Woman wins millions in lottery 4 times!

[alanschu]

OK, what are your chances of winning the next lottery? Not very good, right? What are the chances of someone in the world winning the lottery? People win somewhere every day, right?

 

 

The chances of some person winning the lottery are indeed higher. However, for some person to win the lottery, a specific person must win it.

 

 

If you take a million people, and randomly choose one of those people to receive a million dollars, the probability that someone wins a million dollars is 1. But the probability of any individual winning a million dollars is all the same: 1 in a million.

 

I think this is where you started to gravitate towards when you started to discuss smaller populations when determining multiple lottery winners. It is true that you must have had someone win one lottery in order to win a second. So the probability that some person wins their second lottery with the next lottery event is 0 if no one has yet won a lottery. If one person has won the lottery, then the probability that some person wins their first lottery is 999,999/1,000,000 and the probability that some person wins their second lottery is 1/1,000,000. However, the probability for any individual to win the next lottery is 1 in a million, regardless of whether or not they have already won a lottery. This is because each lottery event is an independent event. The results of the previous lottery wins will not affect the chances of any of those 1,000,000 individuals winning the next lottery. They will all still be 1 in a million.

 

 

And to be technical, the chances of SOMEBODY winning are 1.

 

If there is no chance of a non-winning ticket to be drawn (like in a 50-50 draw), then yes the probability of some person winning the lottery is 1. Most lotteries do not play this way though.

Edited July 22, 2010 by alanschu

[Wrath of Dagon]

Yes, he did use p in the formula instead of a number, I didn't pay attention to that. He should be able to assume a probability of the lottery (it's usually twice the jackpot she won) and replace 100 with the total amount of money she spent playing the lottery, but we can only guess what that is.

 

nvm, that's what he did anyway, I didn't quite realize he was just changing p in my original reply.

 

Edit: You still don't seem to understand that what's relevant is someone in the pool of possible winners winning, not an individual person winning. And individual odds are not 1 in a million, for example if each spends $100,000 playing, the odds go from 1 in a million to 1 in 10.

Edited July 22, 2010 by Wrath of Dagon

[Calax]

Yes, he did use p in the formula instead of a number, I didn't pay attention to that. He should be able to assume a probability of the lottery (it's usually twice the jackpot she won) and replace 100 with the total amount of money she spent playing the lottery, but we can only guess what that is.

 

nvm, that's what he did anyway, I didn't quite realize he was just changing p in my original reply.

 

Edit: You still don't seem to understand that what's relevant is someone in the pool of possible winners winning, not an individual person winning. And individual odds are not 1 in a million, for example if each spends $100,000 playing, the odds go from 1 in a million to 1 in 10.

So basically you're saying that somebody will win four times but the chances of picking that winner are to high? Why does us picking somebody matter at all?

[alanschu]

Yes, it's just as likely to happen on the 1st as on the millionth, but it's hugely more unlikely to happen on one specific one (1, 2, 1000000) than it is on any of the million attempts.

 

This is true, but it still does occur on a specific trial.

 

What you described her "just as likely to happen on the 1st as on the millionth" is essentially a Poisson-distribution.

 

If over the interval of a million trials, the lambda for this poisson distribution is going to be 1. This means that we reasonably expect the event to happen once in a million trials (which makes sense because E[X] where X is a Poisson random variable is lambda). So at 0 trials, we expect the event to happen once from trial 1 to trial 1 million.

 

If we do 300,000 trials and the event doesn't occur, the distribution doesn't occur, we cannot say that we expect the event to happen once in the next 700,000 trials because it failed the past 300,000. Our estimate after 300,000 trials will have to remain that we expect it to happen once in the next million trials (300,001 to 1,300,000).

 

In other words, the Poisson process has no memory.

 

 

The probability that the event happens on the very next trial from where you are (regardless of what trial you're on) is always going to be very low. But there must be a trial that immediately precedes the successful event.

 

In fact, the probability that something that happens "once in a million years" actually occurs only once over that interval is e^-1 = 0.36. It's more likely to NOT occur precisely once during that interval.

Edited July 22, 2010 by alanschu

[Walsingham]

Yes, but what I was saying if an event has a 1 in a million chance for all practical purposes it will not occur in a single trial. However, it will almost certainly occur in millions of trials. I understand some people believe it can still occur in a single trial, but I don't, and this thread illustrates that it's useless to argue about.

