Winning proximity

[nimzov]

Now that makes sense to me.

By the same token, then, the poor lottery loser did not come close to winning, but instead was once in close proximity to the winning lottery ticket.

Well when he gave one ticket away he had a 50-50 chance of winning. That is quite close to winning from my point of view.

 

[dlorde]

I say the whole question is meaningless. There is no “close”. Either a ticket matches all the numbers, and thus wins, or else it doesn't. If it doesn't win, then it doesn't win, period; and there is no point in trying to define by how much it failed to win.

If you got a ticket that was 12 - 14 - 30 - 46 - 52 - 67, it would have failed to win exactly the same as all the other tickets.

This.

 

[nimzov]

This.

What would be your answers to my 3 questions in message #26.

 

[Fnord]

Well when he gave one ticket away he had a 50-50 chance of winning. That is quite close to winning from my point of view.

That's called "Fallacy of Equivocation". "Being close to a winner" and "being close to winning" are not the same concepts.

 

[dlorde]

...That is one of the reason I started this thread, it has noting to do per se with lottery. Lottery was just an example.

But the lottery is a very specific example. The alternative analogies you have presented are not the same thing at all.

A few days ago, a Philippino man graciously allowed a woman to push past him in a queue for lottery tickets. She bought the ticket he would have bought, and he bought the next ticket (that would have been hers). He won $17m.

How close was the woman to winning? Does "one ticket away" really mean anything?

 

[nimzov]

That's called "Fallacy of Equivocation". "Being close to a winner" and "being close to winning" are not the same concepts.

Not at all. If being close to a winner is a problem for you then consider this.

You do not give away the ticket. You burn randomly one of the two ticket.

By burning one of the ticket you have 50-50 chance of winning.

And you are closer to winning, than if you had burned both tickets.

 

[Fnord]

You do not give away the ticket. You burn randomly one of the two ticket. By burning one of the ticket you have 50-50 chance of winning. And you are closer to winning, than if you had burned both tickets.

I still don't buy into this. Only an idiot would burn (or give away) a potentially winning lottery ticket before checking to see if it is indeed a winner.

But to concede the issue, to me, someone who "almost won" is someone who was disqualifed just before receiving the trophy. He did not really win, since his victory was denied. But he did not lose, either, since he placed first.

Schrödinger's Victory, as it were.

 

[Uncayimmy]

Well when he gave one ticket away he had a 50-50 chance of winning. That is quite close to winning from my point of view.

The problem I am having with your posts is the lack of meaningful language. You're in a science forum asking questions where lots people with a science background or interest will answer. We deal with precision of language. For example:

I tell you I have a son. My wife gets pregnant. What are the odds the next child will be a boy? 1 in 2.

I tell you I have two children and one of them is a boy. What are the odds my other child is a boy? 2 in 3. Why? Because the events have already happened, there are four possibilities:
Boy Boy
Boy Girl
Girl Boy
Girl Girl

Since we know that the last pair of events didn't happen, it's not a possibility. Of the remaining pairs of events, the odds are 2 in 3 that my other child is a boy. It all depends on how you ask the question.

So, when you say things like "close to winning" we want that to have meaning. Your scenario implies that one must redeem the ticket in order to win. Otherwise, the person would have "won" before he ever gave way the matching ticket. If he doesn't give away a ticket but gets into a fiery car crash and is burned to death along with the ticket, did he come close to winning? Who knows? Was he on his way to redeem the ticket? Did he even know that his ticket matched the numbers drawn?

What's interesting is that many people do not go through these types of decision trees (at least consciously). What's even more interesting is that people such as yourself will continue to swap analogies rather than engage someone in their decision trees. The reality is that no matter what analogy you give me, I'm going to go through some sort of decision tree before answering. I'll be able to explain and defend my answer. Somebody else may look at it differently and defend their decision tree.

The truly fascinating part is that some people have no idea that on a subconscious level they have their own decision trees. They just call it a gut feeling. I have gut feelings, but to me that's a starting point not an ending point. I start with my "gut" and try to figure out why.

 

[dlorde]

What would be your answers to my 3 questions in message #26.

> Are all letters equal losers ?

Other than the winning letter, yes. All other letters lose.

> Are all letters infinitely far from the letter "i" ?

Spatially, no.

> Did ticket "j" came closer to winning than ticket "u" ?

Spatially, yes.

If you define a win in terms of how close an 'entry' comes to some winning position (in any arbitrarily defined way, e.g. 'closest to the bull wins') , then, of course, closeness is relevant, and it means something to say "I came closer than you". But if you define a win solely in terms of an exact numerical match, there is no 'close' to a win - by definition.

 

[nimzov]

How close was the woman to winning? Does "one ticket away" really mean anything?

I think it does.

For the woman to buy the right ticket and win you need only to consider a single event : not changing place with the man.

For another customer, at the other end of the country to win, you need to consider a lots of events involving traveling to go to this particular store and buying the ticket. In that sense in the situation you propose, the woman is (in term of event leading her to buy the ticket) closer to winning than is another person located 1000 km away.

 

[dlorde]

... I start with my "gut" and try to figure out why.

