Lotto Probability

[Just thinking]

I honestly don't see what reason you could possibly have for not accepting that?

Simply due to the forcing of an event (or prediction).

If the bag contained only a handful of predictions for the next upcoming lottery, then having one of those occur is improbable. When the bag contains all possible outcomes, then having one of them occur is simply a certainty. Addressing the improbable event as whatever happens to be the outcome of numbers is no more than giving the outcome the combined probability of each and every outcome (1/13 million + 1/13 million + 1/13 million ... 13 million times).

One might think that the JREF foundation might conceed the $1,000,000 prize to a 13 million to 1 shot ... but let's say that 13 million entries applied for the JREF prize where each entry predicted the outcome for the next lottery -- and each predicted a different outcome. Would JREF award the prize to whoever happened to get it right? No way -- because nothing improbable happened. There was essentailly no 13 million to 1 shot -- even though one can later say that the "winner" had a 13 million to 1 chance of getting it right.

 

[Rasmus]

Simply due to the forcing of an event (or prediction).

Nobody forces an event.

You have a bag full of little slips of paper, that name different potential lottery results. No even is taking place.

If the bag contained only a handful of predictions for the next upcoming lottery, then having one of those occur is improbable.

You are already combining several events into one. I belive this is premature and not what the situation so far is about.

When the bag contains all possible outcomes, then having one of them occur is simply a certainty.

Yes. But as far as I understand the scenario - that is not being disputed, nor is it the current question.

Addressing the improbable event as whatever happens to be the outcome of numbers is no more than giving the outcome the combined probability of each and every outcome (1/13 million + 1/13 million + 1/13 million ... 13 million times).

Nobody is doing that, though. At least - not so far.

And as far as I can tell, nobody is refuting that if every possible number is played once, then a winner will be drawn for sure.

But what you seem to be saying is that this somehow makes the individual numbers more likely to be drawn, and I really don't see how, or where or why that would be possible. As long as oyu look at an individual line, it's chances are always 13.000.000:1. Even after a line has been drawn (and turns out to be e.g. 14-23-27-33-34-45) it's chances were never any better or worse than 13.000.000:1.

So that 14-23-27-33-34-45 should be drawn was an improbable event. (Hence, whoever had those numbers will become very rich for playing and winning the lottery, EVEN IF 13 million people playes 13.million different lines.)

 

[Just thinking]

Ok, another way to look at the bag of "improbable" events (lottery predictions) as being what I believe to be significantly different from the bag of 13 million unique jellybeans is this ...

One can be asked predict the chances of going in and pick out a specific jeyybean ... the odds are roughly 13 million to 1. And one can then be asked what are the chances that lottery prediction a, b, c, d, e, f will result? Again, the odds are roughly 13 million to 1.

But ... one can also be asked what are the chances of picking 2 specific jellybeans? Well, the odds are 13 million x 13 million to 1 ... something we would agree is extrememly unlikely. But it can be done.

One cannot go in and successfully pick out two winning lotto predictions for the next lottery. Why not? Because these events are somehow tied together; they are not independent events like the jellybeans. When one predicts that the winning lotto combination will be a, b, c, d, e, f they are also saying that it will not be any of the other 12,999,999 commbinations as well. This is what distinguishes the jellybeans from the predictions, although I'm not quite sure how to put this into the proper mathematical language, yet. They become tied together in a way that makes the complete set no longer improbable events.

 

[Just thinking]

But what you seem to be saying is that this somehow makes the individual numbers more likely to be drawn, and I really don't see how, or where or why that would be possible.

No, I am not saying that -- but I can see how you might think that. I am simply saying that given enough players, one should not be surprised at someone's excitement in winning.

As long as oyu look at an individual line, it's chances are always 13.000.000:1. Even after a line has been drawn (and turns out to be e.g. 14-23-27-33-34-45) it's chances were never any better or worse than 13.000.000:1.

But as a group containg all possible outcomes I believe they then simply bacome rare events.

