Keno Math Help

On my last road trip I found 4 machines with a keno game that looks very promising. It's an 8-Spot progressive game where the meter starts at $500 and runs at 1%. You get the meter when you catch 6 out of 8 with the last number drawn (called a Power Hit) completing the six. The bet is $1. The money odds on the payscale looks like this:

.............PAYS.......POW'R HIT
4/8..........2................8
5/8........11..............44
6/8........48...........$500 + 1% meter
7/8......200.............800
8/8......800.............800

The Power Hit is the 20th ball drawn. You get paid 4 times the regular pay when the last ball drawn completes the pay, except in the case of the 6/8 with the power hit, and in the case of the 8/8 because Montana law is $800 max jackpot.

What keno math I know I learned from John Scarne. So using his method I first determined how many 8 spot combinations there were:

80X79X78X77X76X75X74X73/8X7X6X5X4X3X2X1 = 28,987,537,150

In order to make a 6/8 with a Power Hit you would have to have 5/8 by the 19th number drawn. So I did this:

19X18X17X16X15/5X4X3X2X1 = 11,628
61X60X59/3X2X1 = 35,990

11,628 X 35,990 = 418,491,720

28,987,537,150/418,491,720 = 69.2667

With 19 numbers out the last ball would have to be one of the 3 remaining numbers in the 8-spot. There would be 61 numbers left in the jug. So:

61/3 = 20.3333

69.2667x20.3333 = 1408.4205

So 1408 would be the frequency of making 6/8 with the power hit. That is, as long as I did the math right. So that's my question to those of you who know how to do the math. Did I do it right? If I did it wrong then where did I go wrong? If I did it right then this game is shaping up to have a big edge.

It looks like I got the thread title wrong. It should say Keno Math Help.

For those of you who may be wondering how I find stuff like this the short answer is I never quit looking. I consider it part of my job to check every meter on every machine every time I go in a casino. In the case of this game it only took me 6 years to find some out of whack meters.

Quote: mickeycrimm

Did I do it right?

Yes

Hi Mickey,

First, I want to tell you that instead of writing 80x79x78x77x76x75x74x73/(8x7x6x5x4x3x2x1), you can just write "80 choose 8", or "80 ch. 8" or Ch(80, 8) for short. They all mean the same thing; the number of ways to uniquely choose 8 items from 80 possibilities. Also, if you are doing this on your calculator, there is an easier way: just go to google and type "80 choose 8" and it will give you the answer -- this should save you some time.

So, if I understand this correctly (I have never played Keno), you pick 8 different numbers from 1 to 80, they pick 20 different numbers from 1 to 80, and you win the prizes you mentioned for matching the right number, with the bonus for matching the last number.

So, to save time, I would calculate it "backwards" (starting with the last number drawn, since order is not actually important -- the odds would be the same if you needed to hit the first number instead of the last)

You must hit the last number. Probability = 8 / 80 = 1 / 10

There are 79 ch. 19 ways that the other 19 numbers can be drawn. Those 19 numbers need to be 5 hits and 14 misses. The "hits" are chosen from your 7 remaining numbers, and the "misses" are chosen from the 72 that you did not pick. So the total number of combinations that have 5 hits and 14 misses is:

(7 ch 5) * (72 ch 14)

So the probability of hitting it is (1/10) * (7 ch 5) * (72 ch 14) / (79 ch 19) = about 1 / 1408.42274

So, we got the same answer (the difference is minor and likely due to rounding).

Quote: mickeycrimm

For those of you who may be wondering how I find stuff like this the short answer is I never quit looking. I consider it part of my job to check every meter on every machine every time I go in a casino. In the case of this game it only took me 6 years to find some out of whack meters.

I have never even seen a progressive keno machine. Are they common in Vegas? (I know that you don't gamble in Vegas, but I'm curious..)

Quote: AxiomOfChoice

You must hit the last number. Probability = 8 / 80 = 1 / 10

There are 79 ch. 19 ways that the other 19 numbers can be drawn. Those 19 numbers need to be 5 hits and 14 misses. The "hits" are chosen from your 7 remaining numbers, and the "misses" are chosen from the 72 that you did not pick. So the total number of combinations that have 5 hits and 14 misses is:

(7 ch 5) * (72 ch 14)

So the probability of hitting it is (1/10) * (7 ch 5) * (72 ch 14) / (79 ch 19) = about 1 / 1408.42274

So, we got the same answer (the difference is minor and likely due to rounding).

