteen spins. We can multiply this number by fifteen to get the total number of expected additional Free Games per initial spins:

0.1807215*15 = 2.7108225

Okay, so now we are going to do the same thing based on our expected 2.7108225 retriggered games:

2.7108225 * .1807215 = 0.48990390843

And then again with that:

0.48990390843 * .1807215 = 0.08853616918

And again:

0.08853616918 * .1807215 = 0.01600038929

One last time:

0.01600038929 * .1807215 = 0.00289161435

We will now add all these expected extra Free Games to our initial fifteen, which is going to get us close enough:

15 + 2.7108225 + 0.48990390843 + 0.08853616918 + 0.01600038929 + 0.00289161435 = 18.3081545812

Therefore, for each initial set of fifteen Free Games, we are expected to get something along the lines of 18.308155 Free Games. If we had gone one more retrigger occurrence, it would have added about .00052257688 expected Free Games, so we can go ahead and do that.

 18.3081545812+.00052257688 = 18.3086771581

So, we’ll break that off and call it 18.308677 Free Games expected per initial set.

While that may seem low, keep in mind that the probability of ANY retrigger (per spin) is:

(0.0207466) + (0.0011191) + (0.0043865) = 0.0262522 or 2.62522%

So, the probability 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 certainly add up quickly when they are hitting, but we’ve basically accounted for that by going on a basis of expected additional Free Games per initial set of spins.

Base Return Of Free Games

The good news here is that the Extra Draw probabilities are the same, with the significant change being that there are now two WILD spots during the Free Games. Another thing that is the same are the base return tables, so essentially, you have five spots to hit either 2 or 3 spots on a three-spot card. You can also hit four and five (and be awarded as if 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: 16

FOUR-SPOT (BECOMES SIX-SPOT):

2/4 PAYS: 1

3/4 PAYS: 3

4/4, 5/4, 6/4 PAYS: 47

FIVE-SPOT (BECOMES SEVEN SPOT):

3/5 PAYS: 2

4/5 PAYS: 13

5/5, 6/5, 7/5 PAYS: 149

RETURNS BASED ON TOTAL NUMBER DRAWN:

DRAW TWENTY TOTAL:

Choose 3: 2.087671821849037 * 0.9039726 = 1.88719812474

Choose 4: 2.190619507075203 * 0.9039726 = 1.98026001142

Choose 5: 2.428321678321678 * 0.9039726 = 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 = 0.6134570042284534

nCr(5,3)*nCr(72,19)/nCr(77,22) * 16 = 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 = 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) * 16 = 1.2262514858044779

nCr(5,5)*nCr(71,26)/nCr(76,31) * 16 = 0.1471501782965373

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 final few steps will have us multiplying the expected return (in credits) by the total expected number of Free Games and then factoring that against the probability 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 75.71035% (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) * 47 = 0.0147908182154757

ADDED VALUE: (0.0147908182154757+0.287115883006294+2.1533691225472047+0.5112141701903778+0.3322892106237456) * 0.0708943 = 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) * 47 = 3.937198899601288

nCr(6,5)*nCr(70,21)/nCr(76,26) * 47 = 0.7070887819692109

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 = 0.06935475772953615190391

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 = 0.8752770917286489

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) * 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 + 0.12638030755925172755624 + 0.06935475772953615190391 + 0.23386464256347551036254 = 2.62021921492746638982269

FOUR-SPOT ADDED RETURN OF FREE GAMES:

The final few steps will have us multiplying the expected return (in credits) by the total expected number of Free Games and then factoring that against the probability of hitting Free Games in the first place:

(2.62021921492746638982269*18.3086771581) * 0.0055056 = 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 = 0.0105667697409932

ADDED VALUE: (0.0105667697409932+0.254262896892648+ 2.4229758409769986+1.0374287561727268+0.4368121078622008) * 0.0708943 = 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) * 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 = 0.0615347183641124

ADDED VALUE: (0.0615347183641124+1.0983947227994061+7.8456765914243295+2.5410290770476398+0.8158488006574362) *0.0120923 = 0.1494908641884351248852

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 = 3.5126923723992075

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 = 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 final few steps will have us multiplying the expected return (in credits) by the total expected number of Free Games and then factoring that against the probability of hitting Free Games in the first place:

(2.99658864230849032317022*18.3086771581) * 0.0055056 = 0.3020568931666592841333559378737508509792

FIVE SPOT BASE RETURN WITHOUT PROGRESSIVES:

0.3020568931666592841333559378737508509792 + 0.484069 = 0.7861258931666592841333559378737508509792 OR 78.6126%.

