Lotto Probability

[BillyJoe]

It is erroneous to claim that the 6 numbers that actually occured (for Event 1) is an improbable event because there was absolutely no requirement placed upon the outcome. Rare, yes -- improbable, no. BJ, you seem to agree with this later on in your satement, yet you initially claim it to be improbable. Was this just a slip up?

This is what I said....

Both EVENT 1 and EVENT 2 result in a set of six numbers which were improbable (Probability = 1 in 13 million).

You crossed out the "improbable" and substituted "rare". I have reversed that and emphasized the were. Does this make any difference for you? In other words, I'm talking about the probability before the draw. Once the numbers are drawn probability does not come into it. As you said, the numbers that come up in the draw will occur rarely - because, before the draw and before the next one, that set of numbers (and all other sets) was improbable.

We really are saying the same thing in different ways.

 

[BillyJoe]

What exactly do you mean by the second statement? Taken as a group? If you include enough possible examples of improbable events, then having one happen is no longer improbable -- in other words, having an unspecificed one happen is not an improbable event..

"Taken as a group" came right out of one of your posts.
But I can't seem to explain this thing to you. I'll try again:
Throw into a bag all those things that are improbable.
In that bag you will have next weeks lotto numbers 1,2,3,4,5,6
You will also have have next weeks lotto numbers 12, 23, 34, 37, 42, 47
You will also have the following weeks lotto numbers 1,2,3,4,5,6,
etc etc
If you keep going, you will eventually have in your bag all members of the class "Improbable events"
So what I am saying is that members of this class, which are all individually improbable, are occurring all the time.
(Remember that this was in response to mummypyjamas post, so you need to see it in the context of that post)

It's like lottery winners. Each and every winner considers their winning an improbable event [and they are correct - their lotto numbers were in that bag of "improbable events"] -- but with so many players there are winners almost every drawing [Yes. This is what I mean by "improbable events happen all the time". Simple as that]. So having an unspecified winner happen is not an improbable event [Yes, I agree. Nearly every week someone wins. But that is not what my phrase refers to] , hence that argument is fallacious if one looks at these as a string of improbable events that happen often.

I have added my comments above.

Just to be clear ... my example of a rare event (which is not improbable):
The numerical combination of the next 6 lotto balls drawn in my state.

Yes, six lotto numbers will be drawn next week and, yes, it is improbable that those numbers will come up in any future draw (just as it was before this draw) so those numbers are rare (and will continue to be rare).

... my example of an improbable event (which is also rare should it happen):
The next winning lotto numbers in my state's drawing will be 4 - 8 - 19 - 20 - 33 - 40.

Yes.

It's OK -- I'm not at all upset with any of this.

Just in case there is any misunderstanding, I post here out of interest, to hear others point of view, to occasionally get a new angle on things and sometimes just for fun, definitely not to upset people

regards,
BillyJoe

 

[Just thinking]

This is what I said....

You crossed out the "improbable" and substituted "rare". I have reversed that and emphasized the were. Does this make any difference for you?

Actually -- it does. To say that those specific numbers that occured were improbable is (IMHO) invalid. Why? Because there was absolutely no condition placed on any expectation of the drawing. If one looks at the numbers drawn and then says that getting those specific numbers was millions to one, they would be correct. Now, it may appear that I'm contradicting myself -- after all I say it's correct to claim that getting those specific numbers was millions to one, yet no improbable event occured. One must recognize that this was only a rare event. It's not improbable because it can be repeated with ease -- each time another drawing (and different set of numbers come out) is made.

One cannot say that at 1:00 PM (before the drawing) that getting 6 lotto balls has a probability of 1, and later, at 2:00 PM (after the drawing) say that the same drawing had a probability of millions to 1 simply because you now know the numbers that came out -- these are actually two different events. Yes, two different events with only 1 drawing. This may be why you are claiming that one must learn JT Speak -- well, maybe you're right. This is a very subtle difference yet results in distinguishing what is simply a rare event into one that was highly improbable from happeneing.

In other words, I'm talking about the probability before the draw. Once the numbers are drawn probability does not come into it. As you said, the numbers that come up in the draw will occur rarely - because, before the draw and before the next one, that set of numbers (and all other sets) was improbable.