 

It shouldn't make any difference, but I can give an example. I was playing a friendly, not for money evening of casino games (on professional equipment), and I played roulette. I played single numbers and I won 7 times, lost once, and won another two times. I don't know the odds of that offhand, but it's bloody 'weird'. However, the odds on each attempt to win were identical.

[alanschu]

Assuming you were playing with a European Roulette Table:

 

Your odds would have been (1/37)^7 x (36/37) x (1/37)^2 to get the specific order your mentioned. (about 7.5e-15)

 

The odds of you winning 9 times out of 10 attempts would be 10 times more likely (though still quite small)

 

10!

---- x (1/37)^9 * (36/37)

9!

[Wrath of Dagon]

In other words, the Poisson process has no memory.

I never claimed it does, you keep arguing against something you think I'm saying instead of what I'm actually saying.

 

As far as Wals's sequence, those would only be the odds he first predicted that specific sequence, getting some sequence means nothing since you'll always get some sequence. The only way this wouldn't be true if you pulled some pattern that's much rarer, like your example of a 100 Aces from 100 decks of cards. Any sequence of only one value would be much more rare than a sequence of mixed values, thus the odds that you'd pull all the same value instead of mixed would be appropriately lower.

 

Edit: Actually I take that back. If I understand you correctly and you actually got the same number 7 times in a row, those are indeed incredible odds. I would have to assume that either the roulette was fixed, or everyone was too drunk to know the difference.

Edited July 23, 2010 by Wrath of Dagon

[Oblarg]

In other words, the Poisson process has no memory.

I never claimed it does, you keep arguing against something you think I'm saying instead of what I'm actually saying.

 

As far as Wals's sequence, those would only be the odds he first predicted that specific sequence, getting some sequence means nothing since you'll always get some sequence. The only way this wouldn't be true if you pulled some pattern that's much rarer, like your example of a 100 Aces from 100 decks of cards. Any sequence of only one value would be much more rare than a sequence of mixed values, thus the odds that you'd pull all the same value instead of mixed would be appropriately lower.

 

Edit: Actually I take that back. If I understand you correctly and you actually got the same number 7 times in a row, those are indeed incredible odds. I would have to assume that either the roulette was fixed, or everyone was too drunk to know the difference.

 

Dagon, claiming something is too unlikely to happen in one trial is faulty logic - if you believe that, you must believe that it is also impossible in any number of trials because you can break any number of trials into a sequence of single trials, thus leading to a contradiction.

 

In short, anything with a nonzero probability can indeed happen in one trial. It may be very unlikely, but it is possible.

[alanschu]

In other words, the Poisson process has no memory.

I never claimed it does, you keep arguing against something you think I'm saying instead of what I'm actually saying.

 

 

Dagon, the fact that you believe a single trial is impossible gives a very strong indication that you do not believe that the Poisson distribution has no memory. As Oblarg states, you can break any sequence of events into a sequence of single trials.

 

 

I won't dispute that it's much more likely for a "one in a million" event to happen over several hundred thousand trials as opposed to the next trial. However, if the event occurs at time interval T, then the event MUST have happened immediately after the time interval T - 1. Since the Poisson process does not have a memory, this means that it happened on the next trial. Since the Poisson process does not have a memory, the probability of any event happening on the very next trial is going to be the same, regardless of which trial you are actually on.

 

 

To put it more mathematically:

 

For any time interval T, the probability of an event succeeding in the next time interval, T + 1, of a Poisson process is the same regardless of the value of T, where T goes from 0 to infinity.

Edited July 23, 2010 by alanschu

[Calax]

Dagon, maybe the reason we think you're saying something else is because you don't make yourself clear and are only half sure of things.

[Hurlshort]

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

[Istima Loke]

In other words, the Poisson process has no memory.

I never claimed it does, you keep arguing against something you think I'm saying instead of what I'm actually saying.

 

As far as Wals's sequence, those would only be the odds he first predicted that specific sequence, getting some sequence means nothing since you'll always get some sequence. The only way this wouldn't be true if you pulled some pattern that's much rarer, like your example of a 100 Aces from 100 decks of cards. Any sequence of only one value would be much more rare than a sequence of mixed values, thus the odds that you'd pull all the same value instead of mixed would be appropriately lower.

