The Probability of an Event

[erobz]

So, I have these two seemingly wildly improbable events that I observed. I once saw my parents next door neighbor waiting for a subway in DC. We both live 100's of miles away, and we see each other waiting for this train while randomly sightseeing. To me it seems to be a tiny probability, even if you tell me it's not...it's always going to be special! Another event happened to me while I was waiting for a medical test. A worker read out my previous address as they were checking in someone for something while I was in the next room! The events were separate in time some 15 years.

Here is where I'm confused, me knowing that it was my previous address being read makes the latter event unique. The girl next door had no idea we lived at the same house. I think about some of the things that lead up to that event, sequentially (well within the limits of human cognition) and it seems to me as though every moment/reality of my existence led to this, and I conclude I just witnessed a near zero probability event.

So which is the less likely event? Even though I find the first event to be absurd, at least 15 year of seconds (meaningful human time scale in my estimation) - moments of my and all of their life had passed getting to this event. So, I can only conclude that this event, though seemingly more likely (being that we only leave a few towns away from each other) absolutely makes the first event probabilistically insignificant. Then again, the same can be said for the older neighbor that I met on the train (then in their 40's with a family, and me just in my 20's with my girlfriend- now wife -just starting out). Every moment in their lives led to that point in time (including moving next to my parents in the first place, my parents moving there, etc... and we shared the absurdity together as we waited for the train! So where does the clock start to calculate a probability like this- perhaps the beginning of the universe itself? Basically, I'm stuck...every event is either the most absurd thing that has ever happened (my perspective) or no significant event at all (their perspective-ignorance is bliss).

What are the fundamentals of probability that I'm missing? How do mathematicians actually analyze this stuff?

 

[FactChecker]

So, I have these two seemingly wildly improbable events that I observed. I once saw my parents next door neighbor waiting for a subway in DC. We both live 100's of miles away, and we see each other waiting for this train while randomly sightseeing. To me it seems to be a tiny probability, even if you tell me it's not...it's always going to be special! Another event happened to me while I was waiting for a medical test. A worker read out my previous address as they were checking in someone for something while I was in the next room! The events were separate in time some 15 years.

That is bazaar.
But there are some details that you should ignore.

Here is where I'm confused, me knowing that it was my previous address being read makes the latter event unique. The girl next door had no idea we lived at the same house. I think about some of the things that lead up to that event, sequentially (well within the limits of human cognition) and it seems to me as though every moment/reality of my existence led to this, and I conclude I just witnessed a near zero probability event.

So which is the less likely event? Even though I find the first event to be absurd, at least 15 year of seconds (meaningful human time scale in my estimation) - moments of my and all of their life had passed getting to this event. So, I can only conclude that this event, though seemingly more likely (being that we only leave a few towns away from each other) absolutely makes the first event probabilistically insignificant. Then again, the same can be said for the older neighbor that I met on the train (then in their 40's with a family, and me just in my 20's with my girlfriend- now wife -just starting out). Every moment in their lives led to that point in time (including moving next to my parents in the first place, my parents moving there, etc... and we shared the absurdity together as we waited for the train! So where does the clock start to calculate a probability like this- perhaps the beginning of the universe itself?

A lot of that is not relevant to the probability. You would have been surprised by seeing a neighbor of your parents no matter who it was or how that particular person became a neighbor. In any case, your parents would have a neighbor and you would be surprised, so there is nothing lucky about it being that particular person.

Basically, I'm stuck...every event is either the most absurd thing that has ever happened (my perspective) or no significant event at all (their perspective-ignorance is bliss).

What are the fundamentals of probability that I'm missing? How do mathematicians actually analyze this stuff?

Maybe a person could make some rough estimates based on the size of your town, the rate of people from your town visiting DC, etc. I think that any estimates would be very rough with several orders of magnitude of uncertainty.

 

[Dale]

I conclude I just witnessed a near zero probability event

If you accept real-valued random variables then every event is zero probability. Zero probability does not mean impossible.

