Pay table formula

What formula could I use to calculate a pay table that meets the following criteria?
1. EV=0 (EV%=100%), player should break even in long run
2. Payout proportional to probability. An event that occurs twice as often should pay half as much

For example: I want to create a pay table for a game where the player rolls 2 dice. If the sum of the dice is 8-12, player wins, 2-7 player loses. Since rolling an 8 is 5 times more likely than a 12, rolling a 12 should pay 5x what rolling an 8 pays. Overall EV should be 0, player should break even.

How would I calculate the pay table for each total, 8-12?

I'll get into something more to your specifications the post after this one, but the quick and dirty way would be:

7 = PUSH

6 = Lose 20% of bet---8 = Win 20% of bet

5 = Lose 40% of bet---9 = Win 40% of bet

4 = Lose 60% if bet---10 = Win 60% of bet

3 = Lose 80% of bet---11 = Win 80% of bet

2 = Lose 100% of bet---12 = Win 100% of bet

Okay, for something a little bit more complicated, what we would first want to do is determine the overall probability of losing, which would be any roll 2-7:

1 + 2 + 3 + 4 + 5 + 6 = 21/36 Probability of Losing

The rest of it you want to index to the probability of winning on those particular outcomes, with the less likely ways to win paying more. There are fifteen possible combinations of dice left that yield the following probabilities (relative to each other):

Eight: 5/15
Nine: 4/15
Ten: 3/15
Eleven: 2/15
Midnight: 1/15

If we want the payouts to be perfectly proportional to the probabilities, then we have to get an idea of what the base profit for a winning outcome would be. The way I would start with this is by determining what the average profit would need to be for these two to be equivalent:

(21/36)/(15/36) = 1.4x***

With that, we have determined that the average outcome must be 1.4x on the profitable side in order to offset the lower probability of the result happening as opposed to a losing result. The next thing that we want to do is weight the payouts according to their rarity, which would be relative to each other. With that, a result of eight should reflect the lowest possible profit with a result of 12 being five times more profitable than that, everything in between also doing what it should.

Again, what we can do for that is create another equation that solves for x using a longer equation that effectively accomplishes the same thing that we did above:

(21/36) = (x * 5/36) + (1.25x * 4/36) + (1.66666667x * 3/36) + (2.5x * 2/36) + (5x * 1/36)--->Solve for x

x = 21/25

What that means is that the base pay is going to be 21/25 for x, so a result of eight would pay .84 and the rest accordingly:

Eight: .84

Nine: (21/25) * 1.25 = 1.05

Ten: (21/25) * 1.666666667) = 1.4

Eleven: (21/25) * 2.5 = 2.1

Midnight: (21/25) * 5 = 4.2

Okay, let's go ahead and prove this out:

(21/36) - (.84 * 5/36) - (1.05 * 4/36) - (1.4 * 3/36) - (2.1 * 2/36) - (4.2 * 1/36) = 0

It's not exactly an elegant payscale, but the House Edge is 0% and the payouts are perfectly proportional to the likelihood of each winning event occurring.

***What you will notice is that this average matches the payout for a result of ten and the reason why is because ten is the median and mean outcome (3/15) in terms of frequency of winning results.

Quote: Mission146

(snip)
Eight: .84

Nine: (21/25) * 1.25 = 1.05

Ten: (21/25) * 1.666666667) = 1.4

Eleven: (21/25) * 2.5 = 2.1

Midnight: (21/25) * 5 = 4.2

Okay, let's go ahead and prove this out:

(21/36) - (.84 * 5/36) - (1.05 * 4/36) - (1.4 * 3/36) - (2.1 * 2/36) - (4.2 * 1/36) = 0
(snip)
link to original post

I know you were trying to get a "perfectly proportional payout", but what about the following:
8 = 0.8
9 = 1.1
10 = 1.4
11 = 2.2
12 = 4

Please check figures, but I think I have used your information correctly to get a "different, but also correct" pay table.

Note: I used your figures and formula as a "short-cut" (I probably wouldn't have been able to do this without something like your post, above).

----
Update:
Math not checked, but if this is a "live casino style" operation, you probably want something similar to the pay table^^^ below:

0.75
1
1.5
2.5
4/3.75

^^^ (important): It is very unlikely to be an exactly "0% EV pay table" (it is just an example of a style of one, to make it easier for the dealer to payout).

Note 1: All the above strike-through / edits were made around 1510, Pac Time.
Note 2: "Live dealers " (especially ones used to Blackjack payouts) may find this pay table a bit "easier" (?)

