Powerball Odds and Statistics

[avelworldcreator]

Seeing the latest argument in this thread reminds me of the debate earlier on it about the lottery. I noticed no one distiguished between instantaneous probability (single trial) vs probablity over a number of trials. An example would be using a 4-sided dice (yes, I could have used the more common 6-sider but that forces me to use rounded numbers for decimals).  One roll has only a 1 in 4 chance of a three showing up, or a 3/4 chance of it not showing up - 75%. TWO rolls gives only a 9/16 or only 56.25% of it not showing up (3/4)^2. At THREE trials the odds for it NOT appearing drops to 42.1875% (3/4)^3, and so on.  So the actual odds of winning the lottery by an individual is actually based on the number of times x people play it not just the computed chance of a specific number combination showing. That's why you actually see winners almost every year or so with major wins. The number of people playing actually improves the odds of winning for everyone (increases the number of trials). But what would I know? I just tutor in multivariate calculus and statistics. 

[Arcana]

Seeing the latest argument in this thread reminds me of the debate earlier on it about the lottery. I noticed no one distiguished between instantaneous probability (single trial) vs probablity over a number of trials. An example would be using a 4-sided dice (yes, I could have used the more common 6-sider but that forces me to use rounded numbers for decimals).  One roll has only a 1 in 4 chance of a three showing up, or a 3/4 chance of it not showing up - 75%. TWO rolls gives only a 9/16 or only 56.25% of it not showing up (3/4)^2. At THREE trials the odds for it NOT appearing drops to 42.1875% (3/4)^3, and so on.  So the actual odds of winning the lottery by an individual is actually based on the number of times x people play it not just the computed chance of a specific number combination showing. That's why you actually see winners almost every year or so with major wins. The number of people playing actually improves the odds of winning for everyone (increases the number of trials). But what would I know? I just tutor in multivariate calculus and statistics. 

I don't specifically recalling a discussion point where this fact is significant.  I don't recall (I could be wrong) someone asserting what the odds of someone winning are, which is based on how many entries are put in and what their distribution are.  However, this statement:

The number of people playing actually improves the odds of winning for everyone

is false.  I hope you didn't teach that to anyone.  The odds of *someone* winning at all is based on the number of *different* powerball combinations that have been played.  Since there are 292,201,338 possible combinations, the odds of *someone* winning on a particular draw if N *different* combinations have been entered is N/292201338.  If every single combination has been put in at least once, then the odds are 1.0, or a certainty.

What you're talking about is something different.  Suppose there are N different entries in the Powerball, and the rules stated that the Powerball operators were to continue to draw powerball combinations *until* someone won.  In that case, the odds of the first one generating a winner are N/P (where P = 292... ).  The odds that the first draw does not generate a winner would be 1-N/P.  The odds that D successive draws would not generate a winner are (1-N/P) ^ D, meaning the odds of a series of draws not generating a winner get lower over time; the odds of drawing a winner increase with more draws.

However, that's not how powerball is played.  Hypothetically speaking, if you bought tickets using the computer to randomly pick the numbers, then the odds of you having the winning number assuming you never draw a duplicate number series ever are 1 - (1-N/P)^T where T is the number of tickets.  In effect, given those criteria each random draw is in effect an attempt to hit the winning target randomly, and the odds of never hitting the target follow the same formula.  However, the only way to do that is to randomly select your ticket numbers yourself, rejecting duplicates.  The lottery computers will not do that for you.

The more people that play, the greater the chances of *someone* winning.  But it doesn't increase the chances for everyone to win.  Your individual odds are the same no matter what.  However, how many people play does have an effect on your statistical return per ticket, because more players both increases the size of the jackpot if you win and increases the probability you'll have to share that jackpot if you win.  But unless you coordinate your efforts with those other people, it doesn't affect your personal odds of actually winning at all.

[avelworldcreator]

I don't specifically recalling a discussion point where this fact is significant.  I don't recall (I could be wrong) someone asserting what the odds of someone winning are, which is based on how many entries are put in and what their distribution are.  However, this statement:

The number of people playing actually improves the odds of winning for everyone

is false.  I hope you didn't teach that to anyone.  The odds of *someone* winning at all is based on the number of *different* powerball combinations that have been played.  Since there are 292,201,338 possible combinations, the odds of *someone* winning on a particular draw if N *different* combinations have been entered is N/292201338.  If every single combination has been put in at least once, then the odds are 1.0, or a certainty.

What you're talking about is something different.  Suppose there are N different entries in the Powerball, and the rules stated that the Powerball operators were to continue to draw powerball combinations *until* someone won.  In that case, the odds of the first one generating a winner are N/P (where P = 292... ).  The odds that the first draw does not generate a winner would be 1-N/P.  The odds that D successive draws would not generate a winner are (1-N/P) ^ D, meaning the odds of a series of draws not generating a winner get lower over time; the odds of drawing a winner increase with more draws.