Before you can determine the probability of an event, you need to determine the sample space. I don’t know what that would be.

 

[erobz]

A lot of that is not relevant to the probability.

I guess I just want to understand a bit more why? Where is the delineation, if I'm interpreting @Dale and yourself correctly carving out a sample space is a ludicrous proposition if one thinks too long.

You would have been surprised by seeing a neighbor of your parents no matter who it was or how that particular person became a neighbor. In any case, your parents would have a neighbor and you would be surprised, so there is nothing lucky about it being that particular person.

Yeah, completely fair. I only came across this when I tried to delineate what events where necessary in the chain. Meeting anyone I know on a subway in a distant city would have been just as uncanny. But I'm still stuck trying to decide how it ended up so and so after the fact. I have to go from some set of events to another and it's all very fuzzy (dare I say computationally vast) rather quicky on how we got there.

 

[erobz]

If you accept real-valued random variables then every event is zero probability.

Well, I guess I don't because the continuum seems to be illusory? At any rate I don't think that is an issue in practice because I don't believe I could tell the difference between a 1/10^6 and a near zero-probability event (i.e. near zero and zero probability events seem to be indistinguishable to a human)...I mean yeah... is it the coalescence everyones' seconds, picoseconds, or is it whatever is the smallest interval of time these days(zeptoseconds)?

Before you can determine the probability of an event, you need to determine the sample space. I don’t know what that would be.

I think that's what I'm trying to understand. How do we carve out a boundary on all these seemingly dependent probabilities.

 

[erobz]

Maybe a person could make some rough estimates based on the size of your town, the rate of people from your town visiting DC, etc. I think that any estimates would be very rough with several orders of magnitude of uncertainty.

I would say that's a bit arbitrarily starting the "clock" at person A and B know each other etc... It's somewhat unsatisfying (and forgive me for using this word on the is site) philosophically? Computationally, lets go... but its unsatisfying. It one of those things that just over my head.

 

[FactChecker]

and it's all very fuzzy (dare I say computationally vast)

I agree. I don't see how it can be done rigorously. This reminds me of the Drake Equation, which multiplies a lot of factors together to think about the number of extraterrestrial civilizations in the Milky Way Galaxy, It is not for an accurate calculation.

 

[Dale]

the continuum seems to be illusory?

I think that is pretty wrong in terms of modern physics, both in experiments and in experimentally validated theories. But maybe that is a topic for a different thread.

I think that's what I'm trying to understand. How do we carve out a boundary on all these seemingly dependent probabilities.

Well, you were surprised. So the question is what other circumstances would have surprised you as much or more? Would you have been equally surprised to see your parents other next door neighbors? How about your parents? Or your spouses parents, or their neighbors? How about siblings or cousins or their neighbors? How about if you saw them in a train station in a different city, or at a bus stop, or an airport, or on a beach or in line at a movie? What circumstances would generate such surprise in you?

We had a similar thread here: https://www.physicsforums.com/threads/an-observed-extreme-probability-event.1054188/

 

[erobz]

Well, you were surprised. So the question is what other circumstances would have surprised you as much or more? Would you have been equally surprised to see your parents other next door neighbors? How about your parents? Or your spouses parents, or their neighbors? How about siblings or cousins or their neighbors? How about if you saw them in a train station in a different city, or at a bus stop, or an airport, or on a beach or in line at a movie? What circumstances would generate such surprise in you?

I think our surprise is just something we think is rare. I guess it's all very...meh in terms of concrete objectivity. Still if it's simply the recognition of a complex stream of information that is surprise, I can't help but think its governed for lack of a better term. If we could somehow plot historical timelines of every individual and add them with a Fourier Series shouldn't they sum (perhaps in some highly non-linear way) to reality? I know you not from Adam, but (like it or not) our realities are intertwined (albeit ever so lightly) as far as I can tell.

