Author Topic: Powerball Odds and Statistics  (Read 24960 times)

avelworldcreator

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Re: Powerball Odds and Statistics
« Reply #140 on: January 25, 2016, 09:08:59 AM »
I'm sorry, but this is word salad.  You are asserting that for me to meet your criteria of demonstrating you are in error, I am required to:

a) Show that the number on each ticket is not an independent random selection.
b) That if it IS an independent random selection that the probility of selected values does not follow a binomial distribution.

Going to stop right there.  You want me to prove that the numbers on the tickets are not independent random selections, and if they are the selected values do not follow a binomial distribution.  Those are essentially contradictory things.  Moreover, none of these things is actually necessary to prove you wrong.  None of those statements directly supports the statement "the odds of a ticket having the winning sequence is dependent on the number of people who enter."  You keep talking about the probability of a ticket having a number that matches a binomial distribution of other selections.  I have no idea why.

Really? First the dismissive statement "that's just word salad". That's an evasion.

The only statement there that needed better clarification is "2" (I used numbers, not letters for enumeration). That should be the "probability of the selected values BEING UNIQUE does not follow a binomial distribution"

Show me exactly where I talk about A ticket having a number that matches the binomial distribution of other tickets. I speak of the probability of selected values - a plural; a group - when speaking of any distribution .

I gave a list of possible rebuttals. All you had to do demonstrate - not just make an assertion - how ANY of them were untrue. This is a requirement in any branch of mathematics, theoretical or applied. It distinguishes between the amateur and the professional. Objective demonstration of your claims. A professional does not resort to insults, ad hominems, dismissals of the other party's concerns, questions, or viewpoints, or avoids direct requests to substantiate any claim made. The professional accepts admissions of error by another person and doesn't continue to press the subject. The professional aims to be critical without being accusatory. A professional can confident that if he (or she) is called into court to testify on an area they are considered expert that they will be able to present their conclusions and fully support them but are prepared to admit error if that is discovered when those are examined. A professional is prepared also prepared to do the same in the court of their peers. A professional is able to distinguish between what is subjective opinion, assumptions, and emotional appeals or other logical fallacies and what is objective evidence, argument, and logical analysis.
And this point I have no other choice to conclude you are not a professional in any field. You are a highly skilled amateur who has developed a following based on a few past accomplishments, emotional argument, and esoteric language. You sound good on the surface but when pressed, or you feel challenged, the internal rot begins to leak out. I find you arrogant and needlessly argumentative. You avoid taking responsibility for your claims and you depend on browbeating the other party in order to "win". I've had disputes with other people in this forum but a lot of it has been miscommunication or loose statements with no ill-will intended. I've been caught on mistakes and I've readily owned up to them when I recognized I had been sloppy in my work. I don't fear failure. In fact mistakes I can walk away from, even when I've been seriously injured, are things I've laughed about. I came to these forums almost three years ago to find out if there was a way to rescue the game and the community I had come to love and care about. I've spent almost all my free time and other personal resources to this end in the way that seemed the most viable and I began here; the record still exists in the archives of these forums. I tried to limit the number of personal claims since I most recently showed up. I've admitted to only one superlative though I can admit to more. I've gone through direct assessment of my abilities by demonstrably trained and accomplished professionals who had been proven qualified to do so - something you have failed to do - and have done well. That's all that needs to be said on that topic. The only thing you have actually proven to me? That you are a board troll. I do not feed trolls when I discover them. Goodbye.
Missing World Media primary co-founder, senior developer, UI/UX acting lead, and software toolsmith.

avelworldcreator

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Re: Powerball Odds and Statistics
« Reply #141 on: January 25, 2016, 09:19:39 AM »
Lesson learned: Arcana goes to Vegas with me if I ever go.  :O
My win/loss ratio in casinos is quite good too. I play the slots for fun only and not for money - I play blackjack if I'm going for that; my usual take is 10 times what I started with. I only failed once when I let my emotions cloud my judgement. If I find a broken machine like she did I report it; I consider to do otherwise theft and fraud.
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Vee

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Re: Powerball Odds and Statistics
« Reply #142 on: January 25, 2016, 09:45:15 AM »
A professional does not resort to insults, ad hominems, dismissals of the other party's concerns, questions, or viewpoints, or avoids direct requests to substantiate any claim made.