So that 14-23-27-33-34-45 should be drawn was an improbable event. (Hence, whoever had those numbers will become very rich for playing and winning the lottery, EVEN IF 13 million people playes 13.million different lines.)

That is not quite the same thing as listing each one and then combining them in a set.

 

[Just thinking]

Nobody forces an event.

Each "improbable" prediction involves a lotto drawing where one of the 13 million combination of numbers must result.

You have a bag full of little slips of paper, that name different potential lottery results. No even is taking place.

See above.

You are already combining several events into one. I belive this is premature and not what the situation so far is about.

Placing all possible outcomes together is the combining.

Nobody is doing that, though. At least - not so far.

It is something which must be done, otherwise what is being predicted -- a specific one?

 

[ynot]

Lotto balls don’t have a memory.

Previous draws have no effect on the current draw (assuming the draw is completely random).

The odds of a particular person winning the major prize are worse than the odds of that person dying on the day of the draw.

Any number combination is as likely/unlikely to win as any other.

You cannot win the major prize if you don‘t have a ticket.

You can win the major prize if you have a ticket, but it is extremely unlikely that you will.

Lotto should be played as a form of fantasy entertainment, not as a source of income.

Only disposable income should be used to purchase lotto tickets (not food, rent, etc. money).

People that claim to have psychic/paranormal abilities don’t win lotto any more often than “normal” people.

 

[BillyJoe]

I hope those inflections aren't an indication of some serious anger .

If you read back, you will see that scooter's a bit of a jokester. I was replying to him in kind.

Anyway -- I do agree that there are 13,000,000 items (descriptions of events) in the bag.

Do you agree "that there are 13,000,000 items (descriptions of IMPROBABLE events) in the bag"?

....it's whether or not they can still be considered as improbable when all grouped together...

I thought I had made it clear that my intention is to consider them as individual improbable events: individual improbable event X 13,000,000. Is there still an issue here?

Plus, I believe that no matter how they are described, the lottery can proceed as usual.

The real lotto was drawn last night. Our lotto has not yet gotten off the ground.

So, for the sake of continuing this debate, I will conceed that you have 13 million improbable events in the bag this week and each week a lottery occurs.

Remember I am trying to find the point at which we disagree. Therefore, if we don't fully disagree on this point, we may have arrived at our destination - the lotto may never be drawn!

Mind ... I do not fully accept that premise, but will go along with it for now.

So, perhaps we have found our point of departure then.
But, I would like to know how you would argue directly against the following:

1) Find item of type A and throw it into the appropriately labelled bag
2) Repeat this 13 million times.
3) You now have 13 million items of type A in your bag.

How can this work for A = jellybean but not for A = improbable event?
Surely it doesn't matter what A represents.


sorry about the lotto draw,
BillyJoe

 

[Just thinking]

So, perhaps we have found our point of departure then. But, I would like to know how you would argue directly against the following:

1) Find item of type A and throw it into the appropriately labelled bag
2) Repeat this 13 million times.
3) You now have 13 million items of type A in your bag.

How can this work for A = jellybean but not for A = improbable event?
Surely it doesn't matter what A represents.


sorry about the lotto draw,
BillyJoe

No problem (lotto draw).

Well, I tried to explain that when A = jellybeans it is possible to pull one, two or even more jellybeans from the bag. But can one draw more than one winning lotto prediction out of the bag successfully?

Here's what I mean. Let's look at 13 million improbable events that can happen on a given day -- one can be Bozo flight 344 crashing. Another can be that lightning will strike the Sears Tower. Another can be we find Osama Bin Laden. Another can be that N. Korea agrees to UN requests. Etc., etc., etc., each one with odds of 13 million to one. Are these improbable events any different than the 13 million predictions for the next lottery, each being 13 million to one also? Yes. Why? -- because they are independent events. Any one occuring does not have an effect on any other one occuring. No event becomes impossible if any other one happens -- and more than one can happen, too. They are also not all-inclusive, as there are many other events possible in the universe that given day. But the 13 million lotto predictions are not independent -- rather they have some form of dependency on each other. When one predicts that a, b, c, d, e, f will be the next winning set, they are also claiming that the other 12,999,999 outcomes will never happen -- making each of those other predictions an impossible event (should that given prediction win). And, they are all-inclusive -- making one of the events a must.

