Lotto Probability

[Rasmus]

I'm afraid it would still be NO.

Even if we ignore your second question posted with this one, Question 1 still is based on the drawing of 6 lotto balls -- just look at the previous questions that lead to this one; they are based on lotto drawings of 6 balls and the likelyhood to an outcome.

What does it matter?

If 1-2-3-4-5-6 is an unlikely event,
and
if 1-2-3-4-5-7 is an unlikely event,
and
if 1-2-3-4-6-8 is an unlikely event,
and
if any other combination i-j-k-l-m-n is an unlikley event

(you agreed to this much, right?)

then I should still be able to look at all of them, and see a great many unlikely events, right? (At this time, there is no actual drawing of lottery numbers involved. I am just saying that *if there was* a drawing, then these would be unlikely outcomes. - individually, still.)

Besides ... how could it not be a set of 13 million improbable events when you ask "...we have about thirteen million improbable events to throw into our bag ... " Sounds like a set containing 13 million different elements to me.

... yet, you still seem to be saying that it is not a set of 13 million improbable events? At least this is how I understand the last question and your answer to it. (I am seeing the question as just forming a set from 13 million improbable events - without commenting on any properties of the set yet, other than that it contains 13 million items that we have talked about before)

 

[Just thinking]

What does it matter?

If 1-2-3-4-5-6 is an unlikely event,
and
if 1-2-3-4-5-7 is an unlikely event,
and
if 1-2-3-4-6-8 is an unlikely event,
and
if any other combination i-j-k-l-m-n is an unlikley event

(you agreed to this much, right?)

then I should still be able to look at all of them, and see a great many unlikely events, right? (At this time, there is no actual drawing of lottery numbers involved. I am just saying that *if there was* a drawing, then these would be unlikely outcomes. - individually, still.)



... yet, you still seem to be saying that it is not a set of 13 million improbable events? At least this is how I understand the last question and your answer to it. (I am seeing the question as just forming a set from 13 million improbable events - without commenting on any properties of the set yet, other than that it contains 13 million items that we have talked about before)

Because they are no longer unlikely or improbable events.

They are now only rare events -- not unlikely. By giving me the entire set of individual events, seeing one happen is no longer improbable. Seeing a specific one is -- but you will not specify just one any longer. And if you wait until after the drawing to specify one, you have painted a bullseye around the shot.

The list you start at the beginning of this reply indicates that I will accept one set at a time of 6 numbers as an unlikely event to occur -- and that is true -- but not all at once. After all, what then is the improbable event? That one of them will occur? No -- that's a certainty. That the set that happended to come out is? That's painting the bullseye.

Consider the following scenario ... There are exactly 13 million possible lotto combinations. Thirteen million different people play the game one week, and each person buys exactly one ticket and picks a unique combination that no one else has. Is having a winner an improbable event? No -- one of the 13 million possible combinations must be had by one person. So what was the improbable event? That Joe Blow was the winner? You can't say that until after you know who wins -- (If you could somehow see each person as the lotto is drawn, you would not be surprised in seeing someone jump up and scream victory -- but you would if only one person played with one ticket). So what is improbable here is subjective at best. I will agree that Joe Blow winning the lottery twice in a row (or anyone else for that matter) is an improbable event, however. But how often does that happen? So basically what I will accept as a group of 13 million improbable events is any one of the 13 million players hitting the lotto two weeks in a row.

 

[Rasmus]

That Joe Blow was the winner? You can't say that until after you know who wins

No, I can say that before the event. For each of the 13 Million people, and I would be right.

(I am sure each one of them would give me lousy odds if i was to bet them that they wouldn't win the lottery....)

-- (If you could somehow see each person as the lotto is drawn, you would not be surprised in seeing someone jump up and scream victory -- but you would if only one person played with one ticket).

There is a difference between saying "one out of 13 million people will win" and saying "it was unlikely that it should have been Jon Doe who won."

If no improbable event had happened, then why would there be any jumping and screaming going on?

So what is improbable here is subjective at best.

No.

Chances that someone wins: 1
Chances that Jon wins: 1:13.000.000
Chances that Jane wins: 1:13.000.000
Chances that Tom wins: 1:13.000.000
Chances that Sandra wins: 1:13.000.000
Chances that Marc wins: 1:13.000.000

There is nothing subjective anywhere in the process.

I will agree that Joe Blow winning the lottery twice in a row (or anyone else for that matter) is an improbable event, however. But how often does that happen? So basically what I will accept as a group of 13 million improbable events is any one of the 13 million players hitting the lotto two weeks in a row.