Good to know.

If I may be permitted some hyperbole (and why wouldn't I? everyone knows hyperbole is the greatest thing ever), I'd liken this whole conversation to a geometry teacher asking a student for the measure of an interior angle in an equilateral triangle, the student responding with several elegant proofs of the pythagorean theorem, the teacher saying that's wrong because the question was about angles in an equilateral triangle, not the sides in a right triangle, and the student then refusing to accept that they're wrong unless the teacher disproves the pythagorean theorem.

avelworldcreator

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Re: Powerball Odds and Statistics
« Reply #143 on: January 25, 2016, 09:58:58 AM »
Good to know.

If I may be permitted some hyperbole (and why wouldn't I? everyone knows hyperbole is the greatest thing ever), I'd liken this whole conversation to a geometry teacher asking a student for the measure of an interior angle in an equilateral triangle, the student responding with several elegant proofs of the pythagorean theorem, the teacher saying that's wrong because the question was about angles in an equilateral triangle, not the sides in a right triangle, and the student then refusing to accept that they're wrong unless the teacher disproves the pythagorean theorem.
I suspect I'm the target of that example but I can take a friendly poke  ;D
I'd happily buy you a beer while you probably poke at me some more.  8)

[Edit: fixed the spelling error "by" when I meant "buy". Darn those homonyms! :o )
« Last Edit: January 25, 2016, 10:05:05 AM by avelworldcreator »
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avelworldcreator

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Re: Powerball Odds and Statistics
« Reply #144 on: January 25, 2016, 10:03:36 AM »
I remember several RPGs from when i was a kid (i want to say some of the final fantasy games even) having a luck stat that wasn't explained in the manual (yeah, i'm old) and didn't seem to change with leveling or be able to be enhanced. And of course I'd always have a vague idea of what it was probably supposed to do by virtue of knowing what the word 'luck' meant but it always kind of bothered me anyway. I think at some point I decided it must be a holdover from pen and paper systems and was just an 'under the hood' randomizer of some sort and just quit worrying about it.

It's usually just a positive modifier to ability scores as well as defense and attack. The name might change to something other than "luck" but the mechanic remains the same. The example that comes readily to me is the Paragon template in the D&D 3.0 Epic Rule Book.
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Codewalker

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Re: Powerball Odds and Statistics
« Reply #145 on: January 25, 2016, 04:00:22 PM »
Each drawing represents more than one contest.

How so? Unless you care about the amount won in the jackpot (which as far as I know no one in this thread has given more than a passing mention to), there is only one contest that is relevant to the person who wants to know their odds of winning it. Can you show that the other independent contests somehow affect the chance of the ticket that I bought matching the random result?

Saying that you draw 1 of R combinations is no more complicated than saying you draw 1 of R integers and it actually avoids confusion and is more accurate.

The integer equivalency is useful because the probability of a single die roll being a certain number is a well known and accepted value that can be taken as an axiom and used as a baseline to get to some common ground.

Assigning a number to each possible combination allows the problem to be reduced to a single die roll. It's well known that it reduces to 1/s and I don't think that's in dispute here.

Good practice but you ducked my complaint of hypocrisy.

I disagree with it, but it's irrelevant to the discussion. Attacking the person making the argument doesn't change the facts or contribute in any meaningful way.

What is printed on the ticket is a simplified number. It is indeed a form of odds. It's the odds of a specific value being drawn. That is all.

It's the number that is most relevant to an individual playing the lottery, and most importantly, it is not affected by how many other people play.

The odds of being a winner is useless -  if every drawing guaranteed a winner. But the odds of all tickets being a loser exists as a strong possibility and affects the final odds.

I still don't see how that affects an individual's chances at all. The odds of all tickets losing don't affect anything other than jackpot rollover. The odds of a specific number (set of numbers, whatever) coming up remain fixed at 1/292201338.

If you have ever played roulette you know there is a some outcomes you can't put your chips on - the house always wins. If you are calculating odds in that game you can't ignore that possibility.  And that game and the lottery are largely equivalent.  The number of tickets sold affect the odds of there being a winner and in no small way.