Okay, so let me see if I have the overall return:

Baseline Probabilities

1/12.27
1/54.64
1/422.53
1/6232.27
1/230114.61

Probabilities for Power Hit

1/61.34691 (1/10) * (7 choose 3) * (72 choose 16) / (79 choose 19)
1/218.548356837 (1/10) * (7 choose 4) * (72 choose 15) / (79 choose 19)
1/1408.42275398 (1/10) * (7 choose 5) * (72 choose 14) / (79 choose 19)
1/17806.4897533 (1/10) * (7 choose 6) * (72 choose 13) / (79 choose 19)
(1/10) * (7 choose 3) * (72 choose 12) / (79 choose 19)

Base Return

((1/12.27-1/61.34691)*2) + ((1/54.64-1/218.548356837)*11) + ((1/422.53-1/1408.42275398) * 48) + ((1/6232.27-1/17806.4897533) * 200) + (1/61.34691*8) + (1/218.548356837*44) + (1/1408.42275398 * 500) + (1/17806.4897533 * 800) + (1/230114.61 * 800) = 1.1169085830406175

Return w/o 6/8 Power Hit

((1/12.27-1/61.34691)*2) + ((1/54.64-1/218.548356837)*11) + ((1/422.53-1/1408.42275398) * 48) + ((1/6232.27-1/17806.4897533) * 200) + (1/61.34691*8) + (1/218.548356837*44) + (1/17806.4897533 * 800) + (1/230114.61 * 800) = 0.761901538041471

Return if 6/8 Power Hit were 192

((1/12.27-1/61.34691)*2) + ((1/54.64-1/218.548356837)*11) + ((1/422.53-1/1408.42275398) * 48) + ((1/6232.27-1/17806.4897533) * 200) + (1/61.34691*8) + (1/218.548356837*44) + (1/1408.42275398 * 192) + (1/17806.4897533 * 800) + (1/230114.61 * 800) = 0.8982242433211433

Break Even, without the Progressive increase is:

1-0.761901538041471 = 0.23809846195852902

1408.42275398 * 0.23809846195852902 = 335.3432915100337

Proof:

((1/12.27-1/61.34691)*2) + ((1/54.64-1/218.548356837)*11) + ((1/422.53-1/1408.42275398) * 48) + ((1/6232.27-1/17806.4897533) * 200) + (1/61.34691*8) + (1/218.548356837*44) + (1/1408.42275398 * 335.345) + (1/17806.4897533 * 800) + (1/230114.61 * 800) = 1.0000012130519486

Community Chest-Bank Error in your Favor-Collect $165!!!

So I guess the problem is that the advantage is available to anyone who plays the game, potentially limiting the longevity.

Is this game legal within the (idiotic) MT regulations? Mickey, I know you mentioned maximum payback restrictions on some game types.

I am certain that the game is not positive when the progressive is at it's minimum. I'll look over your calculations, Mission.

The game would be legal in Montana. The minimum return for a keno is 80%, and there is no maximum (unlike slot games which have a max of 92%).

edit: Holy Crap and keep your mouth shut Mickey! Your calculations look good, Mission.

Quote: CrystalMath

I am certain that the game is not positive when the progressive is at it's minimum. I'll look over your calculations, Mission.

The game would be legal in Montana. The minimum return for a keno is 80%, and there is no maximum (unlike slot games which have a max of 92%).

I just did, and my calculations were wrong. I forgot to multiply the probability of the top pay by 800, so I actually had the return percentage a little worse than it actually is. That has been fixed.

Particularly, the base pay (if the 6/8 were perfectly graduated) would be 0.8982242433211433 which makes perfect sense.