However, that's not how powerball is played.  Hypothetically speaking, if you bought tickets using the computer to randomly pick the numbers, then the odds of you having the winning number assuming you never draw a duplicate number series ever are 1 - (1-N/P)^T where T is the number of tickets.  In effect, given those criteria each random draw is in effect an attempt to hit the winning target randomly, and the odds of never hitting the target follow the same formula.  However, the only way to do that is to randomly select your ticket numbers yourself, rejecting duplicates.  The lottery computers will not do that for you.

The more people that play, the greater the chances of *someone* winning.  But it doesn't increase the chances for everyone to win.  Your individual odds are the same no matter what.  However, how many people play does have an effect on your statistical return per ticket, because more players both increases the size of the jackpot if you win and increases the probability you'll have to share that jackpot if you win.  But unless you coordinate your efforts with those other people, it doesn't affect your personal odds of actually winning at all.

What I meant is that increasing the number of players increases the probability that *someone* will win because that means the number of draws goes up directly. If we distribute the win chance across the number of players then, yes, the odds of improve for everyone. The increase is, admittedly, tiny with numbers this large but it *is* there. When the overall chances of a win go up the individual players chances do not remain static.

If 292,201,338 different combination are entered on a single trial then the odds are certainly 100%! But in practice, even if you had that number people actually playing the probability of 292,201,338 unique values being selected is not 100%. For that to happen you would have to make sure every player's draw is unique and therefore the trials would not be independent.

I simplified the binomial distribution formula. It's what is used when you have multiple, independent trials with two possible outcomes (which the lottery is).
Wikipedia: Binomial Distribution.
N is the total number of trials in question. If your sample size is equal to the number of trials then N=n and the formula reduces to p^k (obviously the 1-p term and the binomial coefficient reduce to 1 when the sample size and the population are the same value - that was my simplification). Then the next question becomes determining the correct value of p. Should it be chosen to be the instantaneous chance of loss or win? Are we looking for the chance of someone winning every time or losing? In practice p is selected to be the larger value. In this specific case it would be 1-1/292201338 or 292201337/292201338. So our formula is (292201337/292201338)^k. Now the next question is over how many trials does this number drop to 50% or less? When do the chances of winning become equal to, or greater than the chances of losing? We are looking for the solution to the formula (292201337/2992201338)^k<=.5 and that can be calculated by taking the log base p of .5 which is (applying the ceiling function to round) 202,538,534. Notice that is awfully close to the number of possible combinations.

[MegaWatt]

oh crap did some one get into a Math fight with Arcana ? TAKE COVER !

[avelworldcreator]

oh crap did some one get into a Math fight with Arcana ? TAKE COVER !

He has a reputation? Huh. It took me a week to learn calculus (including differential equations) before I was 19. Been at it about 30 years. Never quit my studies. It's even one of my hobbies.

[Aggelakis]

He has a reputation? Huh. It took me a week to learn calculus (including differential equations) before I was 19. Been at it about 30 years. Never quit my studies. It's even one of my hobbies.

Arcana is a she, and yes. It's something of a joke 'round City parts.

[avelworldcreator]

Arcana is a she, and yes. It's something of a joke 'round City parts.

Thank you for that correction. Any insult is the unintentional result of my ignorance.

[Joshex]

He has a reputation? Huh. It took me a week to learn calculus (including differential equations) before I was 19. Been at it about 30 years. Never quit my studies. It's even one of my hobbies.

Indeed, She has a reputation, she worked on City of Heroes, and as far as I can tell her position was dealing with the statistical side of things.

Seriously game statistics are no joke, by definition they are calculus, where dynamic changable values are compared and processed against other dynamic changable values. The work of a Statistical engineer is rough, because you have to plan all the numbers carefully, especially Res as it's a percent and you don't want it to be too easily capped, and when res is coming from multiple sources including your powers, others powers and items, you need to carefully consider how much /this/ power should give at max (with and without all enhancements), and then reconsider it based on level and find the lowest base res for the power compared to all the powers and the amounts of damage that will hit on average at the lowest level then compile the lowest res the power can give based on this, however not all res powers are equal so you need to define a high and low range for both the lowest and highest level situations then from there it's a decision as to the distribution based on what kind of powerset it is and what it's hallmarks should be.

and that's if the power isn't a dual effect res/def, and that's if you already have a well defined stat system, if you are starting from scratch and trying to include a whole bunch of stat customization then you have a serious headache.

I hardly think that is the limits of Arcana's ability or expertise either.