The probability of matching a specific number on a Roulette wheel is a better analogy than Bingo, so long as you confine the analysis to betting on one number and ignore the payout odds. However, since Roulette has a much smaller number of outcomes, most statistical analyses of it focus on how best to improve your chances of beating the house in a situation where it is feasible to win multiple times during a session. In that case the analysis becomes about how much you win relative to how much you lose, taking payout proportions into account.

As I've been onto Arcana about already, show your work.

I'm not really sure how you expect me to "show my work" for X*Y. On the surface it seems like a rigged game to ask for a proof when there isn't even agreement if the methodology for applying the theorems is sound. Let me gather a little more information about what you're considering as a valid starting point and I'll see what I can do.

Show that the chances of continuing to play the lottery over several games does not increase your chances of winning

I never said it didn't. That it increases your chances should be obvious to anyone.

or does not follow a binomial distribution.

I did mistakenly say earlier that it was a simple n*1/v problem. After giving it a moment's thought, it became obvious that is incorrect -- playing the lottery 292 million separate times does not guarantee a win, as it would if all the tickets were purchased for a single draw. Indeed it follows your (tongue in cheek here) beloved binomial distribution; though only as far along as the number of times you individually play, so it stays exceedingly small.

However, that's orthogonal to the point I was trying to make and to the assertion you made that started this whole thing:

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. 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.

Later I think you may have revised that assertion to say it lowered the chances (forgive me, there are a lot of posts to search through), but unless I'm missing something, the last I heard you were still claiming that more people playing somehow altered an individual's chance of matching the jackpot.

So, before I spend any more time on it, is that what you're saying? Below is what I'm saying, do you agree or disagree with them?

* The odds of one individual ticket matching the winning powerball numbers in a single drawing are 1/292,201,338.

* The odds of one person winning the powerball jackpot in a single drawing increase linearly with the number of unique tickets they purchase (n/292,201,338).

* The odds of one person winning the powerball jackpot in their lifetime increase nonlinearly with the number of drawings they participate in during that lifetime. Using binomial terms, this would be a probability mass function of f(1; n, 3.4223x10^-9), where n is the number of drawings they enter.

* The above odds do not change regardless of how many other people play the powerball. Even in the degenerate case of an infinite number of other players, the odds of my ticket matching the draw are still the same, my jackpot winnings as a result would simply approach zero.

Quote
Powerball Lottery
The part I love is close to the bottom. The "Return To Player Jackpot Size" table. He relates number of tickets sold to the odds of winning. Looks great... until you distribute the winning odds across the number of tickets sold.

Yes, because that table is all about the return, not the odds of winning. Of course the jackpot is split if there is more than one winner. But nobody (except perhaps you?) is talking about the size of the jackpot, since the original statement was just "More people playing improves the odds of winning for everyone".

Finally, it will take a few minutes to find everything, but I'm in the process of splitting this off to a separate thread, as it has veered WAAAAAY off-topic, even moreso than is normal for this thread.

Arcana

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Re: Powerball Odds and Statistics
« Reply #146 on: January 25, 2016, 06:49:51 PM »
Good to know.

If I may be permitted some hyperbole (and why wouldn't I? everyone knows hyperbole is the greatest thing ever), I'd liken this whole conversation to a geometry teacher asking a student for the measure of an interior angle in an equilateral triangle, the student responding with several elegant proofs of the pythagorean theorem, the teacher saying that's wrong because the question was about angles in an equilateral triangle, not the sides in a right triangle, and the student then refusing to accept that they're wrong unless the teacher disproves the pythagorean theorem.

Except this started with, and should end with a simple question, repeated beyond the point of reasonableness: do the odds of a powerball ticket having the winning combination change based on the number of additional tickets entered.  There is only one correct answer to this question, and it is "no."  Someone who says "banana" is talking about something else.  Someone who says "yes, because banana" is just plain wrong.  Also crazy, but wrong regardless.

Arcana

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Re: Powerball Odds and Statistics
« Reply #147 on: January 25, 2016, 06:55:28 PM »
How so? Unless you care about the amount won in the jackpot (which as far as I know no one in this thread has given more than a passing mention to), there is only one contest that is relevant to the person who wants to know their odds of winning it. Can you show that the other independent contests somehow affect the chance of the ticket that I bought matching the random result?

