Need Help with Very Large Numbers

[Beady]

I've run across this a few times in my reading, and I'm not sure I understand the concept, or how it should be applied.

As I presently understand it, the concept of Very Large Numbers pretty much says that, with a large enough population, nothing is remarkable. The example I've seen used is that, with a large enough population, it's a certainty that at least one individual in that population will win the lottery more than once. Therefore, if someone in your community wins the lottery more than once, it's no big deal.

Somehow, this just doesn't seem right. If I won the lottery even once, it would be a *damn* big deal. In my world, the certainty that "someone" will win is really irrelevant, since the possibility that I will be affected in any way at all if anyone but myself, a relative or a very close friend wins is pretty negligible.

Seems to me, then, that people who cite Very Large Numbers when discussing mundane and local affairs aren't really contributing anything germane.

So, where am I wrong?

[DaveW]

Because you are reducing it to a small population again (you and your family).

[rppa]

One problem is the vagueness of the phrase "a big deal".

The other is the vagueness of the concept "THE lottery" (I'll
explain in a moment).

Yes, it's a big deal if someone in your neighborhood
wins the lottery. But you aren't surprised if you read in
the paper that somebody somewhere won it.

You would be surprised if you read in the paper that
somebody won their second lottery, or won with
1-2-3-4-5-6. (That's no less likely than any other
combination, but the chances of any PARTICULAR
combination hitting are the same as any particular
person). Those would strike you as "a big deal".

Let's say that a typical lottery is played by 10^7 people,
and that there are 100 lotteries per day played by a
total of 10^9 people. In 100 lotteries per day, a 1 in a million
outcome like 1-2-3-4-5-6 is a big deal.

How about if the population is 10^15 instead of
10^9? Is it a big deal now? It depends. If by
"THE lottery" you mean something that only occurs
a few times per day, winning chances are now
1/10^13 instead of 1/10^7, then it would be an even bigger
deal for somebody to win two.

On the other hand, if there was a separate lottery
for every 10 million people (10^7), so there were
100 million lotteries per day instead of 100, then
every day you would expect to read cases of somebody
winning with 1-2-3-4-5-6, or somebody winning their
second lottery.

Does that help? Or does it make the confusion worse?

[pupdog]

It's like distinguishing between something that is "statistically significant" versus something that is "significant" emotionally, financially, or in some other way to some individual. It's a matter of a definition specified by some field versus a common definition.

[Beady]

Thanks for the replies, gang, but it's not helping.

The way I see it, people tend to think and live both locally and in the short term. It's just not natural to think of the Big Picture or the Long Term. Should I buy my lottery tickets based on Very Big Number theory?

Thinking back, what got me going on this was an article in The Skeptical Inquirer (I think), where the author said something to the effect that you shouldn't be surprised when someone wins "a" lottery two or even three times, since it's bound to happen given a big enough population. That struck me as just plain wrong; it seems to trivialize the significance of the event to no particular purpose. Sort of like saying, "You won a million bucks? Big deal! Someon was bound to, and it just happened to be you."

What's the point? I truly don't get it.

[T'ai Chi]

Originally posted by Beady
Thinking back, what got me going on this was an article in The Skeptical Inquirer (I think), where the author said something to the effect that you shouldn't be surprised when someone wins "a" lottery two or even three times, since it's bound to happen given a big enough population. That struck me as just plain wrong; it seems to trivialize the significance of the event to no particular purpose. Sort of like saying, "You won a million bucks? Big deal! Someon was bound to, and it just happened to be you."


If an author said that it is no big deal for the same person to win a lottery multiple times, I'd question that. If they said that the probability involved is very very small, but yes, rare events do occur', I wouldn't question that.

I'd be interested in seeing data on the number of multiple lottery winners by country.

[drkitten]

Originally posted by Beady

The way I see it, people tend to think and live both locally and in the short term. It's just not natural to think of the Big Picture or the Long Term. Should I buy my lottery tickets based on Very Big Number theory?