I’ll check that when I’m on the computer tomorrow. Quite honestly, I’m a bit happy with myself that you used what I did as proof of concept. That’s why I show a step by step process the way I do on these sorts of questions.

Thanks, I appreciate all the help you folks are giving!

My original intent was to compare the Powerball pay table with the type of pay table I suggested. When I struggled to figure out how to calculate my “fair odds” pay table, I thought I’d try something simpler as a way to learn how to do the calculations. Then I’d try applying the same formulas to Powerball.

I’m curious to also try this experiment for some Blackjack side bets or for video poker pay tables.

Quote: seitz

Thanks, I appreciate all the help you folks are giving!

My original intent was to compare the Powerball pay table with the type of pay table I suggested. When I struggled to figure out how to calculate my “fair odds” pay table, I thought I’d try something simpler as a way to learn how to do the calculations. Then I’d try applying the same formulas to Powerball.

I’m curious to also try this experiment for some Blackjack side bets or for video poker pay tables.
link to original post

Easy enough for me for drawing or card games if you can tell me specifically what you want.

Video poker is a tougher affair because maximum theoretical return to player depends on optimal strategy for the game, unless you want a stud game, which goes back to being fairly easy. VP draw games require programming and are outside of my pay grade.

But, yes, this same concept should work with any fixed odds game.

Quote: ksdjdj

Quote: Mission146

(snip)
Eight: .84

Nine: (21/25) * 1.25 = 1.05

Ten: (21/25) * 1.666666667) = 1.4

Eleven: (21/25) * 2.5 = 2.1

Midnight: (21/25) * 5 = 4.2

Okay, let's go ahead and prove this out:

(21/36) - (.84 * 5/36) - (1.05 * 4/36) - (1.4 * 3/36) - (2.1 * 2/36) - (4.2 * 1/36) = 0
(snip)
link to original post

I know you were trying to get a "perfectly proportional payout", but what about the following:
8 = 0.8
9 = 1.1
10 = 1.4
11 = 2.2
12 = 4

Please check figures, but I think I have used your information correctly to get a "different, but also correct" pay table.

Note: I used your figures and formula as a "short-cut" (I probably wouldn't have been able to do this without something like your post, above).

----
Update:
Math not checked, but if this is a "live casino style" operation, you probably want something similar to the pay table^^^ below:

0.75
1
1.5
2.5
4/3.75

^^^ (important): It is very unlikely to be an exactly "0% EV pay table" (it is just an example of a style of one, to make it easier for the dealer to payout).

Note 1: All the above strike-through / edits were made around 1510, Pac Time.
Note 2: "Live dealers " (especially ones used to Blackjack payouts) may find this pay table a bit "easier" (?)
link to original post

Very nice! It works and also comes out to zero! How did you do the fine tuning?

Quote: seitz

Thanks, I appreciate all the help you folks are giving!

My original intent was to compare the Powerball pay table with the type of pay table I suggested. When I struggled to figure out how to calculate my “fair odds” pay table, I thought I’d try something simpler as a way to learn how to do the calculations. Then I’d try applying the same formulas to Powerball.

I’m curious to also try this experiment for some Blackjack side bets or for video poker pay tables.
link to original post

Since the Powerball is delayed anyway, and you brought it up, let's go ahead and do this for the Powerball!

Fortunately, California lottery conveniently lists the Odds of every prize (I'm sure others do, as well, if not all of them) right here:

https://www.powerball.ca/odds-prizes/

While I prefer they be expressed as probabilities, that is easy enough for us to convert on our own. The first thing to note is that I am going to use the free online scientific calculator found here:

https://web2.0calc.com/

Simply because I like it and it is long enough to spit out the probability for the top prize as a decimal, which saves me a step.

Top Prize: 1/292201338 = 0.0000000034222978

Five Numbers (Nobody Home) = 1/11688054 = 0.0000000855574418

Four + Powerball = 1/913129 = 0.0000010951355175

Four Numbers (Nobody Home) = 1/36525 = 0.0000273785078713

Three + Powerball = 1/14494 = 0.0000689940665103

Three (Nobody Home) = 1/580 = 0.0017241379310345

Two + Powerball = 1/701 = 0.0014265335235378

One + Powerball = 1/92 = 0.0108695652173913

Powerball Only = 1/38 = 0.0263157894736842

The next thing that we have to do is convert losing to a probability:

1 - (0.0000000034222978 + 0.0000000855574418 + 0.0000010951355175 + 0.0000273785078713 + 0.0000689940665103 + 0.0017241379310345 + 0.0014265335235378 + 0.0108695652173913 + 0.0263157894736842) = 0.9595664171647135

And winning:

(0.0000000034222978 + 0.0000000855574418 + 0.0000010951355175 + 0.0000273785078713 + 0.0000689940665103 + 0.0017241379310345 + 0.0014265335235378 + 0.0108695652173913 + 0.0263157894736842) = 0.0404335828352865

We're going to determine what the average win must be in order for this to be a breakeven game:

0.9595664171647135/0.0404335828352865 = 23.7319166365662066337x

Of course, that's not going to be quite as useful as with the Craps game because the probabilities of all winning results are more equally distributed in our Craps example, whereas these are all over the place. The next thing that we have to do is determine how the probabilities of the winning results relate to one another. The step of getting the multiples of x was so easy last time that I didn't even bother to show it as it could just be done mentally. This time, eh, not so much, so let's do that:

Powerball Only: 0.0263157894736842 (This will relate to the base x)

One Number + Powerball: 0.0263157894736842/0.0108695652173913 = 2.4210526315789473684x

Two Numbers + Powerball: 0.0263157894736842/0.0014265335235378 = 18.4473684210526647842x*

Three Numbers (Nobody Home): 0.0263157894736842/0.0017241379310345 = 15.2631578947366833684x*

Three Numbers + Powerball: 0.0263157894736842/0.0000689940665103 = 381.4210526314688692526x

Four Numbers (Nobody Home) 0.0263157894736842/0.0000273785078713 = 961.1842105270531138816x *

Four Numbers + Powerball: 0.0263157894736842/0.0000010951355175 = 24029.7105272948103429583x

Five Numbers (Nobody Home): 0.0263157894736842/0.0000000855574418 = 307580.3684640346504609959x

Grand Prize: 0.0263157894736842/0.0000000034222978 = 7689508.9239996005023291661x

*This is correct. While both results pay $7 normally, Two + PB is actually less likely than three numbers without.

0.9595664171647135 = (x * 0.0263157894736842) + (2.4210526315789473684x * 0.0108695652173913) + (18.4473684210526647842x * 0.0014265335235378) + (15.2631578947366833684x * 0.0017241379310345) + (381.4210526314688692526x * 0.0000689940665103) + (961.1842105270531138816x * 0.0000273785078713) + (24029.7105272948103429583x * 0.0000010951355175) + (307580.3684640346504609959x * 0.0000000855574418) + (7689508.9239996005023291661x * 0.0000000034222978)

x = 4.0515026547526821719

Powerball Only: 4.0515026547526821719

One Number + PB: 4.05150265 * 2.4210526315789473684 = 9.80890115263157894728312626

Two Numbers + PB: 4.05150265 * 18.4473684210526647842 = 74.73956204342118716274797813

Three Numbers: 4.05150265 * 15.2631578947366833684 = 61.83872465789409371928352626

Three Numbers + PB: 4.0515026547526821719 * 381.4210526314688692526 = 1545.32840731495863399832609519668409772194

Four Numbers: 4.0515026547526821719 * 961.1842105270531138816 = 3894.24038065671664886575582322946072744704

Four Numbers + PB: 4.0515026547526821719 * 24029.7105272948103429583 = 97356.43599427339825473475474186463711513177

Five Numbers: 4.0515026547526821719 * 307580.3684640346504609959 = 1246162.67938184454965999908007895833024899521

Five + PB: 4.0515026547526821719 * 7689508.9239996005023291661 = 31154065.81932882200788680097658227714886385259

Okay, so we substitute to confirm:

0.9595664171647135 - (0.0263157894736842 * 4.0515026547526821719) - (0.0108695652173913 * 9.80890115263157894728312626) - (0.0014265335235378 * 74.73956204342118716274797813) - (0.0017241379310345 * 61.83872465789409371928352626) - (0.0000689940665103 * 1545.32840731495863399832609519668409772194) - (0.0000273785078713 * 3894.24038065671664886575582322946072744704) - (0.0000010951355175 * 97356.43599427339825473475474186463711513177) - (0.0000000855574418 * 1246162.67938184454965999908007895833024899521) - (0.0000000034222978 * 31154065.81932882200788680097658227714886385259) = -0.000000000690972774893343870926910347613011042095407989

Oh yeah!!! You know that's right!