FINAL RETURNS BEFORE PROGRESSIVES AND METER CONTRIBUTIONS:

THREE-SPOT: 0.7571035

FOUR-SPOT: 0.792997

FIVE-SPOT: 0.786126

THE PROGRESSIVE METERS:

This is an unusual game in that there are ten different progressive meters all based on a five-spot that is separate from the Bonus Games and from the player 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 vary as do the probabilities of a particular progressive even coming up. In order to figure in the progressives, we have to look at the probabilities of hitting three out of five, four out of five and five out of five both in the base game and during Free Games. Fortunately, we have already done these probabilities:

NORMAL PROBABILITIES * 0.9039726 (PROBABILITY OF NO EXTRA DRAWS)

(nCr(5,3)*nCr(75,17)/nCr(80,20)) * .9039726 = 0.0758749874492596

(nCr(5,4)*nCr(75,16)/nCr(80,20)) * .9039726 = 0.0109311422596391

(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 = 0.0082057745084666

(nCr(5,4)*nCr(72,18)/nCr(77,22)) * .0708943 = 0.0014436084783414

(nCr(5,5)*nCr(72,17)/nCr(77,22)) * .0708943 = 0.0000944907367642

DRAW 30 TOTAL (OUT OF 26 BECAUSE FOUR UNLOCKED EXTRA DRAWS):

(nCr(5,3)*nCr(71,23)/nCr(76,26)) * .0120923 = 0.0020846716669806

(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)) * 0.0130408 = 0.0009994562735049

(nCr(5,5)*nCr(71,26)/nCr(76,31)) * 0.0130408 = 0.0001199347528206

BASE PROGRESSIVE & METER MOVER PROBABILITIES:

3/5 : (0.0758749874492596+0.0082057745084666+0.0020846716669806+0.0031411482881584) = 0.0893065819128652

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 probabilities of moving the Progressive Meters or hitting a progressive during the Free Games. We must remember that the WILD spots also count towards the Progressive five-spot, so we are essentially drawing from seven instead.

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 = 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 = 0.0006617772611241

(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 = 0.0056575219591335

(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 = 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 = 0.0016207588171455

(nCr(7,5)*nCr(69,21)/nCr(76,26)) * .0120923 = 0.000445708674715

(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)) * 0.0130408 = 0.0039950097242352

(nCr(7,4)*nCr(69,27)/nCr(76,31)) * 0.0130408 = 0.0026633398161568

(nCr(7,5)*nCr(69,26)/nCr(76,31)) * 0.0130408 = 0.0010033977912033

(nCr(7,6)*nCr(69,25)/nCr(76,31)) * 0.0130408 = 0.0001976389588734

(nCr(7,7)*nCr(69,24)/nCr(76,31)) * 0.0130408 = 0.0000156856316566

ADDED PROBABILITIES PER FREE GAME:

3/5 : 0.1581890954550239+0.0154837443092076+0.0033119854089495+0.0039950097242352 = 0.1809798348974162

4/5 : 0.0471792039076387+0.0056575219591335+0.0016207588171455+0.0026633398161568 = 0.0571208245000745

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 = 0.011501410011269

TOTAL ADDED PROBABILITIES DURING FREE GAMES * FREE GAME PROBABILITY:

3/5 (0.1809798348974162*18.3086771581) * 0.0055056 = 0.018242813138614555793199790556832

4/5 (0.0571208245000745*18.3086771581) * 0.0055056 = 0.00575779355898468825039996183432

PROGRESSIVE HIT: (0.011501410011269*18.3086771581) * 0.0055056 = 0.00115934503855140059648226887184

TOTAL PROBABILITIES OF METER HITS OR PROGRESSIVE AWARDS:

3/5 0.0893065819128652 + 0.018242813138614555793199790556832 = 0.107549395051479755793199790556832

4/5 0.0138634666884298 + 0.00575779355898468825039996183432 = 0.01962126024741448825039996183432

PROGRESSIVE: 0.0008404745950033 + 0.00115934503855140059648226887184 = 0.00199981963355470059648226887184 OR 1 in 500.05

Simplified:

3/5 : .1075494

4/5 : .0196213

5/5 : .0019998

We also know that there can be multiple progressives hit or meters moved during one initial spin, (because of Free Games) but by taking the average number of Free Spins per occurrence and the probabilities of each result on each initial Free Game, we fully account for all expectations.

Progressive Meter Added Values

Okay, since I’m keeping the actual name of this game secret, we’re just going to call these Progressives 1-10. The way it works is each individual progressive has its own probability of showing up on an initial spin and the value added to the meter (based on one credit bet) can be different.