Hypothetically speaking, you can think of the lottery as n separate attempts to win, one for each ticket, except the problem is that those are not independent drawings because of course the powerball winning sequence is the same for every draw.  That lack of independence means that *sometimes* your analysis will work, and *sometimes* it will not.  For example, if you treat the powerball as n independent attempts to win, your calculations will show that for all n there is always a finite chance of having no winner regardless of the distribution of the n entries.  In practice, however, if the set of all entered sequences is covering, then the odds of a winner are 100% and thus the odds of having no winner are zero.

Ironically, the viewpoint that you can treat the powerball as n separate draws *only* works to consider the case of calculating the odds of any single ticket winning.  In that case, that viewpoint will calculate those odds as being 1/N, where N is the number of different possible draws.  Those odds would be independent of all other draws, which is the correct answer.  But when you try to calculate anything else across all entries, you'll generally get the wrong answer.

Still wrong, but in an ironic way.

Arcana

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Re: Powerball Odds and Statistics
« Reply #148 on: January 25, 2016, 07:02:19 PM »
I gave a list of possible rebuttals. All you had to do demonstrate - not just make an assertion - how ANY of them were untrue. This is a requirement in any branch of mathematics, theoretical or applied. It distinguishes between the amateur and the professional. Objective demonstration of your claims. A professional does not resort to insults, ad hominems, dismissals of the other party's concerns, questions, or viewpoints, or avoids direct requests to substantiate any claim made. The professional accepts admissions of error by another person and doesn't continue to press the subject. The professional aims to be critical without being accusatory. A professional can confident that if he (or she) is called into court to testify on an area they are considered expert that they will be able to present their conclusions and fully support them but are prepared to admit error if that is discovered when those are examined. A professional is prepared also prepared to do the same in the court of their peers. A professional is able to distinguish between what is subjective opinion, assumptions, and emotional appeals or other logical fallacies and what is objective evidence, argument, and logical analysis.
And this point I have no other choice to conclude you are not a professional in any field. You are a highly skilled amateur who has developed a following based on a few past accomplishments, emotional argument, and esoteric language. You sound good on the surface but when pressed, or you feel challenged, the internal rot begins to leak out. I find you arrogant and needlessly argumentative. You avoid taking responsibility for your claims and you depend on browbeating the other party in order to "win". I've had disputes with other people in this forum but a lot of it has been miscommunication or loose statements with no ill-will intended. I've been caught on mistakes and I've readily owned up to them when I recognized I had been sloppy in my work. I don't fear failure. In fact mistakes I can walk away from, even when I've been seriously injured, are things I've laughed about. I came to these forums almost three years ago to find out if there was a way to rescue the game and the community I had come to love and care about. I've spent almost all my free time and other personal resources to this end in the way that seemed the most viable and I began here; the record still exists in the archives of these forums. I tried to limit the number of personal claims since I most recently showed up. I've admitted to only one superlative though I can admit to more. I've gone through direct assessment of my abilities by demonstrably trained and accomplished professionals who had been proven qualified to do so - something you have failed to do - and have done well. That's all that needs to be said on that topic. The only thing you have actually proven to me? That you are a board troll. I do not feed trolls when I discover them. Goodbye.

It must be sad, thinking you're the only sane person in the world and simply lacking the ability to demonstrate it.

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Re: Powerball Odds and Statistics
« Reply #149 on: January 25, 2016, 11:12:45 PM »
So, before I spend any more time on it, is that what you're saying? Below is what I'm saying, do you agree or disagree with them?

* The odds of one individual ticket matching the winning powerball numbers in a single drawing are 1/292,201,338.

* The odds of one person winning the powerball jackpot in a single drawing increase linearly with the number of unique tickets they purchase (n/292,201,338).

* The odds of one person winning the powerball jackpot in their lifetime increase nonlinearly with the number of drawings they participate in during that lifetime. Using binomial terms, this would be a probability mass function of f(1; n, 3.4223x10^-9), where n is the number of drawings they enter.

* The above odds do not change regardless of how many other people play the powerball. Even in the degenerate case of an infinite number of other players, the odds of my ticket matching the draw are still the same, my jackpot winnings as a result would simply approach zero.

I'm glad you took a step back and returned to the original claims. I truly hope that Avelworldcreator can cool off and come back to continue these talks because they have lead to some of the most interesting posts in a long time.