Well,.... I would say "yes, you should." The lottery agents do.

The lottery is, in general, a money-losing proposition. If you listen to the ads for the lottery, they're always careful to stress how the lottery funds various social programs (I think around here, it's care for the elderly). However, that money doesn't come out of thin air. It comes out of the pockets of lottery players like you and me, people who have bought lottery tickets for a dollar and on average, get only about 75 centts back. On average, every time you buy a ticket, you're making a charity contribution of about twenty-five cents, and you don't even get a tax break for doing it.

If, knowing this, you're still comfortable playing the lottery, fine.


Thinking back, what got me going on this was an article in The Skeptical Inquirer (I think), where the author said something to the effect that you shouldn't be surprised when someone wins "a" lottery two or even three times, since it's bound to happen given a big enough population. That struck me as just plain wrong; it seems to trivialize the significance of the event to no particular purpose. Sort of like saying, "You won a million bucks? Big deal! Someon was bound to, and it just happened to be you."


I don't understand what you mean by "trivialize" here. I didn't win a million dollars, and I suspect neither did you. Someone that you never heard of until now may have won a million dollars. So what? What actual significance does the fact that someone neither of us knows has suddenly become wildly rich have to anyone other than them and their immediate circle of acquaintances? I'd love to be able to trivialize this event, if it had any significance in the first place.....

Let me turn this around. What is the conclusion you would like to draw from the fact that a multiple lottery winner exists? What change does that create in your life? What new aspect of reality has suddenly become clear to you? What new fact, previously hidden, has suddenly become apparent?

[Beady]

Originally posted by new drkitten
Someone that you never heard of until now may have won a million dollars...What change does that create in your life? What new aspect of reality has suddenly become clear to you? What new fact, previously hidden, has suddenly become apparent?

First, I do, in fact, know a multiple winner (he owns the local non-chain pizza emporium), which is perhaps why the idea has been nagging me.

Second, as to how it changes my life, that really is the question, isn't it? Does knowing a multiple winner lessen my own chances? After all that's a pretty small population, just him and me, in which to find a multiple winner.

[drkitten]

Originally posted by Beady
First, I do, in fact, know a multiple winner (he owns the local non-chain pizza emporium), which is perhaps why the idea has been nagging me.

Second, as to how it changes my life, that really is the question, isn't it? Does knowing a multiple winner lessen my own chances? After all that's a pretty small population, just him and me, in which to find a multiple winner.

No, knowing a multiple winner doesn't lessen your own chances (assuming that the lotteries are run fairly, et cetera, et cetera). Each drawing is independent and the chances of any given set of numbers being drawn is the same. As the old cliche goes, "Lady Luck has no memory at all."

Part of the problem is that you're not looking at the "small population" properly. It's not really "just him and me" --- it only seems that way in retrospect. How many other people do you know? Unless you're marooned on a desert island with just the two of you -- in which case, who buys the pizzas? -- then there are lot of other people around in your life, and the only reason you're noticing him in particular is because he's a multiple lottery winner. So the chances of you actually knowing a multiple lottery winner are pretty amazingly low.... assume that there are two such in the United States, and each of them "knows" perhaps 300 people. but that still means that nearly two people out of every million in the United States know a multiple lottery winner personally. You "just happen" to be one of the six hundred or so.

Frankly, the number of people who know a multiple lottery winner personally is probably larger than my college graduate class. But no one is impressed by the astonishing coincidence that I graduated in that particular spring from Whatsamatta U. That's the real lesson from the Law of Large Numbers. Rare events happen -- not often (by definition), but also not never. So finding out that a rare event happens shouldn't, by itself be a life-altering event.

[Ashles]

The numbers are irrelevant.

I had a friend at work who guaranteed me that 1-2-3-4-5-6 would not come up the following week.

I asked him to pick some "Random" numbers and guaranteed that they would not come up either.