This is pure speculation on my part, but if I had to guess, I would venture to say that the 500 Progressive base point is supposed to be a Power Hit on Six/Seven rather than Six/Eight. The return for that would be:

1/10*(6 choose 5)*(73 choose 14)/(79 choose 19) = 1 in 3984.1

Return = 0.12549885796 (500/3984.1)

Now, if you had a six/seven that paid 125, the overall contribution to the return from 6/7 would be:

((1/1365.98 - 1/3984.1) * 125) + 0.12549885796 = 0.18563353599

If we assume a return of 800 for 7/7, then:

0.019522044838501 + 0.18563353599 = 0.20515558082

You would have about .695 remaining for hits 3/7, 4/7, 5/7 to have a game that returns 90%, which is quite feasible. If you look at the Wizard's page on Power Keno, you'll see that the returns for such are:

https://wizardofodds.com/games/keno/appendix/6/

0.2537402+0.16701109+0.21164337 = 0.63239466

With a major difference in Montana being the MAX Jackpot of 800, because Wizard's Power Keno analysis returns 6.25% on 7/7 pursuant to Base Pay 1250 and Power Hit 5,000.

The combined pay based on my theoretical (6/7 is meant for 500) and Wizard's pays for 3/4/5, combined, is:

0.63239466+0.20515558082 = 0.83755024082

However, an increase to a lower Base Pay percentage compensates for this discrepancy readily enough. For example, if you increase the Base Pay of 5/7 from 14 to 18:

(0.21164337/14) * 18 = 0.27211290428

Which would increase the overall return by:

0.27211290428-0.21164337 = 0.06046953428

Thereby resulting in a new overall return of:

0.06046953428+0.83755024082= 0.8980197751---Which is consistent with standard returns.

In this scenario, the player is eating:

1 - 0.8980197751 - .01 = 0.0919802249

And would need an increase to the Progressive of:

1/3984.1 * x = 0.0919802249

x = 366.4584227

Which means that the game could never go positive pursuant to the statutory MAX pay of $800, thus, the highest the game could return is:

(1/3984.1 * 300) + 0.8980197751 = 0.97331908987 or 97.332%

Therefore, my guess is the Progressive should be 6/7 not 6/8 with a paytable that is roughly:

0/7 = 0
1/7 = 0
2/7 = 0
3/7 = 1---PH = 4
4/7 = 2---PH = 8
5/7 = 18---PH = 72
6/7 = 125---PH = PROGRESSIVE (Base of 500)
7/7 = 800

Hey, Mickey, do you know if 6/7 is usually a Progressive on these machines, and, if so, is my Paytable close?

EDIT: You could also add ~8.35% to the return by increasing the base on 4/7 from 2 to 3. If you did that, this would add:

(0.16701109/2) - 0.06046953428 = 0.02303601072

To the return compared to increasing 5/7 to 18 from 14, which means PROGRESSIVE @ 800 would pay:

0.97331908987 + 0.02303601072 = 0.99635510059 or 99.635510059%

Thus, another return with Progressive 6/7 could be:

3/7 = 1---PH=4
4/7 = 3---PH = 12
5/7 = 14---PH 56
6/7 = 125---PH-PROGRESSIVE
7/7 = 800

That's the problem with a statutory Max Pay of $800, if you want to have Progressives, you need to be really cute about it and mistakes can be made.

Quote: Mission146

This is pure speculation on my part, but if I had to guess, I would venture to say that the 500 Progressive base point is supposed to be a Power Hit on Six/Seven rather than Six/Eight.

I'm admittedly confused, but Mickey mentioned other games with improper reset values (due to technician error), such as the video poker where aces full resets at $100 rather than $10.

So for this game, I assumed the reset was supposed to be something like $100 $200 rather than $500.

EDIT: a $200 reset would be more logical, I suppose

Quote: gpac1377

I'm admittedly confused, but Mickey mentioned other games with improper reset values (due to technician error), such as the video poker where aces full resets at $100 rather than $10.

So for this game, I assumed the reset was supposed to be something like $100 $200 rather than $500.

EDIT: a $200 reset would be more logical, I suppose

That's definitely another possibility, and the more obvious one, I was merely speculating on what else it could possibly be. The reason that I think it could be the 6/8 Progressive was meant to be a 6/7 Progressive is also because 6/8 is just a darned weird Progressive point...although, again, they might have to do silly crap like that because of the $800 statutory Maximum.

I was just kind of going with what I am familiar with, and I've certainly seen Progressive Keno games where 6/7 is the Progressive winner...not just in special circumstances, (Bonus hits, such as this game) but also in general.

Am I missing something? How are you calculating the return on the game without knowing the denomination and wager amount to qualify for the progressive?

Quote: DRich

Am I missing something? How are you calculating the return on the game without knowing the denomination and wager amount to qualify for the progressive?