One interesting thing is that, relatively speaking, the Powerball actually ($4) pays about what it should if this were a 100% game indexed to the probability of outcome. This is a bit more relative, but the five numbers without Powerball at $1,000,000 isn't ridiculously far off from 1.25M. Everything else is worlds away, except for the PB, which occasionally pays as it, 'Should,' were this a 100% game with pays indexed to probability of outcome.

Quote: Mission146

Quote: seitz

Thanks, I appreciate all the help you folks are giving!

My original intent was to compare the Powerball pay table with the type of pay table I suggested. When I struggled to figure out how to calculate my “fair odds” pay table, I thought I’d try something simpler as a way to learn how to do the calculations. Then I’d try applying the same formulas to Powerball.

I’m curious to also try this experiment for some Blackjack side bets or for video poker pay tables.
link to original post

Since the Powerball is delayed anyway, and you brought it up, let's go ahead and do this for the Powerball!

Fortunately, California lottery conveniently lists the Odds of every prize (I'm sure others do, as well, if not all of them) right here:

https://www.powerball.ca/odds-prizes/

While I prefer they be expressed as probabilities, that is easy enough for us to convert on our own. The first thing to note is that I am going to use the free online scientific calculator found here:

https://web2.0calc.com/

Simply because I like it and it is long enough to spit out the probability for the top prize as a decimal, which saves me a step.

Top Prize: 1/292201338 = 0.0000000034222978

Five Numbers (Nobody Home) = 1/11688054 = 0.0000000855574418

Four + Powerball = 1/913129 = 0.0000010951355175

Four Numbers (Nobody Home) = 1/36525 = 0.0000273785078713

Three + Powerball = 1/14494 = 0.0000689940665103

Three (Nobody Home) = 1/580 = 0.0017241379310345

Two + Powerball = 1/701 = 0.0014265335235378

One + Powerball = 1/92 = 0.0108695652173913

Powerball Only = 1/38 = 0.0263157894736842

The next thing that we have to do is convert losing to a probability:

1 - (0.0000000034222978 + 0.0000000855574418 + 0.0000010951355175 + 0.0000273785078713 + 0.0000689940665103 + 0.0017241379310345 + 0.0014265335235378 + 0.0108695652173913 + 0.0263157894736842) = 0.9595664171647135

And winning:

(0.0000000034222978 + 0.0000000855574418 + 0.0000010951355175 + 0.0000273785078713 + 0.0000689940665103 + 0.0017241379310345 + 0.0014265335235378 + 0.0108695652173913 + 0.0263157894736842) = 0.0404335828352865

We're going to determine what the average win must be in order for this to be a breakeven game:

0.9595664171647135/0.0404335828352865 = 23.7319166365662066337x

Of course, that's not going to be quite as useful as with the Craps game because the probabilities of all winning results are more equally distributed in our Craps example, whereas these are all over the place. The next thing that we have to do is determine how the probabilities of the winning results relate to one another. The step of getting the multiples of x was so easy last time that I didn't even bother to show it as it could just be done mentally. This time, eh, not so much, so let's do that:

Powerball Only: 0.0263157894736842 (This will relate to the base x)

One Number + Powerball: 0.0263157894736842/0.0108695652173913 = 2.4210526315789473684x

Two Numbers + Powerball: 0.0263157894736842/0.0014265335235378 = 18.4473684210526647842x*

Three Numbers (Nobody Home): 0.0263157894736842/0.0017241379310345 = 15.2631578947366833684x*

Three Numbers + Powerball: 0.0263157894736842/0.0000689940665103 = 381.4210526314688692526x

Four Numbers (Nobody Home) 0.0263157894736842/0.0000273785078713 = 961.1842105270531138816x *

Four Numbers + Powerball: 0.0263157894736842/0.0000010951355175 = 24029.7105272948103429583x

Five Numbers (Nobody Home): 0.0263157894736842/0.0000000855574418 = 307580.3684640346504609959x

Grand Prize: 0.0263157894736842/0.0000000034222978 = 7689508.9239996005023291661x

*This is correct. While both results pay $7 normally, Two + PB is actually less likely than three numbers without.