I just don't see how that makes any sense whatsoever, since we could easily calculate the odds of the same person winning twice. (I am too tired right now, but aren't they also 1:13.000.000? Anyone could win the lottery in the first week, and then it's just the same chance that it should be them again ...)

Rasmus.

 

[Just thinking]

I just don't see how that makes any sense whatsoever, since we could easily calculate the odds of the same person winning twice. (I am too tired right now, but aren't they also 1:13.000.000? Anyone could win the lottery in the first week, and then it's just the same chance that it should be them again ...)

Rasmus.

It's 13 million x 13 million for a single specified person to win twice in a row. It's (13 million x 13 million)/(13 million) or just 13 million to 1 for any unspecified person to win twice in a row. It's (13 million)/(13 million) or 1 for any unspecified person to win once.

 

[Just thinking]

If no improbable event had happened, then why would there be any jumping and screaming going on?

That is a subjective improbability -- you must expect a winner under the conditions I described.

 

[Just thinking]

Chances that someone wins: 1
Chances that Jon wins: 1:13.000.000
Chances that Jane wins: 1:13.000.000
Chances that Tom wins: 1:13.000.000
Chances that Sandra wins: 1:13.000.000
Chances that Marc wins: 1:13.000.000

There is nothing subjective anywhere in the process.

But you will accept any player that wins as the event, no? That's painting the bullseye.

Please specify the improbable event (singular).

Guess what you get when you add up all the 1/13,000,000 13 million times? This is what's happening when you accept anyone who wins.

 

[BillyJoe]

What if I had done this thing with jellybeans: A bag marked "Jellybeans". if you find a jellybean throw it into the bag. If you throw in 13 million jellybeans, you would say, wouldn't you, that there are 13 million jellybeans in the bag. Why is it any different with improbable events. They are just items thrown into their respective bags.

It seems we have found our point of disagreement, but I'm really struggling to see how you can disagree. It's almost a syllogism - undeniably true!



BJ

 

[BillyJoe]

JT,
Consider this scenario:

There is a bag labelled "Jellybeans"
You pick up one jellybean and throw it into the bag labelled "Jellybeans".
You repeat this process 13 million times.
You now have 13 million jellybeans in the bag labelled "Jellybeans".

Do you agree with the last statement?

I think you would have to answer "yes".
If so, consider a slightly changed scenario:

-----------------------------------------

Same as the first scenario except that each jelly bean is a different colour.
(If you object that there are not 13 million different colours, consider that each jellybean reflects a different wavelength of light and that you can measure the wavelength reflected by each jellybean, if you wish, by means of a measuring instrument)

Do you still agree with the last statement?

Again, I think you would have to answer "yes".
If so, consider another slightly changed scenario:

------------------------------------------

Same as the first scenario except that each jelly bean has a different colour pattern.

Do you still agree with the last statement?

You have to answer "yes" surely.
If so, consider yet another slightly changed scenario:

-----------------------------------------

Same as the first scenario except that each jelly bean has a unique set of six differently coloured stripes and that there are 48 different colours.

Do you still agree with the last statement?

If you answer "yes" then, consider that the colour pattern is actually a code for each of the 13 million sets of six lotto numbers.


BJ

 

[Just thinking]

What if I had done this thing with jellybeans: A bag marked "Jellybeans". if you find a jellybean throw it into the bag. If you throw in 13 million jellybeans, you would say, wouldn't you, that there are 13 million jellybeans in the bag. Why is it any different with improbable events. They are just items thrown into their respective bags.

It seems we have found our point of disagreement, but I'm really struggling to see how you can disagree. It's almost a syllogism - undeniably true!

Yes, I would say that there are 13 million jellybeans in the bag -- and I will say that you have a set of 13 million events as described. The difference here is whether or not they qualify (as a group) as being improbable events. Individually, we can call each item a jellybean -- and we can call each individual event improbable, but something happens when we take all possible events together and force one of them to occur.

If 13 million people play the lottery as I described earlier I'm positive you would not be surprised that a winner will step forward after the drawing. He/She would say that their winning was very improbable -- but someone had to win, and the best you could do was say that Joe Blow had a 13 million to 1 shot. Yet, so did everyone else -- so Joe Blow winning was no more improbable than anyone else for an event that had to happen. Joe Blow is amazed and happy -- but are you (who knows the entire scenario) really surprised and stunned?