I will gladly agree that I may be saying things with some confusion, as I mentioned the degree of subtlty with all this -- so I will accept any and all criticisms. But I must once again emphasize that unless one specifices before the draw a specific outcome, there is no improbable event occuring in drawing 6 numbers. (Even if you say "The next 6 numbers I will draw are highly improbable to occur." -- That is simply a wrong way to look at it.) So speaking of the probability of just drawing 6 unspecified numbers before the draw has little meaning -- P will always be 1.

We really are saying the same thing in different ways.

Are we?

 

[Just thinking]

"Taken as a group" came right out of one of your posts.

Which one -- I would like to see the context under which I used it.

But I can't seem to explain this thing to you. I'll try again:
Throw into a bag all those things that are improbable.
In that bag you will have next weeks lotto numbers 1,2,3,4,5,6

OK -- I can easily follow your start here ...

You will also have have next weeks lotto numbers 12, 23, 34, 37, 42, 47

Do you mean the following week's numbers? Taken as written, you have two sets of numbers for one week.

You will also have the following weeks lotto numbers 1,2,3,4,5,6,
etc etc
If you keep going, you will eventually have in your bag all members of the class "Improbable events"

OK, it looks like I was correct in my assumption of what you meant. But I would not classify those as improbable events. Rare outcomes, yes -- but it was certain that outcomes would occur for each and every week.

So what I am saying is that members of this class, which are all individually improbable, are occurring all the time.

And it is exactly here we differ. They are rare outcomes -- not at all improbable in occuring.

Yes, six lotto numbers will be drawn next week and, yes, it is improbable that those numbers will come up in any future draw (just as it was before this draw) so those numbers are rare (and will continue to be rare).

Now we see that improbable involves matching rare events -- quite different from just having one occur on its own.

Just in case there is any misunderstanding, I post here out of interest, to hear others point of view, to occasionally get a new angle on things and sometimes just for fun, definitely not to upset people

regards,
BillyJoe

Me too -- sometimes getting too involved makes one (especially me) appear upsetting.

 

[Number Six]

I haven't read the whole thread but it seem like we're going over the same point over and over. I think some are making it more complicatd than it is We have:

1. A large probability (i.e., 1) that a bunch of balls will be drawn.

2. A very small probability that any pre-specificed set of balls will be drawn.

That's it. It's not that complicated. It's like getting struck by lightning...there's a very good chance it'll happen to someone but a very small chance it'll happen to you. Is there more to this discussion than that? Am I missing something?

 

[Just thinking]

I haven't read the whole thread but it seem like we're going over the same point over and over. I think some are making it more complicatd than it is We have:

1. A large probability (i.e., 1) that a bunch of balls will be drawn.

2. A very small probability that any pre-specificed set of balls will be drawn.

That's it. It's not that complicated. It's like getting struck by lightning...there's a very good chance it'll happen to someone but a very small chance it'll happen to you. Is there more to this discussion than that? Am I missing something?

I believe that BJ would consider both to be improbable; (that a person gets struck by lightning whether specified or not -- because after the event one can say "Gee ... what were the chances of me getting struck?"). So to BJ the event of lighting striking someone is an improbable one -- I would only consider that a rare event.

Of course, that is purely my opinion.

 

[Just thinking]

As commented by BJ:

Originally Posted by Just thinking :

It's like lottery winners. Each and every winner considers their winning an improbable event [and they are correct - their lotto numbers were in that bag of "improbable events"] -- but with so many players there are winners almost every drawing [Yes. This is what I mean by "improbable events happen all the time". Simple as that]. So having an unspecified winner happen is not an improbable event [Yes, I agree. Nearly every week someone wins. But that is not what my phrase refers to] , hence that argument is fallacious if one looks at these as a string of improbable events that happen often.

The problem I see here is (I think) that if one week Joe plays the lottery, he gets to say that his winning it would be very improbable. True. But given the number of players, there is bound to be a winner each week. That's true also. But, this week Joe actually wins! Is his winning an improbable event? Yes. Is there being a winner at all an improbable event? No. So why can't one say that improbable events like this (someone specific winning) happen often? Because Joe's winning is the improbable event -- and guess what ... Joe does not win the lottery often. Nor do the people who should win every other week. This is why when one looks at groups of rare events and takes issue with one from the group occuring each week or whatever, one can see that the true improbable event (a specific person winning) does not count as an improbable event occuring often -- even with a winner each week.