0.9595664171647135 = (x * 0.0263157894736842) + (2.4210526315789473684x * 0.0108695652173913) + (18.4473684210526647842x * 0.0014265335235378) + (15.2631578947366833684x * 0.0017241379310345) + (381.4210526314688692526x * 0.0000689940665103) + (961.1842105270531138816x * 0.0000273785078713) + (24029.7105272948103429583x * 0.0000010951355175) + (307580.3684640346504609959x * 0.0000000855574418) + (7689508.9239996005023291661x * 0.0000000034222978)

x = 4.0515026547526821719

Powerball Only: 4.0515026547526821719

One Number + PB: 4.05150265 * 2.4210526315789473684 = 9.80890115263157894728312626

Two Numbers + PB: 4.05150265 * 18.4473684210526647842 = 74.73956204342118716274797813

Three Numbers: 4.05150265 * 15.2631578947366833684 = 61.83872465789409371928352626

Three Numbers + PB: 4.0515026547526821719 * 381.4210526314688692526 = 1545.32840731495863399832609519668409772194

Four Numbers: 4.0515026547526821719 * 961.1842105270531138816 = 3894.24038065671664886575582322946072744704

Four Numbers + PB: 4.0515026547526821719 * 24029.7105272948103429583 = 97356.43599427339825473475474186463711513177

Five Numbers: 4.0515026547526821719 * 307580.3684640346504609959 = 1246162.67938184454965999908007895833024899521

Five + PB: 4.0515026547526821719 * 7689508.9239996005023291661 = 31154065.81932882200788680097658227714886385259

Okay, so we substitute to confirm:

0.9595664171647135 - (0.0263157894736842 * 4.0515026547526821719) - (0.0108695652173913 * 9.80890115263157894728312626) - (0.0014265335235378 * 74.73956204342118716274797813) - (0.0017241379310345 * 61.83872465789409371928352626) - (0.0000689940665103 * 1545.32840731495863399832609519668409772194) - (0.0000273785078713 * 3894.24038065671664886575582322946072744704) - (0.0000010951355175 * 97356.43599427339825473475474186463711513177) - (0.0000000855574418 * 1246162.67938184454965999908007895833024899521) - (0.0000000034222978 * 31154065.81932882200788680097658227714886385259) = -0.000000000690972774893343870926910347613011042095407989

Oh yeah!!! You know that's right!

One interesting this is that, relatively speaking, the Powerball only actually ($4) pays about what it should if this were a 100% game indexed to the probability of outcome. This is a bit more relative, but the five numbers without Powerball at $1,000,000 isn't ridiculously far off from 1.25M. Everything else is worlds away, except for the PB, which occasionally pays as it, 'Should,' were this a 100% game with pays indexed to probability of outcome.
link to original post

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym

Quote: tuttigym

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym
link to original post

I'll address only your last question as all of your other questions are a waste of my time: No, this could not be performed with real PB ticket purchases because the question stipulates that the Powerball have a return-to-player of 100% AND that the payouts be perfectly proportionate to the probabilities associated with each type of win.

You might not be aware of this, but the Powerball does not have a RTP of 100%. Even if it did, the payouts aren't structured to be perfectly proportionate to the probability of each result. Actual Powerball tickets, believe it or not, yield an expected loss for players.

I'm not sure how it escaped your attention, but the entire premise of this thread is how to make zero house edge games wherein the payouts correspond perfectly to the probabilities of winning outcomes.

Quote: tuttigym

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym
link to original post

I hate to break it to you, but this is mostly basic 6th grade math these days if last night's homework help is a reasonable representation.

Quote: Dieter

I hate to break it to you, but this is mostly basic 6th grade math these days if last night's homework help is a reasonable representation.
link to original post

I'm just proud to know that after all of those years of college, after all the different low-level management jobs, after over a decade of participation on this forum and after more than seven years of writing about gambling for compensation...that I have finally realized my dream of being able to do math at a sixth grade level.

I knew I would get there one day, but I could never imagine just how magical it would feel when I did.

Quote: Mission146

Quote: Dieter

I hate to break it to you, but this is mostly basic 6th grade math these days if last night's homework help is a reasonable representation.
link to original post

I'm just proud to know that after all of those years of college, after all the different low-level management jobs, after over a decade of participation on this forum and after more than seven years of writing about gambling for compensation...that I have finally realized my dream of being able to do math at a sixth grade level.

I knew I would get there one day, but I could never imagine just how magical it would feel when I did.
link to original post

Well, in fairness, you are using a lot more significant digits than we were last night.
You may have advanced beyond writing out long division exercises and intuitively understand "dividend" and "divisor" as well.

Should we put a star for the day by your name on the bulletin board for doing the problem so quickly?

Quote: Dieter

Well, in fairness, you are using a lot more significant digits than we were last night.
You may have advanced beyond writing out long division exercises and intuitively understand "dividend" and "divisor" as well.