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Re: Powerball Odds and Statistics
« Reply #150 on: January 26, 2016, 12:13:11 AM »
Except this started with, and should end with a simple question, repeated beyond the point of reasonableness: do the odds of a powerball ticket having the winning combination change based on the number of additional tickets entered.  There is only one correct answer to this question, and it is "no."  Someone who says "banana" is talking about something else.  Someone who says "yes, because banana" is just plain wrong.  Also crazy, but wrong regardless.

Yeah. Just thought I'd try a quasi parable since neither direct refutation nor analagous examples have worked over 8 pages.

Arcana

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Re: Powerball Odds and Statistics
« Reply #151 on: January 26, 2016, 01:31:30 AM »
I'm glad you took a step back and returned to the original claims. I truly hope that Avelworldcreator can cool off and come back to continue these talks because they have lead to some of the most interesting posts in a long time.

Which ones were the interesting ones?  I know why I decided to jump into the original powerball discussion: numerical analysis is something I do both professionally and casually, so on the official forums you basically couldn't keep me away from that.  But once we determined that no, you probably shouldn't buy powerball tickets unless you have money to burn, the rest drifts off into general probability and statistics.  I didn't think even I could make that interesting.

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Re: Powerball Odds and Statistics
« Reply #152 on: January 26, 2016, 04:36:51 PM »
As near as I can tell (there's a lot to go through), the issue here seems to centre around the question of whether Powerball picks a winning combination from those selected by players.  As I understand it, it does not.  If it did, however, then obviously the number of players would affect your probability of winning--though not in a trivial-to-calculate manner, since it actually depends on the number of unique combinations selected rather than the number of players.  I don't think there's any reasonable way to calculate number of combinations selected as a function of the number of players, except perhaps by statistical analysis if one had access to sufficient data, as there are issues of psychology involved.


Arcana

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Re: Powerball Odds and Statistics
« Reply #153 on: January 26, 2016, 06:49:27 PM »
As near as I can tell (there's a lot to go through), the issue here seems to centre around the question of whether Powerball picks a winning combination from those selected by players.  As I understand it, it does not.  If it did, however, then obviously the number of players would affect your probability of winning--though not in a trivial-to-calculate manner, since it actually depends on the number of unique combinations selected rather than the number of players.  I don't think there's any reasonable way to calculate number of combinations selected as a function of the number of players, except perhaps by statistical analysis if one had access to sufficient data, as there are issues of psychology involved.

It is a possibility, but that's contradicted by the fact that when powerball was replaced with a lottery with explicitly defined rules, the problem persisted.  That suggests the problem is a more general one.

Also, it has been repeated in the thread that *if* powerball were to select the winning combination from the set of all entries, and not the set of all possible draws, then the odds of winning would be one in n, where n was the total number of different sequences entered.  n would be some function of all entries, but the reality of what that value is can't be determined purely from a statistical function because as you say, psychology is involved.  It can be *estimated* by a statistical function because many people elect  to have the computer randomly select their ticket.  Those tickets would be distributed in random fashion which if converted to a histogram and sorted would then have a binomial character.  But the binomial calculation isn't really useful directly, because it calculates the likelihood of numbers being picked multiple times.  It does not directly calculate n, the number of likely different numbers selected.

Actually, this related problem has an analog in game design.  Suppose City of Heroes were to create a set of something, let's say enhancements, and then allowed you to buy them in the Paragon store but only allowed you to buy unopened "packs" of them each containing a random one.  If you wanted the complete set, on average how many of those do you have to buy?

The interesting thing is that this number escalates a bit faster than I think most developers (who generally are not math-proficient) realize**.  Suppose the set of things we're talking about is an enhancement set with five things in it.  The best way I can think of to decompose this problem is to consider step by step what the odds of getting a different one are, and then convert those odds into an expectation of number of purchases, and then sum them.  So the first pack you buy is guaranteed to get you an enhancement you don't have yet, because you don't have any yet.  So you have to buy one pack to get the first one.  Now that you have one, the odds of getting a *different* one on the next buy is 4 in 5, or 4/5.  You could get any random one of the five, but only four of them are ones you want now.  So that means that the statistical average number of packs you have to buy to get unique enhancement number two is 5/4 or 1.2 packs.  Then the next one would be 5/3, then 5/2, and then 5/1.  The sum of all of those is about 11.4.  That's the *average* number of packs you'd have to buy.  If you have a thousand players going after it, the odds are good there will be players that take twice as many, and even three times as many.  Those odds can be calculated.