The 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 factor in the probabilities of simultaneously hitting 3/5 or 4/5 as well as corresponding meter move amount.

Progresive 1: 

3 Hits: 4

4 Hits: 8

Probability: .005

Progressive 2:

3 Hits: 2

4 Hits: 6

Probability: .016

Progressive 3:

3 Hits: 2

4 Hits: 4

Probability: .026

Progressive 4:

3 Hits: 2

4 Hits: 3

Probability: .031

Progressive 5:

3 Hits: 2

4 Hits: 3

Probability: .041

Progressive 6:

3 Hits: 2

4 Hits: 3

Probability: .052

Progressives 7-10:

3 Hits: .5

4 Hits: 1

Probability: .828 (Combined)

Armed with this information, we must now take the meter increase probabilities & values and multiply them by the probabilities of having that particular meter.

Progressive 1:

((.1075494*4)+(.0196213*8)) * .005 = 0.00293584

Progressive 2:

((.1075494*2)+(.0196213*6)) * .016 = 0.0053252256

Progressive 3:

((.1075494*2)+(.0196213*4)) * .026 = 0.007633184

Progressive 4:

((.1075494*2)+(.0196213*3)) * .031 = 0.0084928437

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 = 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: 0.7571035 + .1106375 = 0.867741

FOUR-SPOT: 0.792997 + .1106375 = 0.9036345

FIVE-SPOT: 0.786126 + .1106375 = 0.8967635

HITTING THE JACKPOTS:

The final thing we need to look at is the actual returns of the base jackpots along with the probability of hitting them. The jackpots have different starting values, so we will have to look at them all individually.

Also, the probability for each jackpot to be available on a given game is different, but overall, this will follow the same methodology as the previous section.

Progressive 1:

(.0019998 * 200) * .005 = 0.0019998

Progressive 2: 

(.0019998 * 125) * .016 = 0.0039996

Progressive 3:

(.0019998 * 100) * .026 = 0.00519948

Progressive 4:

(.0019998 * 50) * .031 = 0.00309969

Progressive 5:

(.0019998 * 20) * .041 = 0.001639836

Progressive 6:

(.0019998 * 10) * .052 = 0.001039896

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 = 0.0004139586

TOTAL BASE PROGRESSIVE CONTRIBUTION:

0.0019998 + 0.0039996 + 0.00519948 + 0.00309969 + 0.001639836 + 0.001039896 + 0.002069793 + 0.0012418758 + 0.0008279172 + 0.0004139586 = 0.0215318466

RETURNS WITH METER MOVE ADDED AND JACKPOTS:

THREE-SPOT: 0.7571035 + .1106375 = 0.867741 + 0.021531 = 0.889272

FOUR-SPOT: 0.792997 + .1106375 = 0.9036345 + 0.021531 = 0.9251655

FIVE-SPOT: 0.786126 + .1106375 = 0.8967635 + 0.021531 = 0.9182945

Advantage Play

The easiest way to advantage play this would be to fix in on Progressives 7-10 and play those if they make the game positive.

The best returning version of this game returns .9251655 with no increases to the jackpot AND including meter moves. Of course, you’re not going to hit every meter that you move. The first thing that we will do is calculate the probability of each individual jackpot hitting and then determine what the jackpot value would have to be (both including and not including meter move in the return) for the individual jackpots:

Progressive 1:

.0019998 * .005 = 0.000009999 or 1 in 100,010

Progressive 2: 

.0019998 * .016 = 0.0000319968 or 1 in 31,253

Progressive 3:

.0019998 * .026 = 0.0000519948 or 1 in 19,233

Progressive 4:

.0019998 * .031 = 0.0000619938 or 1 in 16,131

Progressive 5: 

.0019998 * .041 = 0.0000819918 or 1 in 12,196

Progressive 6:

.0019998 * .052 = 0.0001039896 or 1 in 9,616

Progressive 7:

.0019998 * .207 = 0.0004139586 or 1 in 2,416

Progressive 8:

.0019998 * .207 = 0.0004139586 or 1 in 2,416

Progressive 9:

.0019998 * .207 = 0.0004139586 or 1 in 2,416

Progressive 10:

.0019998 * .207 = 0.0004139586 or 1 in 2,416

Positive Points

The next step involves some simple algebra and solving for, “X.” The four-spot has the best return, so we’re going to use it. What we are going to want is for the added progressive return for each meter (by itself) to bring us up to 100%.