Should we put a star for the day by your name on the bulletin board for doing the problem so quickly?
link to original post

(Quote clipped, relevance)

I honestly would have preferred not to, but the improbability of the top jackpot made it necessary that I do so.

I don't want the other kids in class to bully me at lunch, but I would be honored if you would print out my post from the last page and hang it up on your fridge.

Quote: Mission146

Quote: tuttigym

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym
link to original post

I'll address only your last question as all of your other questions are a waste of my time: No, this could not be performed with real PB ticket purchases because the question stipulates that the Powerball have a return-to-player of 100% AND that the payouts be perfectly proportionate to the probabilities associated with each type of win.

You might not be aware of this, but the Powerball does not have a RTP of 100%. Even if it did, the payouts aren't structured to be perfectly proportionate to the probability of each result. Actual Powerball tickets, believe it or not, yield an expected loss for players.

I'm not sure how it escaped your attention, but the entire premise of this thread is how to make zero house edge games wherein the payouts correspond perfectly to the probabilities of winning outcomes.
link to original post

So, the first two questions which would take about 30 seconds to answer are "a waste of your time," but the last question which is unproveable is important to address. I personally do not care how the RTP is calculated in a game that has odds of 292+ million to one odds. I obviously cannot speak for the population that plays, but I am pretty confident that they don't care either.

In case you did not notice, the "expected loss" for players is $2/ticket. My guess is that .000001% of players gives a rat's patoot about the RTP calculations.

tuttigym

Quote: tuttigym

So, the first two questions which would take about 30 seconds to answer are "a waste of your time," but the last question which is unproveable is important to address. I personally do not care how the RTP is calculated in a game that has odds of 292+ million to one odds. I obviously cannot speak for the population that plays, but I am pretty confident that they don't care either.

In case you did not notice, the "expected loss" for players is $2/ticket. My guess is that .000001% of players gives a rat's patoot about the RTP calculations.

tuttigym
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(Quote clipped, relevance)

Please assign a priority ranking to your questions in the future and I may reconsider which ones I choose to spend more time on.

The last question is mathematically provable, which was the last step in the long post that you quoted---in case you missed that.

I understand that you may not care about it, which, I assume, is why you did not start a thread asking:

Quote: seitz

What formula could I use to calculate a pay table that meets the following criteria?
1. EV=0 (EV%=100%), player should break even in long run
2. Payout proportional to probability. An event that occurs twice as often should pay half as much

For example: I want to create a pay table for a game where the player rolls 2 dice. If the sum of the dice is 8-12, player wins, 2-7 player loses. Since rolling an 8 is 5 times more likely than a 12, rolling a 12 should pay 5x what rolling an 8 pays. Overall EV should be 0, player should break even.

How would I calculate the pay table for each total, 8-12?

And:

Quote: seitz

Thanks, I appreciate all the help you folks are giving!

My original intent was to compare the Powerball pay table with the type of pay table I suggested. When I struggled to figure out how to calculate my “fair odds” pay table, I thought I’d try something simpler as a way to learn how to do the calculations. Then I’d try applying the same formulas to Powerball.

I’m curious to also try this experiment for some Blackjack side bets or for video poker pay tables.

Though I could understand why you thought my answer was meant for you rather than the person who:

A. Created the thread.

B. Asked the question about 100% EV games.

C. Brought up the Powerball and applying the same methodology to it.

That said, my answer wasn't actually meant for you. Instead, it was meant for the person who was asking interesting questions---which you have so far failed to do.

Mission, you do this to yourself. Sigh.

Quote: unJon

Mission, you do this to yourself. Sigh.
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I know. The next time that I answer a direct question posed by a new forum member, I would do well to remember that there are forum members about who have a categorical disdain for math and will attempt to derail any such thread at any cost.

Quote: Mission146

Quote: ksdjdj

Quote: Mission146

(snip)
Eight: .84

Nine: (21/25) * 1.25 = 1.05

Ten: (21/25) * 1.666666667) = 1.4

Eleven: (21/25) * 2.5 = 2.1

Midnight: (21/25) * 5 = 4.2

Okay, let's go ahead and prove this out:

(21/36) - (.84 * 5/36) - (1.05 * 4/36) - (1.4 * 3/36) - (2.1 * 2/36) - (4.2 * 1/36) = 0
(snip)
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I know you were trying to get a "perfectly proportional payout", but what about the following:
8 = 0.8
9 = 1.1
10 = 1.4
11 = 2.2
12 = 4

Please check figures, but I think I have used your information correctly to get a "different, but also correct" pay table.