The denomination and wager amount both appear to be $1 based on my understanding of MickeyCrimm's OP.

Quote: Mission146

The denomination and wager amount both appear to be $1 based on my understanding of MickeyCrimm's OP.

Thank you. I just re-read it and you are correct.

Quote: CrystalMath

Yes

Thanks, Crystal, for the confirmation.

Quote: CrystalMath

I am certain that the game is not positive when the progressive is at it's minimum. I'll look over your calculations, Mission.

The game would be legal in Montana. The minimum return for a keno is 80%, and there is no maximum (unlike slot games which have a max of 92%).

edit: Holy Crap and keep your mouth shut Mickey! Your calculations look good, Mission.

The game is called Power Hit Plus. It's another Summit game. Its on a few thousand machines in Montana. But the progressive starts at $200. There are some machines where it starts at $350. But on my last road trip I found four machines in one joint where the meter starts at $500.

Al is coming back from Washington in mid April. He found a five-spot keno game in a little one horse town not far from where I found this game. It's a game with a 50 cent bet. 3/5 pays 5 for 1, 4/5 pays 32 for 1, and the solid five pays 400 for 1.

We're going to tag team both games.

The meter is for either 6/8 or 7/10. I figured the 6/8 was the better play. And the meter normally resets to $200.

Quote: DRich

Am I missing something? How are you calculating the return on the game without knowing the denomination and wager amount to qualify for the progressive?

Its a $1 bet.

Quote: Mission146

Hey, Mickey, do you know if 6/7 is usually a Progressive on these machines, and, if so, is my Paytable close?.

No, the only 6/7 play I had was on a Spielo game called Chambers of Gold. You might have it in your jurisdiction. Spielo has a few exploitable progressive/banking games like Hot "N Hotter, Three Stooges, etc.

The Chambers of Gold game doesn't get any action from the ploppies anymore so I rarely get a play on it these days. Make 6/7 five times and you get the meter that starts at $100 and runs at 4%. I would play when 3 or four 6/7's had been banked and the meter was up at a few hundred dollars.

Quote: CrystalMath

I am certain that the game is not positive when the progressive is at it's minimum. I'll look over your calculations, Mission.

The game would be legal in Montana. The minimum return for a keno is 80%, and there is no maximum (unlike slot games which have a max of 92%).

edit: Holy Crap and keep your mouth shut Mickey! Your calculations look good, Mission.

The 7/8 with the power hit is a key hit. I put the frequency at about 18,000. I think the game is a 3% edge up through the 6/8 with a power hit. And the 7/8 hits add about another 7%.

Quote: AxiomOfChoice

I have never even seen a progressive keno machine. Are they common in Vegas? (I know that you don't gamble in Vegas, but I'm curious..)

Eleven years in Nevada and I only had one keno play. They just aren't there in Nevada. The only play was at John Ascuagua's Nugget in Sparks where they had a double pay promotion for a top line hit. I had to find a game where a big chunk of the payback was in the top line hit on a five-spot. I think it was called Top Hat Keno or something like that.

Probably the biggest machine play of all time was a dollar eight-spot progressive in Florida. We all knew Tuna Lund was in Florida working some big keno play. Al and me were in Yerrington working a Draw Till U Win video poker play and we met a guy who was one of Tuna's partners on the play. He told Al that him and Tuna made $8,000,000 in four years on the play. They wound up partners because they were both on the play and didn't want to butt heads. They were paying people $30 an hour to play the game.

Quote: mickeycrimm

No, the only 6/7 play I had was on a Spielo game called Chambers of Gold. You might have it in your jurisdiction. Spielo has a few exploitable progressive/banking games like Hot "N Hotter, Three Stooges, etc.

The Chambers of Gold game doesn't get any action from the ploppies anymore so I rarely get a play on it these days. Make 6/7 five times and you get the meter that starts at $100 and runs at 4%. I would play when 3 or four 6/7's had been banked and the meter was up at a few hundred dollars.

I've not seen that particular game, but the Spielo Pick N' Play machines are quite old, in fact, Spielo doesn't even manufacture/distribute them anymore, I don't believe. The Progressive games are Lucky Bells Keno, Frost N' Fire Keno, Big Catch Keno and Classic Keno...but only a few of the units on the last one, for whatever reason.