 

Edit: Actually I take that back. If I understand you correctly and you actually got the same number 7 times in a row, those are indeed incredible odds. I would have to assume that either the roulette was fixed, or everyone was too drunk to know the difference.

 

This is what you don't understand: The significance of the aces or generally the significance of any pattern you see in a bunch of cards is completely unrelated to probabilities and is a construction of the human mind. The human mind sees patters. It, like, feeds on them. When you say: "Oh my god I got 12 aces out of 4 decks. The chance of this happening is one in a trillion!" this is your mind seeing a pattern. It sees order and finds it incredible. The chance of getting 12 aces out of 4 decks is exactly the same as getting any other combination. The moment you set 4 decks of cards down and draw four cards from each deck is the moment that will very certainly reveal a combination that has a one in a trillion chance of happening (actually (1/(52*51*50*49))^4 which is about 6 /10000000000000000000000000000). There is no such thing as a "rarer pattern". This is something your mind sees. It loads significance to something that from a probabilistic viewpoint doesn't have any. Probabilities don't care if the card shows number one or number 2 or a queen. And so, the moment you draw the cards, is the moment your so called miracle happens. And I'm not discussing here. I'm just telling you how probabilities work.

[Pidesco]

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

 

 

He's Texan, but you don't have to be mean about it...

[Wrath of Dagon]

This is what you don't understand: The significance of the aces or generally the significance of any pattern you see in a bunch of cards is completely unrelated to probabilities and is a construction of the human mind. The human mind sees patters. It, like, feeds on them. When you say: "Oh my god I got 12 aces out of 4 decks. The chance of this happening is one in a trillion!" this is your mind seeing a pattern. It sees order and finds it incredible. The chance of getting 12 aces out of 4 decks is exactly the same as getting any other combination. The moment you set 4 decks of cards down and draw four cards from each deck is the moment that will very certainly reveal a combination that has a one in a trillion chance of happening (actually (1/(52*51*50*49))^4 which is about 6 /10000000000000000000000000000). There is no such thing as a "rarer pattern". This is something your mind sees. It loads significance to something that from a probabilistic viewpoint doesn't have any. Probabilities don't care if the card shows number one or number 2 or a queen. And so, the moment you draw the cards, is the moment your so called miracle happens. And I'm not discussing here. I'm just telling you how probabilities work.

No, if you can define a set of sequences, you can calculate the probability of the set instead of an individual sequence. Thus a set of all possible sequences is much larger than of the set of sequences with the same rank repeating, thus you're much less likely to see something from that set than from the universal set.

 

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

That's funny, you yourself admitted you don't know much about math, yet feel entitled to critisize both my understanding of math and of English.

 

Dagon, claiming something is too unlikely to happen in one trial is faulty logic - if you believe that, you must believe that it is also impossible in any number of trials because you can break any number of trials into a sequence of single trials, thus leading to a contradiction.

 

In short, anything with a nonzero probability can indeed happen in one trial. It may be very unlikely, but it is possible.

I'm not claiming the probability is actually 0, I'm saying it's practically 0 for a single trial. For a million trials it is indeed significant. What's theoretically possible isn't necessarily practically possible, but we just keep going around in circles on this.

Edited July 23, 2010 by Wrath of Dagon

[Hurlshort]

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

That's funny, you yourself admitted you don't know much about math, yet feel entitled to critisize both my understanding of math and of English.

 

See, that isn't actually a criticism. It is a simple statement, one which you decided not to refute. My math skills may be weak, but I am a confident English teacher, and your writing has flaws. It could be a number of reasons, but being an English language learner is nothing to be ashamed of.

 

I also recall something about you being from another country and becoming a US citizen after birth, but I could easily be mistaking you for another poster.

[Wrath of Dagon]

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

That's funny, you yourself admitted you don't know much about math, yet feel entitled to critisize both my understanding of math and of English.

 

See, that isn't actually a criticism. It is a simple statement, one which you decided not to refute. My math skills may be weak, but I am a confident English teacher, and your writing has flaws. It could be a number of reasons, but being an English language learner is nothing to be ashamed of.