Note: I used your figures and formula as a "short-cut" (I probably wouldn't have been able to do this without something like your post, above).

----
Update:
Math not checked, but if this is a "live casino style" operation, you probably want something similar to the pay table^^^ below:

0.75
1
1.5
2.5
4/3.75

^^^ (important): It is very unlikely to be an exactly "0% EV pay table" (it is just an example of a style of one, to make it easier for the dealer to payout).

Note 1: All the above strike-through / edits were made around 1510, Pac Time.
Note 2: "Live dealers " (especially ones used to Blackjack payouts) may find this pay table a bit "easier" (?)
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Very nice! It works and also comes out to zero! How did you do the fine tuning?
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I think it was just a fluke using "trial and error^^^ " for the second one, as it took me 2 attempts to get to "original and wrong one that I posted" and only 1 more attempt to get to the "correct one" .

^^^: I have read somewhere that "trial and error" in general starts at about 0-2 years of age, but I guess it is a bit later for math ? (I only mention this, because I think some of your math was compared to a "6th grader", and mine may be even simpler, in parts at least )

Also, the first one was "a bit easier", because after leaving the "10 = 1.4 " alone, all I had to do was make sure that the "average" was 4.2 / 36 for all the others (see "proof" below).

"Proof": ((0.8 x 5) + (1.1 x 4)) / 2 = 4.2

Note: I stated "proof" like this above, because it is an incomplete proof, IMO (but I think you get the gist of where I am going with it).

Quote: ksdjdj

I think it was just a fluke using "trial and error^^^ " for the second one, as it took me 2 attempts to get to "original and wrong one that I posted" and only 1 more attempt to get to the "correct one" .

^^^: I have read somewhere that "trial and error" in general starts at about 0-2 years of age, but I guess it is a bit later for math ? (I only mention this, because I think some of your math was compared to a "6th grader", and mine may be even simpler, in parts at least )

Also, the first one was "a bit easier", because after leaving the "10 = 1.4 " alone, all I had to do was make sure that the "average" was 4.2 / 36 for all the others (see "proof" below).

"Proof": ((0.8 x 5) + (1.1 x 4)) / 2 = 4.2

Note: I stated "proof" like this above, because it is an incomplete proof, IMO (but I think you get the gist of where I am going with it).

I like it; using 1.4 as sort of a semi-constant was also a really good idea.

I'm also certainly not above using trial-and-error. I've definitely also done it if I had an idea as to my starting points, and also knew what result I wanted to arrive at, to test what would get me there and then try to reverse-engineer to determine a formula that would cause that.

Quote: Mission146

Quote: ksdjdj

I think it was just a fluke using "trial and error^^^ " for the second one, as it took me 2 attempts to get to "original and wrong one that I posted" and only 1 more attempt to get to the "correct one" .

^^^: I have read somewhere that "trial and error" in general starts at about 0-2 years of age, but I guess it is a bit later for math ? (I only mention this, because I think some of your math was compared to a "6th grader", and mine may be even simpler, in parts at least )

Also, the first one was "a bit easier", because after leaving the "10 = 1.4 " alone, all I had to do was make sure that the "average" was 4.2 / 36 for all the others (see "proof" below).

"Proof": ((0.8 x 5) + (1.1 x 4)) / 2 = 4.2

Note: I stated "proof" like this above, because it is an incomplete proof, IMO (but I think you get the gist of where I am going with it).

I like it; using 1.4 as sort of a semi-constant was also a really good idea.

I'm also certainly not above using trial-and-error. I've definitely also done it if I had an idea as to my starting points, and also knew what result I wanted to arrive at, to test what would get me there and then try to reverse-engineer to determine a formula that would cause that.
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Trial and error got me through most of my school career. Then I learned about Newton’s Method, which is really the smartest way to implement trial and error in many circumstances. Worth spending a few minutes going down the Internet rabbithole on it, if you are not familiar with it.

Quote: unJon

Trial and error got me through most of my school career. Then I learned about Newton’s Method, which is really the smartest way to implement trial and error in many circumstances. Worth spending a few minutes going down the Internet rabbithole on it, if you are not familiar with it.
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I know how I'm spending part of this evening. (This is not a joke)

Thanks, UnJon!