Me too. When a curry or a kebab isn't the cause, a 'gut' feeling is often the result of some mental processing you are doing but are not consciously aware of. This 'intuition' can be very effective, or very misleading (e.g. common-sense understandings of probability) and is probably not entirely rational, so it makes sense to try to establish how it arose. I don't think we can ever be certain, but a 'good enough' explanation makes us feel better and can avoid costly mistakes.

 

[A Laughing Baby]

The scenario you lay out is incorrect on its face. This is a binary outcome situation. You win or you lose. This is akin to asking "how close to being male/female are you?" Gender construct questions aside, it's a binary outcome. There's no in-between.

 

[nimzov]

The problem I am having with your posts is the lack of meaningful language. You're in a science forum asking questions where lots people with a science background or interest will answer. We deal with precision of language.

English is not my first language and I am doing my best to write with clarity. And I am not really asking questions, more hoping for opinions on the subject of closeness and proximity, that is what I wanted to explore. As Vorticity suggested maybe this is not a real mathematical or science subject, more like what Hofstadter explores.

So, when you say things like "close to winning" we want that to have meaning.

This is the question I was exploring, I do not have a definitive answer.

What's interesting is that many people do not go through these types of decision trees (at least consciously). What's even more interesting is that people such as yourself will continue to swap analogies rather than engage someone in their decision trees.

My two main analogies, were the lottery and the dart thrown at 5x5 grid.

 

[A Laughing Baby]

Your two main analogies are different in very important ways. A lottery is a binary outcome where the only input you have is whether or not to be entered into a random draw (in terms of placement) from a distribution. Throwing a dart and trying for only one letter is a binary outcome, but there is a literal physical proximity to victory that (ostensibly) you could change through your own physical actions. One can be changed, adjusted, the other cannot.

 

[dlorde]

I think it does.

For the woman to buy the right ticket and win you need only to consider a single event : not changing place with the man.
...

So who is closer, the woman who pushed in, or the guy on the other side of the country who got the ticket that was one digit different to the winning number, or the guy who stayed too long in bed and missed his place in the queue and would have bought the winning ticket, or the guy who bought a ticket in another shop that would have been the winner if the ticket distributor's van loader had tossed the ticket block into a different stack, etc. ?

There are potentially millions of ways someone else could have bought the winning ticket. Are they all 'closer' than some guy buying a losing ticket in a shop in another city? Perhaps there is there some universal hierarchy of 'closeness' we can agree on?

Or is it just that we like to hear stories that appeal to our non-rational and instinctive (or intuitive) misunderstandings of probability and causality? After all, why do people actually do the lottery at all?

 

[nimzov]

This is a binary outcome situation. You win or you lose. This is akin to asking "how close to being male/female are you?" Gender construct questions aside, it's a binary outcome. There's no in-between.

Binary outcomes have probabilities, being born with 13 toes is also a binary outcome. Is still has low probability.

 

[A Laughing Baby]

Binary outcomes have probabilities, being born with 13 toes is also a binary outcome. Is still has low probability.

Agreed. But having a higher probability of being born with 13 toes (and being born with 10 toes) doesn't mean you're closer to having 13 toes than any other person with 10 toes.

 

[nimzov]

Your two main analogies are different in very important ways. A lottery is a binary outcome where the only input you have is whether or not to be entered into a random draw (in terms of placement) from a distribution. Throwing a dart and trying for only one letter is a binary outcome, but there is a literal physical proximity to victory that (ostensibly) you could change through your own physical actions. One can be changed, adjusted, the other cannot.

The dart is trown randomly by whatever physical mechanism you like. The outcome is a random letter. It is binary in the sense that the owner of a ticket wins by having the right letter or does not win. I think that the analogy of spatial-proximity if the dart lottery an time-proximity of the lottery in the OP, as in the 4th choice of the poll is correct.

 

[SOdhner]

This is akin to asking "how close to being male/female are you?" Gender construct questions aside, it's a binary outcome. There's no in-between.

I'm not sure where to start with that, especially because I don't want to derail the conversation. Hmm.

I guess it all comes down to how you define the terms. Gender can be defined in a lot of ways, some of which are binary and some are not. Some are also thought of as binary in general because people fail to think about some of the less common situations.

Same thing here. "Close" and even "winning" can be defined in different ways. I picked the one where the machine spit the winning ticket out next, but that only applies by certain definitions of "close".

Actually, to really assess the situation I would need to take into account the method by which the machine picks numbers as well as the way the winning numbers are chosen. In any real and satisfying way I don't think any are "close", but I would still FEEL like I was close in some situations.

 

[A Laughing Baby]

ETA: regarding the gender thing, I agree, there is definitely in-between there, and I don't want to start a derail on the validity of the gender binary. Bad choice of example on my part. Mea culpa.


Ah, if the choice is in the letter and not in throwing the dart at the target, then the same thing holds. It's the same as cow pie bingo. You win or you don't win. If the dart (or cow pie) lands in the square adjacent to your chosen one, it doesn't make you nearer to winning. You still lost.

Probability only works in one direction--forward. I think that's where you're getting mixed up. It's very easy and tempting to look back and look at how similar your numbers were to the winning one, etc. But it doesn't change the fact that you didn't win, and you still had the same probability of winning as anyone else did. It just so happens that the random draw from the distribution happened to be closer to your position in the distribution than someone else's.