Now, let's have the exact same game played -- only this time Joe Blow is the only player with one ticket. (He doesn't know this little difference.) Again, Joe Blow wins. Is your feeling of having a winner just as confident as it was earlier? I know it wouldn't be for me. Why? Because the chances for a winner were 13 million to 1 against it. Were the chances of Joe Blow winning either lottery any different for these two scenarios? No. So why would I call the second scenario an improbable event and not the first? Because the first one forces an outcome with a winner, guaranteed. Even though one can say before the drawing that it was improbable for Joe to win, it is also true that it was improbable for Mike to win, or Pete, or June, or Mary, or ... etc. This makes Joe's winning in the first scenario a rare outcome as compared to all the other possible outcomes -- but in no way should we be surprised that someone (Joe Blow) steps forward to claim the winning amount. By stating that whoever wins (in this case, Joe) was an improbable event, we have allowed all the probabilities of each player (13 million) to sum up to that claim. And each week there will be a winner (given the first scenario) just like Joe -- so what's really so improbable with this? The event of this week's winner to win again next week would be an improbable event (13 million to 1) because we have narrowed the selection of winners to just one person -- this week's winner.

Edit to add: If you had predicted in the first scenario that Joe Blow specifically was going to win before the drawing, then that would be an improbable event.

 

[Just thinking]

JT,
Consider this scenario:

There is a bag labelled "Jellybeans"
You pick up one jellybean and throw it into the bag labelled "Jellybeans".
You repeat this process 13 million times.
You now have 13 million jellybeans in the bag labelled "Jellybeans".

Do you agree with the last statement?

Yes.

Same as the first scenario except that each jelly bean is a different colour.
(If you object that there are not 13 million different colours, consider that each jellybean reflects a different wavelength of light and that you can measure the wavelength reflected by each jellybean, if you wish, by means of a measuring instrument)

Do you still agree with the last statement?

Yes -- and we can say that each has a different color.

Same as the first scenario except that each jelly bean has a different colour pattern.

Do you still agree with the last statement?

Yes.

Same as the first scenario except that each jelly bean has a unique set of six differently coloured stripes and that there are 48 different colours.

Do you still agree with the last statement?

Yes.

If you answer "yes" then, consider that the colour pattern is actually a code for each of the 13 million sets of six lotto numbers.

Ok -- so each jellybean represents one set of 6 different lotto balls.

Now what?

 

[BillyJoe]

Yes.
Ok -- so each jellybean represents one set of 6 different lotto balls.

Now what?

So each jellybean could represent one set of 6 different lotto balls in next week's lotto draw.
In other words, each of these jellybeans could represent an improbable event.
You could therefore relabel the bag "Improbable events"
So, if you agree that

You now have 13 million jellybeans in the bag labelled "Jellybeans".

Then you must also agree that...

You now have 13 million improbable events in the bag labelled "Improbable events".

But I think you answered "no" to that question before?


BJ

 

[Just thinking]

So each jellybean could represent one set of 6 different lotto balls in next week's lotto draw.
In other words, each of these jellybeans could represent an improbable event.
You could therefore relabel the bag "Improbable events"
So, if you agree that

You now have 13 million jellybeans in the bag labelled "Jellybeans".

Then you must also agree that...

You now have 13 million improbable events in the bag labelled "Improbable events".

But I think you answered "no" to that question before?


BJ

Afraid so -- did you read my post (369) as to why?

 

[BillyJoe]

Afraid so -- did you read my post (369) as to why?

Not yet - that will have to wait till tonight

But how do you reconcile:

You now have 13 million jellybeans in the bag labelled "Jellybeans".

To which you agree

You now have 13 million improbable events in the bag labelled "Improbable events".

To which you disagree.

But the statements are isomorphic.


BJ

 

[Just thinking]

But how do you reconcile:

You now have 13 million jellybeans in the bag labelled "Jellybeans".

To which you agree

You now have 13 million improbable events in the bag labelled "Improbable events".

To which you disagree.

But the statements are isomorphic.


BJ

Yes ... it does seem confusing -- and trust me, I am having difficulty putting my thoughts on this into words. But I do strongly feel there is a difference. Whereas one can take the hypothetical bag of jellybeans and go in and withdraw a jellybean, I don't think that one can go into the other bag of events and pick an improbable event. Yes, if there were only a handful of lotto combinations in there then you could do it -- but now we have each and every possible combination available to us. It's like going into a bag of lotto tickets that total all possible combinations and choosing the one that will match the next pick of numbers. If only a few people pick tickets, then having a winner is an improbable event. But if 13 million people pick all the tickets, then we are certain that a winner will happen. These now become the same scenarios as described in post 369. Since one of your questions refers to doing this week after week, implying that the specific winner each week is the improbable event, I'm concluding that this winner can only be described as whoever wins. And by arguing that whoever wins was an improbable event, you have allowed all the probabilities of each player (13 million) to sum up to that event.