The jellybeans are more like the 13 million independent events -- which I feel are quite different from the lotto ball predictions. This is why I believe the two are different and that the all inclusive set then becomes a group of rare events as opposed to improbable.

 

[BPScooter]

Yes, a jokester, but not trying to provoke, maybe get a layman-ish take on some of this, but not intending to be an ass or anything.

OK, after looking back at some of this, we need to get clear on what "probable" or "improbable means, let me try with just an example. Do take me back to a prior post if I've missed something.

Jellybean draw=event. Prediction of event=piece of paper with that bean's colors described unambiguously by a unique player.

OK. I have a bag with 1 bean. I tell you there is a red bean in the bag, and ask you to choose a bean. What color do you predict? 100 percent certainty. No probability at all. Prize to you.

2 beans. I tell you there is a red one and a green one. You predict and pick one. Probability .50, right? But I tell your buddy to do it as well before I pick. Well, he might be right, you right, he might be wrong and you wrong, or one or the other of you might predict right. So now we have a different probability for the idea of "winner" vs. any independent player getting a win.

This is what makes me nuts. The moment that a lottery gets set up with a big prize, people get all freaked out about these random number sets.

What do the experts say about the word "probable"? It must have something to do with the certainty or likelihood, even to stretch the metaphor of "betting on it." It is certain that someone will win a lottery of x numbers if there are x! number (or whatever exhausts the combinations) of unique players. So the meta-lottery would be easy to predict. Someone will win.

(Edited to add, oh poo--I guess one needs to tell whether the beans go back in the bag again between draws. For this purpose, let's just say predictions are made and beans go back in bags and can happen more than once)

 

[69dodge]

Instead of calling each lotto event "improbable", let's call it "an event whose probability is 1 in 13 million". Does that make things any clearer?

Just thinking is using "improbable" in a vaguer way than that, to mean something like "I'd be surprised if it happened".

BillyJoe is obviously not using the word in the same way, because his ultimate goal is to demonstrate that improbable events happen all the time, and presumably he doesn't go around perpetually surprised at everything that happens.

As I said much earlier in the thread, I don't think it's especially useful to think about whether we'd be surprised. Whatever happens, happens. Being surprised serves no purpose. We should just focus on what to conclude based on the event that happened. For example, should we conclude that the lottery was rigged? Well, getting the particular set of numbers that we did get has a low probability in a fair lottery, and it has a high probability in a lottery that was rigged to produce those numbers. So that would seem to point to rigging. But, on the other hand, we can't forget how low the prior probability was that the lottery would be rigged to produce those particular numbers. After all, even if the lottery was rigged, which isn't terribly likely a priori, it could have been rigged to give any numbers. What were the prior chances, even assuming it was rigged, that it was rigged for those particular numbers? Something like 1 in 13 million. Which is just as low as the probability that those numbers would result from a fair lottery. So, there's no reason to conclude that the lottery was rigged. Which is basically what Just thinking means when he says that he wouldn't be surprised if those numbers came up.

It's not the case that, assuming a fair lottery, the numbers that came up were probable. Their probability was 1 in 13 million, which certainly deserves the name "improbable". The only reason we don't conclude that the lottery was rigged to produce those numbers is that such rigging was even less probable.

 

[BillyJoe]

JT,

Despite saying that we have not yet had a lotto draw, you keep talking about the draw.
It's before the draw. No numbers have yet come up. Maybe the world will end and we will never have the draw. Forget about the draw. What we do have is 13 million improbable events in the bag. Surely.

Well, I tried to explain that when A = jellybeans it is possible to pull one, two or even more jellybeans from the bag. But can one draw more than one winning lotto prediction out of the bag successfully?