Progressive 1:

With Meter: (1 - 0.9251655) = x * 0.000009999--->7484.19842/4 = $1,871.05

Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.000009999--->18549.054905/4 = $4,637.26

Progressive 2:

With Meter: (1 - 0.9251655) = x * 0.0000319968--->2338.812006/4 = $584.70

Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0000319968--->5796.579658/4 = $1,449.14

Progressive 3:

With Meter: (1 - 0.9251655) = x * 0.0000519948--->1439.268927/4 = $359.82

Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0000519948--->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 * 0.0000619938--->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 * 0.0000819918--->2262.079867/4 = $565.52

Progressive 6:

With Meter: (1 - 0.9251655) = x * 0.0001039896--->719.634463/4 = $179.91

Without Meter: (1 - (0.9251655 -0.1106375)) = x * 0.0001039896-->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 8:

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 9:

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 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 * 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 are figuring out the average that we would want Progressives 7-10 at since they all have the same probability, but are not likely to have one positive by itself. By using the average, you may even hit one of them and have the play continue, but it enables you to essentially play positively more often by treating it as one progressive.

Advantage Play:

As you can see, we did look at Progressives 7-10 as a whole, but this is not the sort of play where I would look at the entire thing when determining if I want to play.

The reason why is because, almost always, you’re going to end up hitting a progressive that forces you to quit playing as it causes the overall return of the game to drop back below 100%. In the meantime, you’ve also 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 movement when determining whether or not something is a play, with exception ONLY to the progressive, or progressives, if you’re going after 7-10, combined, that you’re actually targeting.

Ideally, you’ll be playing for something somewhat likely (such as the 7-10 combination play) that will be above 100% without breaking your back trying to squeeze every little tenth or hundredth of a percentage to justify playing it.

Or, perhaps you’ll have an individual jackpot that puts it above 100% by itself (ignoring meter move on everything else) and then any increases over the progressive base are just gravy. 105% when you’re ignoring other contributions is always better than 105% after adding every little scrap of value you can justify.

Calculating Other Keno Games And Help

The first thing that I want to reiterate is that this is, by far, the most difficult video keno game that I’ve ever analyzed. Not only does this game have several moving parts, (ten different conditional progressives and three sets of numbers with interrelated probabilities) but there are also Free Games that come with two partially Wild numbers (they can be used for Progressives and Base Pays, but do not substitute for additional draws or extra Free Games).

The math used in this is going to be useful to you for MOST Video Keno games and Progressives as some aspect of this game also covers (in full) what you would need to know how to do in analyzing other Video Keno games.

In most cases, any Video Keno Progressive is going to be related to your base picks in some way, so then it becomes a comparatively simple matter of just determining the base return of the game based on the seed amount, figuring out the contribution of the meter (that will add to the progressive as you play) and figuring out the return of the Free Games.

Cleopatra Keno, for one example, often has a Progressive for events such as hitting 6 out of 6, 7 out of 7, 8 out of 8...and so on. That just has to do with those base events (or hitting it during Free Games) so that’s all that needs to be figured out. Of course, Cleopatra Keno is an example of a game that’s already been fully analyzed on Wizard of Odds.

With a convenient calculator---making it a simple matter of just inserting the paytable and converting the Progressive dollar amount to credits.

But, new Keno games seem to be always coming out lately, so knowing how to do the sort of analysis included on this page is going to give you the advantage of being one of the only people to know the play points for semi-complicated to very complicated Video Keno variants.

If you encounter a game and need help with calculating it, then you should be able to figure it out based on what is included above. I could name ten different Keno games that the methodology above would cover, and more than that, you would only need to know a segment of what I’ve done above because, again, most Keno games do not have so many moving parts.

However, if you find yourself with a question as to methodology, then feel free to leave a comment 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 for free.

If you would like me to analyze a Video Keno game FOR YOU, then feel free to shoot me a message and we can discuss my rates. I recommend trying to do it yourself because I can already tell you it’s not going to be cheap. These types of analyses usually take multiple hours---even for a relatively simple game. Most of it involves just repeating a variant of the same formulas used for other parts---as you can see above.

Conclusion

Anyway, I hope that this has helped you to learn some of the methodologies that can be used in analyzing Video Keno games.

If you have reached this conclusion and do not understand what was done above, feel free to leave any questions in the comments. I tried to describe what I was doing and why with as much detail as possible, as well as to clearly separate the different aspects of the game, but it’s still an insanely complicated Keno game. Don’t be afraid to read it multiple times if you find yourself confused or to take a break and come back to it.

I certainly did. 

I always say that 90% of gambling math is NOT figuring out the answers to things---figuring out answers is easy. 90% of gambling math, at least in my opinion, is figuring out what questions need to be asked and then figuring out how to ask them correctly. I spent as much time figuring out HOW one could go about figuring out this game as I spent actually doing the figuring.