 

I also recall something about you being from another country and becoming a US citizen after birth, but I could easily be mistaking you for another poster.

No, you're not mistaken, but why don't you point out the flaws in my writing instead of critisizing my math, which you admit you don't understand?

[Hurlshort]

I haven't said anything about your math skills. I was responding to Calax, who claimed you were being unclear and not explaining things fully. My theory is that something is being lost in translation, hence the 400 post thread that has gone nowhere.

[Wrath of Dagon]

OK, but you're assuming that it's I who's misunderstanding and not Calax, which you really can't judge unless you understand the issues involved. The real reason the posts have gone nowhere is there's a fundamental disagreement on what very small probabilities mean in practice, and this issue doesn't seem like it will get resolved here.

[Serrano]

Again welcome to the internet.

[alanschu]

As far as Wals's sequence, those would only be the odds he first predicted that specific sequence, getting some sequence means nothing since you'll always get some sequence. The only way this wouldn't be true if you pulled some pattern that's much rarer, like your example of a 100 Aces from 100 decks of cards.

 

I think something is being lost in understanding somehow.

 

The odds of him actually getting the sequence that he did is not made irrelevant because he didn't predict it in advance. The probability that he got the sequence that he did is still the same.

 

If I have 5 coin flips, the probability that I get H, H, T, H, T is exactly the same as the probability I get T, T, T, T, T.

 

Yes the probability of getting some sequence is 1. But I can make that assertion at the beginning or at the end. This doesn't not refute that drawing 100 Aces from 100 decks of cards is the same probability of drawing any unique ordered sequence of cards. The probability of you drawing a unique sequence is 1, but the unique sequence that you get is still just as rare as getting 100 Aces.

 

 

Any sequence of only one value would be much more rare than a sequence of mixed values, thus the odds that you'd pull all the same value instead of mixed would be appropriately lower.

 

Of course it is, because if you're going to use any non-Ace value, the probability of drawing those cards is 51/52 instead of 1/52. This isn't what the example illustrated though.

 

I have a deck of cards beside my computer right now. I just drew 5 cards from it in this order:

Jack of Clubs

3 of Hearts

4 of Spades

7 of Spades

King of Diamonds

 

So yes, this is some sequence, which I would have gotten regardless. But the probability that I would get this specific sequence is still 1 / ( 52 * 51 * 50 * 49) or 1 / 6 497 400

 

While you're quick to accept that this is indeed some sequence of cards, you seem to only think that the probability of this event happening matters if and only if it was predicted to happen before the event occurred. This is not the case. The probability of my drawing that sequence of cards is still the same. There is a 1 / 6 497 400 chance that I would have been correct if I had predicted a sequence of cards. We all seem to agree on this, except that you seem to feel that because I didn't explicitly predict it before hand, I have not actually drawn a unique sequence of cards, but just some sequence of cards.

 

I'm saying I did both (since a unique sequence of cards is a subset of some sequence of cards). This still demonstrates that an event that is particularly rare still occurred. In fact, this event is so rare that I suspect it would take me a significant amount of time to draw these same 5 cards in this order again. Yet, this sequence still did come up during my drawing of cards.

Edited July 23, 2010 by alanschu

[Wrath of Dagon]

See my edit in the post you're quoting, I took most of that back already. I still want Wals to verify we understood him correctly, because indeed what he seems to be recounting would be like someone buying a single ticket and winning the lottery, then buying another ticket and winning again. As far as the aces example, I stand by what I said about a certain set of sequences being far more likely than another set.

 

While you're quick to accept that this is indeed some sequence of cards, you seem to only think that the probability of this event happening matters if and only if it was predicted to happen before the event occurred. This is not the case. The probability of my drawing that sequence of cards is still the same. There is a 1 / 6 497 400 chance that I would have been correct if I had predicted a sequence of cards. We all seem to agree on this, except that you seem to feel that because I didn't explicitly predict it before hand, I have not actually drawn a unique sequence of cards, but just some sequence of cards.