We were both, of course, correct.

Our brains are hardwired to see patterns where sometimes there are none.

It's like in casinos where they show the previous winning roulette numbers. They are irrelevant to the next spin. If you see two eights in a row on the list, that is no reason not to put it on eight again - the ball has no memory.

If you toss a coin and it comes up heads, can you guarantee the next time it will come up tails?

Each event in lotteries is irrelevant to the others.

It feels like small odds, because it IS small odds. But people do win it, just like people get struck by lightning.

[Jyera]

Beady

Based on what you described, I think you are right.
That the author DID "trivialize the significance of the event to no particular purpose."

I felt you already have the answer to your own question.
That is ... "no particular purpose"

The author is just saying to the effect of ...
"It is just impersonal mathematics and statistics..."
He is NOT wrong to take a mathematical and statistical perspective. He has every right to ignore the "purpose", as part of his position to view it purely by mathematics(statistics).

But by doing so I he is excluding and missing many other
practical and real life factors.

Here's some other significant real life factors.
(1) Most of the time, the purpose of buying lottery is to win.
(2) Lottery is a losing game.
(3) Lottery is personal and emotional.
(4) If you don't ever buy lottery, don't be surprise that others get to win twice, but you don't even win once.

But it is also natural and true that, significant winning in lottery can only be self-sustainable in a very large population.

In a large city of 3 mill people. 3 mill people can fork out 1 dollar each without grieve , to make a happy $1 mill winner.

If there are only 3 people in a closed town, and they decided to set up a self sustainable lottery. It'll never work with a significant first prize of $1mill.. A person's winning is another person's disaster. So very large numbers matters.

With globalisation and internet lottery. The number of contributor increases. The only catch is ... the system is always designed for the lottery operator to be the ultimate winner.

[Number Six]

Here's the way I explain all this...

In a large population it is likely that _someone_ will win the lottery and so when someone wins almost no one is surprised. But the chance that any single person will win is small, so if you win or if you know someone that wins you're surprised. But it's not really an amazing event because as long as _someone_ wins (which will probably happen), _someone_ will feel surprise at being the person that won. If we're almost sure someone in the future will think a particular amazing thing has happened to them then can we truly say the particular thing is amazing when it finally does happens? No, because amazing things are unusual whereas we knew before the lottery that it was likely someone would win.

As far as happening to know someone that won two lotteries that kinda falls into a different categories. There are lots of "amazing" people you could know. You could know the guy that won two lotteries. You could know the guy that ended up taking a gun to work and shooting people. You could know the guy that happened to get struck by lightning. Etc. The number of people that you _could_ know that had an amazing thing happen to them is large and so that chances that you'll know one aren't that large.

And it goes beyond that...that just talks about knowing people that had amazing things happen to them. But also, an amazing thing could happen to you. Or, the plane crash could have happened a block away form you, where you didn't know anyone but still, it was just a block away, which seems amazing. Or, the new rock sensation could have happened to go to your high school, although you don't know her. Or, the famous serial killer could just have happened to have the same name as your boss. And on and on and on.

Millions of possibilities are constantly occurring without you being aware of them. And then, when one of the strange things happen you notice it and go "Wow, how strange." But you're calling it strange _after_ the fact. You don't notice all the non-strange things but you notice the strange thing. You're unconsciously ignoring the misses and noticing the hits. It's data mining.

[pupdog]

Does knowing a multiple winner lessen my own chances?
It depends on how the question is put (what are you really asking?) If you know 10 people, "your odds of winning the lottery" are better than "your odds of winning the lottery, given one of these 10 other people will win" and, I reckon, they should be they same as "your odds of winning the lottery, given one of these 10 other people has won (if the question is posed after that person has, in fact, won)." For a multiple winner, the structure of the question is similar--if asked after the fact that an acquaintance won, when p = 1, or contingent on one of those people winning twice (has not yet won).