We're not drawing anything out of the bag. Not jellybeans. Not lotto numbers. We've got a bag of 13 million jellybeans. And we've got a bag of 13 million improbable events. Period.

Here's what I mean. Let's look at 13 million improbable events that can happen on a given day -- one can be Bozo flight 344 crashing. Another can be that lightning will strike the Sears Tower. Another can be we find Osama Bin Laden. Another can be that N. Korea agrees to UN requests. Etc., etc., etc., each one with odds of 13 million to one. Are these improbable events any different than the 13 million predictions for the next lottery, each being 13 million to one also? Yes.

For me, the answer is "no".

Why? -- because they are independent events. Any one occuring does not have an effect on any other one occuring. No event becomes impossible if any other one happens -- and more than one can happen, too. They are also not all-inclusive, as there are many other events possible in the universe that given day. But the 13 million lotto predictions are not independent -- rather they have some form of dependency on each other. When one predicts that a, b, c, d, e, f will be the next winning set, they are also claiming that the other 12,999,999 outcomes will never happen -- making each of those other predictions an impossible event (should that given prediction win). And, they are all-inclusive -- making one of the events a must.

I think and important thing to remember is that probability applies only before the event. Once the time for the event to occur is over, probability no longer applies. The probability that the numbers 2,13,15,34,37,42 will come up in next weeks lotto draw is 1 in 13 million. After next weeks lotto draw, probability no longer applies - all you can say is that the probability of 2,13,15,34,37,42 coming up in this week's lotto draw WAS (ie before the draw) 1 in 13 million.

The second point I want to make is that all probabilities are dependent on something. The probability of a set of six lotto numbers coming up is 1 in 13 million because there are 13 million different sets of six lotto numbers each will an equal chance of coming up. If you say that the probability of a plane striking the Sears tower today is 1 in 13 million, you must have based this on something. For example you might have considered reports of planes hitting buildings of a certain size - the size of the Sears building. If there are 13 million of these types of buildings and one gets struck every day, then the chance fo the Sears building being struck is 1 in 13 million. Or something like that.

The jellybeans are more like the 13 million independent events -- which I feel are quite different from the lotto ball predictions. This is why I believe the two are different and that the all inclusive set then becomes a group of rare events as opposed to improbable.

All we have done is put jellybeans in a bag and improbable events in another bag. The world might end tomorrow. Anything. But, what we do know is that we have, right now, 13 million jellybeans in one bag and 13 million improbable events in another. Period.


If you cannot agree, then we have found our point of disagreement, which was the whole point of the exercise. And, of course, you were correct - it seems we do not agree as you stated (I thought we were saying the same thing in different ways )

BJ

 

[BillyJoe]

Instead of calling each lotto event "improbable", let's call it "an event whose probability is 1 in 13 million". Does that make things any clearer?

Hmmm....I thought that was already clear, but.....

Just thinking is using "improbable" in a vaguer way than that, to mean something like "I'd be surprised if it happened"

Is that right JT?

BillyJoe is obviously not using the word in the same way, because his ultimate goal is to demonstrate that improbable events happen all the time, and presumably he doesn't go around perpetually surprised at everything that happens.

That's about right.
(Although I'd be a bit surprised to be dealt a 13 heart hand in Bridge )

 

[BillyJoe]

OK. I have a bag with 1 bean. I tell you there is a red bean in the bag, and ask you to choose a bean. What color do you predict? 100 percent certainty. No probability at all. Prize to you.

Actually, the probability is 1.

2 beans. I tell you there is a red one and a green one. You predict and pick one. Probability .50, right? But I tell your buddy to do it as well before I pick. Well, he might be right, you right, he might be wrong and you wrong, or one or the other of you might predict right. So now we have a different probability for the idea of "winner" vs. any independent player getting a win.