I guess the measure of "reality" is the deal breaker, as there might be nothing of substance there to "measure". I was just trying to reconcile that things like peoples "relative positions" in spacetime make up at least part of what we consider " a reality"- whatever it is. Their positions being significantly more measurable than their "reality" gave me some false sense of hope.

 

[erobz]

I'm going to tap out because I know if I keep it up, I'm going to get kicked out for being borderline metaphysical.

If anyone wants to set some independent probability boundary carved out and calculate I won't argue...promise. I think the current consensus is "sure, it's a bit fuzzy and you're not crazy for thinking so". I'm ok with that.

 

[FactChecker]

I have a couple of opposing thoughts on this:
1) Any person witnesses a million events every day. There might be a lot of strange coincidences that are just not noticed.
2) To have two events that are so noticeable that they hit you between the eyes seems very unlikely. I can't say that I have experienced anything like it.

 

[Hornbein]

The number of events any person witnesses in a lifetime is finite. This however is not useful so we may as well use the continuum as that is what we are used to. What we care about are classes of events. What is the measure of some class?

Someone witnesses an outlandish event. Sometimes circumstances allow us to calculate a probability, sometimes not. In the latter case, what can we do?

A basic problem is, what is the measure of the set of outlandish events witnessable in your lifetime? We have no idea, and I say that moving in that direction seems hopeless. I suggest the time-honored practice of limiting one's scope to something tractable so that progress can be made. Specifically I suggest coincidences. That is, two independent events in the same class that happen within a limited period of time.

In a grocery store in northern Michigan my bill came to $77.77. The checkout lady said, "the person just before you had the same total." Unquestionably a coincidence. Now what's the class? This is entirely subjective, a matter of opinion. Would any total do? Does the total have to be of all the same digit? Or is only $77.77 acceptable? I'd just not worry about such an unanswerable question, estimate the answer for each possibility, and let the reader choose. Done.

What one finds is that the answer is different for me than it is for the checkout person. I witness far fewer checkouts than does she. Same coincidence, different sample spaces, but good estimates can be made for each case.

Note that the interval between the two events is key. Having it be minimal makes all this more impressive. If the checkout person had seen $77.77 ten years ago it wouldn't amount to anything.
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This particular example is distributed discreetly in time. Most coincidences are not like that. Suppose instead that there were a second checkout stand that also got $77.77 at about the same time. The smaller this interval, the less likely the whole shebang is. And we can still make good estimates.

The nice thing about this is that we can exclude all potential events that didn't happen. Those complicate things too much. Instead we're calculating conditional probabilities. Given a checkout that totals $x, what's the chance that the next is also $x? Given a checkout that totals $77.77, what's the chance that the next is also $77.77? And we haven't given up completely on the unconditional probability of this pair of observations. Our conditional probability is an upper bound on that.
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Usually we can't estimate a probability this accurately. But all is not lost. We can use the empirical distribution. Suppose you witness some event. You have never before seen an event in that class so this captures your attention. Then later you witness another event in that same class after some interval. If you convince yourself the events are independent then we can make a reasonable estimate of the probability of such a coincidence. Once again we're calculating a conditional probability and can exclude all potential events that didn't happen. The class is well-defined : events never before seen in one's lifetime. The resulting estimate depends on the length of one's life, how observant one is, how much novelty one experiences in one's daily life, and the interval between the events.

To simplify and remain objective I suggest fixing that interval before hand, such as a maximum of one day. Or instead of events never before seen in one's lifetime it could be events not witnessed in the past X years. Whatever you like. It is good practice to choose these numbers before the study begins. So the quantity we calculate is one day divided by X years. Easy as pie. It's subjective whether the two events are in the same class but there is no avoiding that.

Perhaps the most interesting result from all this is that two of these quantities are known and two unknown. The knowns are the length of one's life for each event and the interval between the two events. Though we can't estimate the two unknowns, we can estimate their product. By knowing how many such coincidences a person has observed we can estimate how observant they are times the amount of novelty in their life. Seems informative to me.