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.
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.