This is why, BJ, I still argue that even taken as a group, improbable events do not happen all the time.

 

[BillyJoe]

I haven't read the whole thread but it seem like we're going over the same point over and over. I think some are making it more complicatd than it is We have:

1. A large probability (i.e., 1) that a bunch of balls will be drawn.

2. A very small probability that any pre-specificed set of balls will be drawn.

That's it. It's not that complicated. It's like getting struck by lightning...there's a very good chance it'll happen to someone but a very small chance it'll happen to you. Is there more to this discussion than that? Am I missing something?

I think we are all agreed on those two bits. But there's two more bits:

3. The surprise we would all (except JT) get if we were ever to be dealt 13 hearts in a hand of bridge, even though it is no less likely than any other hand.

4. The legitimacy or otherwise of the statement: "Improbable events happen all the time". (or can you see a way for that statement to make sense?)

Any thoughts?

BJ

 

[BillyJoe]

JT,

Let's take it one bit at a time:

BJ: You will also have have next weeks lotto numbers 12, 23, 34, 37, 42, 47

JT: Do you mean the following week's numbers? Taken as written, you have two sets of numbers for one week.

Well, there are 13 million sets of six lotto numbers to consider for next week's lotto draw. Every one of the thirteen million sets are improbable. Throw them all into the bag labelled "Improbable Events"

I will wait for your response before continuing.
BJ

 

[Just thinking]

JT,

Let's take it one bit at a time:

An excellent approach -- !

Well, there are 13 million sets of six lotto numbers to consider for next week's lotto draw. Every one of the thirteen million sets are improbable. Throw them all into the bag labelled "Improbable Events"

I will wait for your response before continuing.
BJ

Ah hah! I think I have something here. (Right off the bat, too.)

You would group all the players in a lotto game (given that each plays with only one ticket) as having an unlikely chance for any one of them to win the lottery, right? And (I believe) you would say to the single winner after the drawing that his/her winning was an improbable event, right?

Well, because there was no pre-determined or specified person as being the winner, no improbable event actually took place. Now, even though whoever does win can replace 'Joe' in my example a few posts earlier, you are willing to accept anyone who wins in your reasoning. Therefore, if it's not Joe, it can be Bill ... or Pete ... or Alice ... or Kim ... or Mike ... or Frank ... or Ginger ... or --- etc. etc. etc. What do all these players do to the winning of the lottery -- they make the event of a winner not an improbable event. Because no one is singled out, you have given the probability for a winner, P(winner), a very high value -- maybe even 1. This is because all those ors between each player mentioned means we add up each one's chance of winning to get the net probability of having a winner (or close to it). Yes, any one person has a very unlikely chance of winning -- but you open it up to anyone, and then look at the chance of that person winning and claim that as an improbable event. That's not valid. Do you now see why?

Here's hoping,
JT

 

[BillyJoe]

JT,

Hey, I didn't even mention lotto winners in my very short post! All I have mentioned so far is lotto numbers. I am interested in whether you can find any fault with what I wrote:

Well, there are 13 million sets of six lotto numbers to consider for next week's lotto draw. Every one of the thirteen million sets are improbable. Throw them all into the bag labelled "Improbable Events"

Then we can move on - one step at a time!
BJ

 

[BillyJoe]

PS: It's way past my bed time but I'll look in the morning before my morning run.

 

[Just thinking]

JT,

Hey, I didn't even mention lotto winners in my very short post! All I have mentioned so far is lotto numbers. I am interested in whether you can find any fault with what I wrote:

Well, there are 13 million sets of six lotto numbers to consider for next week's lotto draw. Every one of the thirteen million sets are improbable. Throw them all into the bag labelled "Improbable Events"

Then we can move on - one step at a time!
BJ

Well, yes. It's the statement "Every one of the thirteen million sets are improbable." -- I would change improbable to rare or unique; after all -- what exactly are you claiming to be improbable?

Have a good night.