Exactly, that's the crux of the matter. The event that happened is that you drew a sequence of cards, and the probability of that event is 1, because you have to draw some sequence. Had you predicted that exact sequence, then the event would be you drew a specific sequence, with the huge odds. Edited July 23, 2010 by Wrath of Dagon

[Oblarg]

This is what you don't understand: The significance of the aces or generally the significance of any pattern you see in a bunch of cards is completely unrelated to probabilities and is a construction of the human mind. The human mind sees patters. It, like, feeds on them. When you say: "Oh my god I got 12 aces out of 4 decks. The chance of this happening is one in a trillion!" this is your mind seeing a pattern. It sees order and finds it incredible. The chance of getting 12 aces out of 4 decks is exactly the same as getting any other combination. The moment you set 4 decks of cards down and draw four cards from each deck is the moment that will very certainly reveal a combination that has a one in a trillion chance of happening (actually (1/(52*51*50*49))^4 which is about 6 /10000000000000000000000000000). There is no such thing as a "rarer pattern". This is something your mind sees. It loads significance to something that from a probabilistic viewpoint doesn't have any. Probabilities don't care if the card shows number one or number 2 or a queen. And so, the moment you draw the cards, is the moment your so called miracle happens. And I'm not discussing here. I'm just telling you how probabilities work.

No, if you can define a set of sequences, you can calculate the probability of the set instead of an individual sequence. Thus a set of all possible sequences is much larger than of the set of sequences with the same rank repeating, thus you're much less likely to see something from that set than from the universal set.

 

If I recall correctly, English is WoD's second language, so that could be the reason for the misunderstandings.

That's funny, you yourself admitted you don't know much about math, yet feel entitled to critisize both my understanding of math and of English.

 

Dagon, claiming something is too unlikely to happen in one trial is faulty logic - if you believe that, you must believe that it is also impossible in any number of trials because you can break any number of trials into a sequence of single trials, thus leading to a contradiction.

 

In short, anything with a nonzero probability can indeed happen in one trial. It may be very unlikely, but it is possible.

I'm not claiming the probability is actually 0, I'm saying it's practically 0 for a single trial. For a million trials it is indeed significant. What's theoretically possible isn't necessarily practically possible, but we just keep going around in circles on this.

 

In this case, there is no difference between theory and practice. A chance is a chance, no matter how low. It's nonzero, therefore it is possible.

 

See my edit in the post you're quoting, I took most of that back already. I still want Wals to verify we understood him correctly, because indeed what he seems to be recounting would be like someone buying a single ticket and winning the lottery, then buying another ticket and winning again. As far as the aces example, I stand by what I said about a certain set of sequences being far more likely than another set.

 

While you're quick to accept that this is indeed some sequence of cards, you seem to only think that the probability of this event happening matters if and only if it was predicted to happen before the event occurred. This is not the case. The probability of my drawing that sequence of cards is still the same. There is a 1 / 6 497 400 chance that I would have been correct if I had predicted a sequence of cards. We all seem to agree on this, except that you seem to feel that because I didn't explicitly predict it before hand, I have not actually drawn a unique sequence of cards, but just some sequence of cards.

Exactly, that's the crux of the matter. The event that happened is that you drew a sequence of cards, and the probability of that event is 1, because you have to draw some sequence. Had you predicted that exact sequence, then the event would be you drew a specific sequence, with the huge odds.

 

Unfortunately, Dagon, that's not how statistics work. It is still a unique sequence of cards, whether or not he predicted beforehand what sequence he would draw. Thus, something with an "impossibly low" probability did occur. He did not only draw "a sequence of cards," he drew "that sequence of cards." The probability of him drawing that sequence of cards might be somewhat meaningless as you can't use it to make any meaningful predictions of conclusions, but it is still the probability of that event occurring, and that event did occur.

Edited July 23, 2010 by Oblarg

[Wrath of Dagon]

OK, I understand that's your position, there's no point in repeating it ad infinitum.

[Nepenthe]

Again welcome to the internet.

[Wrath of Dagon]

Unfortunately, Dagon, that's not how statistics work. It is still a unique sequence of cards, whether or not he predicted beforehand what sequence he would draw. Thus, something with an "impossibly low" probability did occur. He did not only draw "a sequence of cards," he drew "that sequence of cards." The probability of him drawing that sequence of cards might be somewhat meaningless as you can't use it to make any meaningful predictions of conclusions, but it is still the probability of that event occurring, and that event did occur.

That's just wrong, sorry, the event that occurred had a probability of 1 of occurring, as I said.