Quote: ksdjdj

^^^: I have read somewhere that "trial and error" in general starts at about 0-2 years of age, but I guess it is a bit later for math ? (I only mention this, because I think some of your math was compared to a "6th grader", and mine may be even simpler, in parts at least )

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Please know it is intended as a non-pejorative comparison.
Mission is much better at working with numbers longer than 5 digits than any number of 6th graders I've met.
A lot of it is multiplying and dividing - 6th grade math. There's just like a whole worksheet's worth of it there.

Quote: Dieter

Quote: ksdjdj

^^^: I have read somewhere that "trial and error" in general starts at about 0-2 years of age, but I guess it is a bit later for math ? (I only mention this, because I think some of your math was compared to a "6th grader", and mine may be even simpler, in parts at least )

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Please know it is intended as a non-pejorative comparison.
Mission is much better at working with numbers longer than 5 digits than any number of 6th graders I've met.
A lot of it is multiplying and dividing - 6th grade math. There's just like a whole worksheet's worth of it there.
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And all to solve one problem, which the worksheet didn't even specify an approach for, because the worksheet basically just gave me some initial parameters and said, "Figure out how to do something with this."

I definitely took your meaning, Dieter. As I alluded to earlier in the thread, I could come up with a way to combine everything that I want to do into a single formula, but then other people would look at that and not know how to recreate it for something else.

There are posts on here where I look at the left side of the equal sign and ask myself, "What the hell is that?" I then look at the result, do the problem my way, then say, "Oh, yeah, that's the right answer. I still don't know what XYZ did in this formula, though, but I agree with XYZ."

With that, even when I can do a single (though more complicated) formula, which I freely admit isn't always the case---sometimes the ONLY way I can figure out an answer is by doing as many simple steps as possible---I often choose not to because anyone with the sixth grade mathematical comprehension that you referred to can then copy my method for a different proposition.

For example, suppose that the OP wanted to do this for Mega Millions should that jackpot ever get really high and it occurs to him or her, the OP could then come back to this thread and apply the same step by step thing that I did with Powerball.

Actually, the basic concept/methodology should work with literally any fixed odds game as long as one knows the probabilities.

Quote: Mission146

Quote: unJon

Trial and error got me through most of my school career. Then I learned about Newton’s Method, which is really the smartest way to implement trial and error in many circumstances. Worth spending a few minutes going down the Internet rabbithole on it, if you are not familiar with it.
link to original post

I know how I'm spending part of this evening. (This is not a joke)

Thanks, UnJon!
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Fair warning it’s like trial and error for solving well behaved math equations. Not clear you would use it ever analyzing gambling situations. But I still find it super cool.

Quote: unJon

Quote: Mission146

Quote: unJon

Trial and error got me through most of my school career. Then I learned about Newton’s Method, which is really the smartest way to implement trial and error in many circumstances. Worth spending a few minutes going down the Internet rabbithole on it, if you are not familiar with it.
link to original post

I know how I'm spending part of this evening. (This is not a joke)

Thanks, UnJon!
link to original post

Fair warning it’s like trial and error for solving well behaved math equations. Not clear you would use it ever analyzing gambling situations. But I still find it super cool.
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I appreciate the disclaimer, but I can assure you that I enjoy learning about all manner of things that do not have gambling applications!

Quote: Dieter

Quote: tuttigym

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym
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I hate to break it to you, but this is mostly basic 6th grade math these days if last night's homework help is a reasonable representation.
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And I hate to break it to you but your kids 6th grade math ability does not guarantee or translate to success in the real world. This is precisely why so many here are broke, and depressed when compared to people that possess the traits necessary to actually make money. Information is a commodity you can get the answer to anything in seconds. I’m More impressed with Mdawgs stamina and dedication to his adventures thread

Quote: Seedvalue

Quote: Dieter

Quote: tuttigym

Do you actually type this stuff out on your keyboard, or do you "speak" to your computer which then types out this stuff? How long does it take? How does one prove these numbers are real, and can one perform such will real PB ticket purchases?

tuttigym
link to original post

I hate to break it to you, but this is mostly basic 6th grade math these days if last night's homework help is a reasonable representation.
link to original post

And I hate to break it to you but your kids 6th grade math ability does not guarantee or translate to success in the real world. This is precisely why so many here are broke, and depressed when compared to people that possess the traits necessary to actually make money. Information is a commodity you can get the answer to anything in seconds. I’m More impressed with Mdawgs stamina and dedication to his adventures thread
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Thank goodness. If future success hinges on long division abilities, "You want fries with that?" is going to be the most common conversation in their career.