[avelworldcreator]

Given I'm one of the math people for CoT I'm familiar with the issues (the math team consists of three people, one guy who also does advanced statistical analysis of map data for a corporation who's clients include national governments, a professor of theoretical physics, and myself. I'm considered the second most skilled person on the subject). In fact I developed a clamping function for certain game stats. I based it on a formula from relativity I've spent a few decades analyzing (I'll try to attach the image here). I'm also internationally published in statistical population modeling with game applications. She tried to rebut one of my statements (and I admit I made my statement a little loosely - I'm not offended) and I corrected her. It just happened to be a specific area of statistical analysis I've spent a couple of decades on. That's not a weakness on her part, in fact she saved me some effort on calculating some values on my return argument. I'd love to have her on our team but we don't have a budget for her (we are all working as volunteers).

Statistics isn't calculus but it is an area where calculus is applied. I suspect that's what you meant.

Oh, and the formula:

[worldweary]

CoT sounds like it's only going to be for smart people.Guess I'm out.

[avelworldcreator]

CoT sounds like it's only going to be for smart people.Guess I'm out.

Nah. It's for people who want to have fun. Of course there are going to be a few "eggheads" around. We're doing the hard work so everyone else can have a good time. In our UI we have both "simple" and "advanced" modes. We are aiming to work with Able Gamers and people with learning disabilities as much as we can without disadvantaging the quality of the experience for everyone else. We are group of volunteers who loved CoH and have spent a lot of our personal time and cash to get it to where it is now and hope to see it be enjoyed and for it last for years to come. It's part of a strategy to get back somehow what we lost with CoH's closing.

[umber]

He has a reputation? Huh. It took me a week to learn calculus (including differential equations) before I was 19. Been at it about 30 years. Never quit my studies. It's even one of my hobbies.

Guess I'm a little surprised of someone working on a spiritual heir to City of Heroes project not knowing one of the prominent CoH community members.  If you don't mind my asking, do you have any experience with CoH?  It would worry me if CoT was trying to recapture the flavor of something the developers hadn't experienced themselves.

Another worry, has nobody from CoT already approached Arcana for at least an extra set of eyeballs looking at design decisions?  Pick her brain as much as she is willing to let you!

[Shibboleth]

CoT sounds like it's only going to be for smart people.Guess I'm out.

Looks like that math degree I didn't get--I got tired of going to school and realized that despite having more hours in upper division math than I had in all of computer science, I was (a) one semester away from the Comp Sci degree (b) three semesters* away from the math degree due to the haphazard way I had taken my classes and (c) three semesters from the dual degree--may get some use.

Never could understand programmers who had an aversion to math...its just way too handy.

*Three semester because the I was three courses short, one of which was a pre-requisite for the other and not offered until the following semester--the school had a horrible habit of offering certain classes only every other year in a single semester.

[avelworldcreator]

Due to some difficult personal circumstances I was away from these forums for an extended period (theft of my equipment,etc. when I'm self-employed and it was a major part of my livelihood - it interrupted my work with CoT as well, but it didn't take much to get back on track when I did). Did I have any experience with CoH? I had transferred my level 50 character that I had created in 2005 to the beta based on a rumor that it would be the last server to close down. My last memories was seeing it kissing the pavement as I saw the the server connection being lost. Remember seeing the "construction" projects for the player arenas. I remember using the map hack to badge hunt and did so in all three "sides". I remember jacking the difficulty level for this character to max (a MA regen scrapper) when I was first learning the game and forgetting I had done it for months. Oddly I still did well doing mostly solo stuff despite having so many of my targets in missions conning red and purple and UV.  I even remember buying extra RAM to play the game better and actually taking a personal loan to do it. I think we have only about two people with us that have never played the game. If you look at my posting history on this forum you will find I wrote our original IP agreement trying to protect the contributions of our volunteers. I also took care of doing the actual initial set up of Missing Worlds Media as a corporation, establishing our domain name,  and creating and managing our original forums. And of course I had multiple characters on several servers at the end. I miss them all. How much experience in game are you expecting or consider "adequate"? I started playing in early 2005 both on the regular and beta servers.

I'd love to have Arcana on the team! We are quite stretched at the moment and every extra hand is welcome. We need artists, writers, and good coders. Currently I've been working on: The costume creator UI, the web page redesign, the chat system, the game loader and user authentication system, trying my hand at mission writing, trying to figure out how to best do combat simulations to tweak settings, creating UI related art assets, assisting our PR person, and looking into corporate level set up issues such as communications services.

[Vee]

The number of people playing actually improves the odds of winning for everyone
What I meant is that increasing the number of players increases the probability that *someone* will win because that means the number of draws goes up directly. If we distribute the win chance across the number of players then, yes, the odds of improve for everyone.

K I'm not a math but why would we distribute the added win chance?

The argument struck me as less about math than about language - i.e. the increase in the odds of somone winning vs the increase in "everyone's" odds, i.e. each person's odds.