 

[Number Six]

"Improbable" and "rare" aren't necessarily the same thing. "Improbable" is a more precise term since it implies that a probability is associated with the event. OTOH, "rare" is more subjective. Also, the criterion for how small the probability must be before it is considered "improbable" is subjective.

The probability for drawing a single set of balls in the lotto next week can be computed. Whether it's improbable depends on how low you think a probability must be before it qualifies as improbable. And whether it is rare depends on what you call rare.

Look at it this way, suppose they started having millions of lotto drawings every day. The probability of drawing a particular set of balls on each draw would stay the same but the occurrence of the event itself would be less rare.

 

[Just thinking]

"Improbable" and "rare" aren't necessarily the same thing. "Improbable" is a more precise term since it implies that a probability is associated with the event. OTOH, "rare" is more subjective. Also, the criterion for how small the probability must be before it is considered "improbable" is subjective.

The probability for drawing a single set of balls in the lotto next week can be computed. Whether it's improbable depends on how low you think a probability must be before it qualifies as improbable. And whether it is rare depends on what you call rare.

Look at it this way, suppose they started having millions of lotto drawings every day. The probability of drawing a particular set of balls on each draw would stay the same but the occurrence of the event itself would be less rare.

To me, improbable implies a degree of unlikelyness -- one specific event out of many similar events, or a situation of many unusual conditions occuring all at once in one place (a lightning strike), or a matching of two or more rare events simultaneously or head-to-head. It can also be predicting the exact outcome when millions of equally probable events can occur. It is also rare in occurance, as its name implies.

Rare just seems to mean one of many, or something unique. Having something rare happen often is not unusual, nor improbable -- the pattern of billiard balls after a break, or the exact combination of cards in a bridge hand. Making these unique/rare events happen is easy to do often. Getting one to repeat becomes improbable -- as described above.

Your example of millions of lotto drawings each day simply makes having a particular set come up more often -- but it does not make it any less rare in overall occurance compared to all the drawings. There are degrees of subjectivity involved to be sure, and I have always claimed rare and improbable as being different, but I think that simply placing a time factor (or overall numbers) as a distinction in changing something from being rare to no longer rare is misleading -- after all, are 4-leaf clovers any less rare if we look at my backyard as compared to all of my home state? Does including the entire state make finding one any less difficult?

 

[BillyJoe]

Well, yes. It's the statement "Every one of the thirteen million sets are improbable." -- I would change improbable to rare or unique; after all -- what exactly are you claiming to be improbable?

Okay, we have a misunderstanding here, but we will have to back up a bit to find the exact point where our thinking diverges.
Let's start with two very simple and seemingly straightforward questions...

1) Would you say that it is IMPROBABLE that next week's lotto numbers will come up 1,2,3,4,5,6 ?

2) Would your answer be the same if I substituted any other of the 13 million sets of six lotto numbers in the above question?

I really just want yes/no answers.
I'm trying to home in on the point where we disagree.

BJ


[Edit: I have edited this twice, the second time back to the original wording. I think I know what the problem is now but let's see...]

 

[Just thinking]

Let's start with two very simple and seemingly straightforward questions...

1) Would you say that it is IMPROBABLE that next week's lotto numbers will come up 1,2,3,4,5,6 ?

Yes.

2) Would your answer be the same if I substituted any other of the 13 million sets of six lotto numbers in the above question?

Yes -- but you must specify which group of 6; just saying any group that happens to come up is no good.

 

[Molinaro]

Rarely have I seen a thread take such an improbable turn

Although, since this is the internet.. the probability of this hapening is most likely 1.

 

[Just thinking]

Rarely have I seen a thread take such an improbable turn

Although, since this is the internet.. the probability of this hapening is most likely 1.

Stick around ... this may become interesting.

 

[BillyJoe]

Yes -- but you must specify which group of 6; just saying any group that happens to come up is no good.

Yes, that's why I said to substitute each set into the first question.
Okay, so far so good....

What if I rephrase those questions slightly:

1) Would you say that next week's lotto numbers coming up 1,2,3,4,5,6 is IMPROBABLE ?

2) Would your answer be the same if I substituted any other of the 13 million sets of six lotto numbers in the above question?

This may not sound like much of a difference, and I think it isn't, but there's something going wrong here, so we'll go slowly so we can hopefully catch it.

BJ