But if you don't calculate them, as a game designer you could simply guestimate, and guestimate wrong.  You could design a system where out of ten thousand players there are hundreds of them that take so long to get something relative to the average that the cost becomes oppressive.  There are ways to design safety values for that, but those options aren't always used (for example, if you are okay with about 12 packs being necessary but you want to make sure it doesn't take more than 20, you can change the pack design so that the players get something such that twenty of them guarantees they will be able to get all the enhancements; that sets a floor on the worst case scenario.  Allowing players to trade four unwanted cards for a wanted one would be another way to guarantee the floor).

The lottery is actually an extremely simple problem statistically.  Real world design problems have a much higher chance of getting the terms and conditions confused, and then the math misapplied.  That's why statistics is one of the most error-prone math disciplines.  In fact there have been arguments going on for a decade now about how statistics is misused in scientific papers that actually goes back to a problem (or rather a class of them) I first realized and discussed with my college professors thirty years ago.  The problem has been around that long, and presumably smart people have failed to address it for at least that long.  Variations of it even cropped up on the City of Heroes forums.  I would often have to confront players that decided to use statistical computations improperly and try to explain why their calculations were inappropriate, and would usually get a textbook read back to me.  Even when the real world ended up disagreeing with their calculations, they would stubbornly presume the problem somehow had to be with the game and not their calculations.



** I think most developers assume that the number of packs you have to buy in this situation is roughly proportional to the number of things, while it it actually roughly proportional to the number of things times the log of the number of things.  In other words, it increases as n log(n), not just n.  That extra log(n) factor is what makes developer intuition seem reasonably close for small numbers like 5, but get quickly out of whack when it comes to bigger numbers.  Star Trek Online has a collection game like this where the number of things you need to acquire to complete the set is over a hundred, and you can only draw about one thing per day.  I would bet the devs thought it would take something slightly longer than a hundred days to collect them, when it will actually take over 500.

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Re: Powerball Odds and Statistics
« Reply #154 on: January 26, 2016, 08:57:54 PM »
Well, I posited that point of confusion as an explanation as it seemed the most likely explanation for the ongoing discussion.  It would also explain why, for example, the issue of winning the lottery being the matching of two random variables (the combination on the ticket and the winning combination) was raised as being relevant, when it actually has no effect on the probability of a win (since the probability of the winning combination matching the one on the ticket is constant across all possible combinations).  I believe you're right in saying this has been specifically addressed, but after a while it's hard to remember.  :)

Though this does make me curious as to density of particular combinations selected.  I would guess, for example, that the numbers up to 12 (and to a lesser extent up to 31) are over-represented as people play dates as part of their selection.

Arcana

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Re: Powerball Odds and Statistics
« Reply #155 on: January 26, 2016, 11:22:19 PM »
Though this does make me curious as to density of particular combinations selected.  I would guess, for example, that the numbers up to 12 (and to a lesser extent up to 31) are over-represented as people play dates as part of their selection.

Maybe, maybe not.  Some people pick dates.  Some people have lucky digits, and pick numbers ending in 7 (including 7) or 8 (if you're Asian).  Some low numbers might be avoided more than higher ones, like 13.  Even if you pick dates, some people use the year as part of the date, so numbers from 32 to 69 would still be in play, particularly at the higher end of the range.  And of course if you're picking numbers at all as opposed to letting the computer pick a random sequence, you might be thinking about deliberately trying to pick uncommon numbers to avoid what you believe the common numbers are. 

The winning sequence for the recent mega jackpot powerball was 4, 8, 19, 27, 34 and powerball 10.  That's an overrepresented set of low numbers but you had to have 34 in there as well.  The jackpot increased by about $600-$700 million which suggests about 900 million tickets purchased, approximately.  There were 26,110,646 winners total and you'd expect to see about that many winners out of about 600 million tickets sold (there's about a one in 25 chance of winning something in general).  There were three jackpot winners which is what you'd expect out of about 876 million tickets entered.  That's relatively good agreement between the numbers given the very rough nature of these estimates, although it is possible that the discrepancy between the monetary estimate and the winning frequency estimate suggests the winning numbers drawn were statistically less likely to be selected by players than average.  I'd be careful about concluding that from these very rough numbers though.