 

[BPScooter]

I'm only casually following this, because to me it seems like nobody's doing anything too strange. Maybe it's all just in the description. If we have our "all possible outcomes" jellybean bag, and ask the bag to be emptied, then it seems clear that a winner will be selected. Which winner is not determined, or determinable.

Here's where I think lottos are a little different. What if I say, "step right up, a winner is in this bag, give me a dollar"--well, maybe I'll get only half as many people and so, what does that mean, I have a 50/50 chance that the winner will not be chosen? OK, that happens, we return the beans to the bag. Step right up, a winner in the bag (OK, now we'll add part of the money I took from you to make a bigger prize). Wow! Bigger prize! 75% of the beans go next week. No winner. OK, now we have a BIG PRIZE! In fact, more people want to play so I get out a duplicate bag of beans... wow, 2 people played the same "lucky numbers" they always play, and have to split the pot. I think the odds of a particular winner winning are variable, but the odds of a particular number being a winner are easy to figure out.

What if in the first place, I had an infinite number of beans representing the eventual winner, and no restrictions on how many could play? What are the chances that 2 or more people would take the same bean?

My gut feeling is that lottos are just sort of a bad bet. I suspect people just lock into some lucky number combination and play repeatedly, although that probably wouldn't really improve their odds at all versus just picking random numbers every time or letting the computer pick them. Or play a whole bunch of tickets on a particularly big lotto.

There might be a dissertation here--the interplay between real statistics, folk knowledge of statistics, and human psychology in a lotto. Ha ha... glad it's not me who has to try to sort it out! Thanks for a fun thread to think about.

 

[BillyJoe]

Afraid so -- did you read my post (369) as to why?

I've read it now. I agree with all of it - see I told you we agree!
Including this bit....

Yes, I would say that there are 13 million jellybeans in the bag -- and I will say that you have a set of 13 million events as described. The difference here is whether or not they qualify (as a group) as being improbable events. Individually, we can call each item a jellybean -- and we can call each individual event improbable, but something happens when we take all possible events together...

This is why I qualified my previous question thusly:

Just to be clear, "thirteen million improbable events" means "improbable event X thirteen million", not "a set of thirteen million improbable events".

"improbable event X thirteen million" means each event taken individually X thirteen million (because that's how many there are).
"a set of thirteen million improbable events" means the thirteeen million improbable events taken together as a group.

So, with this qualification, do you now agree with the following statement:


You now have thirteen million improbable events in the bag labelled "Improbable Events"


Then, finally, we might be able to move on to the actual lotto draw.
(maybe)

BJ

 

[BPScooter]

I think what you are hinting at is 13 million psychics, with 13 million unique predictions, and only one bean in the bag.

How is that different than one bean-picker picking 13 million times?

Where we get all messy is the interaction between people and numbers. One solitary guy playing 1$ per week on the same numbers probably has the same chance each week if he changes numbers, goes to a psychic for better numbers, etc. That's what's intriguing me about the question. The big "Powerball" thing in the US was what, 10 years ago they invented it? Of course as the potential jackpot grew, the chances of winning were increased because more people were betting on the same jelly-bean bag. Lots of people didn't want state lotteries. I thought it was a great big hypocritical money-grab in my state. The only legal gambling was by buying an effing keno card. No thanks, if I want to gamble I'll go to Nevada. But now every state is doing it. Indian tribes have made a nice retribution.

 

[BillyJoe]

I think what you are hinting at is 13 million psychics, with 13 million unique predictions, and only one bean in the bag.

All I am trying to do is convince JT there are 13 million @#$%^&* items in that bag!
Otherwise there'll be no &^*%$#@ lottery at all.

 

[Just thinking]

All I am trying to do is convince JT there are 13 million @#$%^&* items in that bag!
Otherwise there'll be no &^*%$#@ lottery at all.

Hi BJ,

Sorry to take a bit longer this time to reply -- I hope those inflections aren't an indication of some serious anger .

Anyway -- I do agree that there are 13,000,000 items (descriptions of events) in the bag ... it's whether or not they can still be considered as improbable when all grouped together like that with having one of them forced to occur. Plus, I believe that no matter how they are described, the lottery can proceed as usual.

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

Again -- Sorry for the delay.

 

[Rasmus]

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

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

If you take one event, that you previously agreed was improbable,
and a second event, that you also previously agreed was improbable,

then you have to events that are (both, individually) improbable.

Nobody so far has been addressing any combined probabilities or introduced any other events to the question (such as drawings, or predictions/bets on particular numbers, or individual players, etc.)