The probabilities are as follows:

..Probability of me winning: 0.5
Probability of my me losing: 0.5
.............................total: 1.0

Probability of my buddy winning: 0.5
..Probability of my buddy losing: 0.5
......................,,...........total: 1.0

..Probability of me and my buddy both winning: 0.25
....Probability of me and my buddy both losing: 0.25
Probability of me winning and my buddy losing: 0.25
Probability of me losing and my buddy winning: 0.25
......................................................total: 1.00

It is certain that someone will win a lottery of x numbers if there are x! number (or whatever exhausts the combinations) of unique players.

X!/(X-N)!, where N is the number of numbers that have to be chosen,

So the meta-lottery would be easy to predict. Someone will win.

Interesting term, meta-lottery.

BJ

 

[Rasmus]

Interesting term, meta-lottery.

Speaking of which, maybe you could go a step back, and explain what exactly you are looking at when you speak of the "events" you want to put into your second bag?

I have no problem agreeing that they are all equally improbable - but perhaps restating what it is you want to group together might help?

Frankly, I am not even certainanymore, if you are even looking at just potential outcomes for any one lottery? It might make the concept easier to understand, because unlike a jeallybean, and event is hard to stuff into a bag

 

[BillyJoe]

Frankly, I am not even certain anymore, if you are even looking at just potential outcomes for any one lottery?

As I said previously, an improbable event (by definition) can't have occurred yet. Probability does not apply once the event has occurred. After the lottery draw, all you can say is that those events WERE improbable (ie before the draw). So, yes, the improbable events are potential outcomes of a future lottery draw.

It might make the concept easier to understand, because unlike a jeallybean, and event is hard to stuff into a bag

Yes, it's just a metaphorical way of categorizing things - in this case improbable events. You might also have a metaphorical bag labelled "Emotions" in which you might metaphorically stuff happy, sad, annoyed, angry, frustrated etc.

BJ

 

[BPScooter]

Yep, I totally agree and see how the math works. What makes me pull my hair out is people that don't see how a "ten times ten to the eleventh power" bad bet is any more lucky if they use the same lucky numbers every day or if they go buy a random card. Maybe the definition of "win" is where the house has its odds right. If we all said "unique winner on a specific draw" that is indeed amazing.

What the lottos, and casinos, and every "house" does is to play these people for fools. Get you to bet a big $ on a game that on the average pays low $. I like games of chance but I also like th money I earn. I guess I won't win, because I don't play.

 

[Just thinking]

Sorry for the break in responce ... I just needed a few days off from this.

JT,

Despite saying that we have not yet had a lotto draw, you keep talking about the draw.
It's before the draw. No numbers have yet come up. Maybe the world will end and we will never have the draw. Forget about the draw. What we do have is 13 million improbable events in the bag. Surely.

I keep referring to the draw because that is what these events in the bag are all about -- predicting the next lotto outcome. You can look back at the very first question of this series you started and see that's true.

We're not drawing anything out of the bag. Not jellybeans. Not lotto numbers. We've got a bag of 13 million jellybeans. And we've got a bag of 13 million improbable events. Period.

The events are predictions of what will be drawn in the next lotto -- it doesn't get any more direct than that.

For me, the answer is "no".

How so? In the first case we know that one prediction (and only one prediction) will actually result in being true (lotto outcome) whereas with the independent events no one event must occur. And even two or more can occur.

I think and important thing to remember is that probability applies only before the event. Once the time for the event to occur is over, probability no longer applies. The probability that the numbers 2,13,15,34,37,42 will come up in next weeks lotto draw is 1 in 13 million. After next weeks lotto draw, probability no longer applies - all you can say is that the probability of 2,13,15,34,37,42 coming up in this week's lotto draw WAS (ie before the draw) 1 in 13 million.

Yes -- here we agree.

The second point I want to make is that all probabilities are dependent on something. The probability of a set of six lotto numbers coming up is 1 in 13 million because there are 13 million different sets of six lotto numbers each will an equal chance of coming up. If you say that the probability of a plane striking the Sears tower today is 1 in 13 million, you must have based this on something. For example you might have considered reports of planes hitting buildings of a certain size - the size of the Sears building. If there are 13 million of these types of buildings and one gets struck every day, then the chance fo the Sears building being struck is 1 in 13 million. Or something like that.