The meter operation is quite unique and lends itself well to growing Progressives for those inclined to look for them. On the $0.25 denomination, bets range from $0.25-$2.50, but $2.25 + $2.50 NEVER qualify for the Progressive, for whatever reason, nor does $0.25. The Optimal Strategy, regardless of where the meter is at, is always to bet $0.50. It's nice, though, because you have people building up that Progressive on the short-coin end and on the high end.

The nickel denomination is $0.05-$2.50, same parameters otherwise.

Additionally, bets of $0.75-$2.00 don't qualify for the Progressive if the Base Pay would be greater, but they still add to the meter.

The result, of course, is Progressive Meters that can increase quickly because of non-qualifying bet amounts AND the fact that the necessary result for the Progressive can actually hit a few times, (Base Pay greater than Progressive for certain bet amounts) but the Progressive is not paid out.

Do they have these games in Vegas? I've never seen them. The only Keno machines that I've seen are part of the multi-game machines that also have VP, slots, and sometimes blackjack on then.

The Spielo Pick N' Plays are such machines, but no, I did not see any of those machines in Vegas.

Quote: mickeycrimm

The game is called Power Hit Plus. It's another Summit game. Its on a few thousand machines in Montana. But the progressive starts at $200. There are some machines where it starts at $350. But on my last road trip I found four machines in one joint where the meter starts at $500.

Al is coming back from Washington in mid April. He found a five-spot keno game in a little one horse town not far from where I found this game. It's a game with a 50 cent bet. 3/5 pays 5 for 1, 4/5 pays 32 for 1, and the solid five pays 400 for 1.

We're going to tag team both games.

Thanks Crystal, Mission and Axiom for the help analyzing this play. Al and me played the game a week ago and made $2200 in lesd than two hours. I put the edge at 11%. On turbo the game plays at 48 games per minute. At a bet of $1 a game its a $320 an hour play on $2900 an hour in action. The game has some nasty variance but we ran good hitting 8 power 6's in less than two hours.

I've been on this road trip for two weeks and won't make it home for another week. I'll tell you guys the story then.

Quote: mickeycrimm

Thanks Crystal, Mission and Axiom for the help analyzing this play. Al and me played the game a week ago and made $2200 in lesd than two hours. I put the edge at 11%. On turbo the game plays at 48 games per minute. At a bet of $1 a game its a $320 an hour play on $2900 an hour in action. The game has some nasty variance but we ran good hitting 8 power 6's in less than two hours.

I've been on this road trip for two weeks and won't make it home for another week. I'll tell you guys the story then.

Wow, good job man.

I don't understand how paytables like this can survive. With VP it's different -- most people play terribly so they can still make money from VP with a 100%+ paytable. But there is no skill in Keno. If everyone sitting at the machine makes money, won't they soon notice that they are losing money on that machine?

Quote: AxiomOfChoice

Wow, good job man.

I don't understand how paytables like this can survive. With VP it's different -- most people play terribly so they can still make money from VP with a 100%+ paytable. But there is no skill in Keno. If everyone sitting at the machine makes money, won't they soon notice that they are losing money on that machine?

Multi-game units with that not being the most popular game because ploppies don't realize its value at the correct time?

Quote: AxiomOfChoice

Wow, good job man.

I don't understand how paytables like this can survive. With VP it's different -- most people play terribly so they can still make money from VP with a 100%+ paytable. But there is no skill in Keno. If everyone sitting at the machine makes money, won't they soon notice that they are losing money on that machine?

First off, you're welcome, Mickey, pleasure for me to be able to assist a legend in his own right!

I would say that games like this survive simply because people don't realize the value. Some players are still going to lose because of the amount of money they put in the machine (not enough) and Variance, so they'll tap out. I imagine the majority of players on that machine are playing some other game, then you have the players who are playing some other amount of numbers on the game who do not have the edge. Many of the players who do eventually win on this machine (one way or the other) will often cash out on the first good hit and leave or go on to play something else.

Without a fundamental understanding of Keno Math, your average layman is not going to look at this machine and see any kind of advantage, it's just a paytable to them. Thus, I would say that the amount of players playing this specific game at this specific number of picks are not going to swing the machine's average hold in a drastically obvious enough manner to be a cause for alarm to the operators, at least, not always.

Crap I'm in Washington this week.