[avelworldcreator]

K I'm not a math but why would we distribute the added win chance?

The argument struck me as less about math than about language - i.e. the increase in the odds of somone winning vs the increase in "everyone's" odds, i.e. each person's odds.

"Distribute evenly" is just a fancy way of saying I'm dividing by the number of people and giving them an equal "share". Think about it.  If you had a cake and the winning chance is the portion that has chocolate icing, and the losing part is the part without, if you spread the frosting more (increase the chances of a win) the chance of a random slice getting frosting gets better. Everyone gets their own cake and each one only gets one random slice of the cake. Not only does the chance of frosting on their slice go up, but chances of more than one person getting frosting goes up too.

[Vee]

"Distribute evenly" is just a fancy way of saying I'm dividing by the number of people and giving them an equal "share". Think about it.  If you had a cake and the winning chance is the portion that has chocolate icing, and the losing part is the part without, if you spread the frosting more (increase the chances of a win) the chance of a random slice getting frosting gets better. Everyone gets their own cake and each one only gets one random slice of the cake. Not only does the chance of frosting on their slice go up, but chances of more than one person getting frosting goes up too.

That's a false analogy. Spreading more icing increases my odds because it reduces the number of losing pieces. That would be more akin to the increased odds from having multiple tickets with different number combinations. [Edit: No it wouldn't, it'd be like if they decided once 100 million tickets were sold they'd draw one less number.]

More people playing definitely increases the odds that someone will win, even multiple people. But it does nothing for the odds of each individual number combination being the winning combination. That'd still be one in 600whatever million. Just as if I roll a die the odds of my getting a six aren't affected by other people, even several hundred million of them, rolling dice as well.

[avelworldcreator]

That's a false analogy. Spreading more icing increases my odds because it reduces the number of losing pieces. That would be more akin to the increased odds from having multiple tickets with different number combinations.

More people playing definitely increases the odds that someone will win, even multiple people. But it does nothing for the odds of each individual number combination being the winning combination. That'd still be one in 600whatever million. Just as if I roll a die the odds of my getting a six aren't affected by other people, even several hundred million of them, rolling dice as well.

I admit it wasn't the best overall image. Quite the hack in fact. But it's not a false analogy. The model was to demonstrate the chances of an individual having the winning number if it got picked; not to demonstrate the increase in odds that a particular number would be the one selected.

Try dice tossing for example using a standard 6-sided die. If I only toss a single die the chances of a 1 showing up is 1 in 6 but the odds of it not showing up is 5 in 6. If I double the number of dice that odds of not showing becomes 25 in 36 which means that if you slice the cake into 36 pieces 25 of them lack frosting. At three dice it goes to 125 in 216 meaning our cake now has 125 pieces without frosting and 91 with. At 4 it is 625 in 1296 (yeah, I'm really doing these numbers in my head at 4am!) which comes to 671 WITH frosting and 625 without. Only 4 trials to more than break even. Is a lottery a fair system though? Nope. It has a lot of similarity to a Ponzi scheme but without the lie you will actually be guaranteed to win if you play. If you got people to play, let the money pool for a while, then split the wealth equally among the actual players, then the system would be fair. Of course this would be less the vig (the public use portion of the monies). Can you attract people to invest in such a system? Yes. Let them win an extra portion (say 10%) lottery style. A definite return with a slight chance of even greater return after a period of investment? With community improvements? Would you play? I know I would.

[Vee]

Yeah we all agree that more people playing increases the chance someone will win. It was the part about increasing everyone's odds where the quibble came in. Which is why I said it's more of a language disagreement than a math one . You were using 'everyone' in the aggregate sense rather than to mean everyone who played had a better chance of being the winner. So really not even a disagreement at all, just a different use of terms.

The cake example doesn't work for a lottery though since the lottery has a static number of losing number combinations no matter how many people play. You can add all the icing you want but it always goes on the same piece of cake. The more people who play the more icing there is but there's always only the one piece of cake with icing (setting aside of course non-jackpot wins. But I think those payouts are static and that territory was covered in the original conversation.) And of course the more players the higher the likelihood that the iced piece will need to be divided.

[umber]

How much experience in game are you expecting or consider "adequate"?

I've gotta say, that whole paragraph comes across a little defensive over an honest question.  I'd think just about anyone else participating in this thread would find a dedicated CoH fan not having heard of Arcana almost as odd as, say, not knowing what IOs are.  Its just part of the cultural background noise.

If there was one fault I'd assign to the CoT team its that their customer facing has been a bit rough.  I'm sure they are an excellent team of developers, programmers, and graphics designers but the public relations could use some attention as well.

I'd love to have Arcana on the team!

 
Good to hear it!  Arcana, anyone from the CoT team approach you if for nothing else than a rough consulting session?