Yes, but it has no effect on any of the other 13 million events occurring.

All we have done is put jellybeans in a bag and improbable events in another bag. The world might end tomorrow. Anything. But, what we do know is that we have, right now, 13 million jellybeans in one bag and 13 million improbable events in another. Period.

It's having all the possible outcomes (predictions) together where one must occur that prevents me from labeling them as improbable events. If, like I pointed out earlier, Joe wins the lottery among 13 million unique players, one should not be so surprised. If Joe bought the only ticket to a 13 million to one lottery and won, then yes, that's an improbable event. Expecting Joe to win in the first scenario is a prediction -- and if that coming true is what you consider to be the improbable event, please keep in mind that all the other players are too making similar predictions. This is why JREF will not award the big prize to a lottery prediction if 13 million unique entries apply -- absolutely nothing improbable will happen; someone will be correct. Once we collect all possible outcomes together and force one to occur, we have nothing more than a set of unique and rare outcomes -- looking afterwords at the outcome and saying that that specific outcome was improbable only is another way of predicting that one of the combinations was going to happen. Yes, its odds were 13 million to 1, but so were all of the other outcomes -- and one had to happen. So what's really improbable here? That Joe won? Not if you waited for whoever wins to replace Joe's name with.

If you cannot agree, then we have found our point of disagreement, which was the whole point of the exercise. And, of course, you were correct - it seems we do not agree as you stated (I thought we were saying the same thing in different ways )

BJ

Yes -- it does seem this is the point of disagreement -- but cheer up, you found it!

 

[Just thinking]

Instead of calling each lotto event "improbable", let's call it "an event whose probability is 1 in 13 million". Does that make things any clearer?

Just thinking is using "improbable" in a vaguer way than that, to mean something like "I'd be surprised if it happened".

Well, yes that's true -- but not all inclusive. Of couse I'd be surprised as improbable means unlikely to occur -- but then does. Improbable also implies a degree of rarity -- having a winner each week for a lotto drawing is not rare, hence I do not consider there being a winner an improbable event, no matter who it is -- even me. Why? Because someone must win (in an all-inclusive set of unique players). Now, saying that my winning was improbable may be true -- but accepting that as the improbable event then means that in order for improbable events to occur often, I should win the lottery often. Well, no one does that. And opening that up to whoever wins week after week as being the often occurrng improbable events is accepting the entire playing field as the improbable winner -- that's not valid, as I pointed out in the previous post to BJ with the JREF scenario.

Consider the formation of snowflakes during a snowstorm. Each one is unique and rare in design and structure. Are these formations improbable events, with billions upon billions occuring each and every second? I would argue no, as their continued formation is highly likely (OK, a certainty) during a snowstorm -- even though the odds of one forming any specific way is trillions upon trillions to 1.

 

[macgyver]

Wow, it's been a while since I've visited this thread...I have some catching up to do. I'm surprised that anybody is still here...

However, I just came across a little blurb in Discover magazine that seemed appropriate for discussion.

It was referring to the Monty Hall TV gameshow where three doors are hiding potential prizes. The participant chooses a door, then another door is revealed that shows nothing behind it. The participant is then given the opportunity to stick with their first choice, or switch to the other unopened door.

Apparently it is always better to switch than to stay with the door you originally picked.

Anybody care to explain that one?

 

[Just thinking]

Anybody care to explain that one?

It's much easier to see why it's better to switch if instead of 3 doors (where you initially pick 1) we use 20. Now, you go and pick one door -- your odds of getting the winning door is 1/20 or 5%. Not good, eh? Then they open all but one other door to show that the winning prize wasn't behind any of those. But you knew initially that the winning door was most likely one of the remaining 19 -- or a 95% chance. Since 18 of them had nothing, the 95% chance of the winning door is now focused all on the one not opened, and still 5% that it's your initial pick. Now, would you switch if offered the chance? In the case with only 3 doors the odds are not so extreme, but you go from 33% to 67% by switching.