Isn’t everything absurdly improbable?

Question

Isn’t every event by definition improbable in the sense that each event precedes an infinite series of causes that could have (theoretically atleast) been different?

We think of someone winning five lotteries as extremely improbable and a rare event. But the probability of five different, specific people winning five lotteries is the same.

The probability of getting 100 straight heads is the same as any other sequence, yet the other sequences aren’t seen as improbable.

When it comes to other types of mundane events, such as a person walking across the street, isn’t that also an “absurdly improbable” event? Once you factor in the details and specifics of that event, such as the way in which the person is walking, what clothes he’s wearing, what time it is, it also seems very improbable event.

So when, exactly, is an event probable?

Answer 1

The probability of getting 100 straight heads is the same as any other sequence, yet the other sequences aren’t seen as improbable.

Yes they are, if you put a piece of paper in a sealed envelope predicting that sequence ahead of time. And conversely, the more coin tosses you make, the higher the probability that any given sequence will happen at random. That's why the total number of tests is as vitally important as the number of tests which meet your hypothesis.

If anyone thinks one is more improbable than the other, with a fair coin, then by definition they don't know anything about statistics or probability and therefore should be ignored. Sorry if this sounds harsh, but a "popular" opinion doesn't count for anything compared to actually knowing the subject.

Answer 2

Isn’t every event by definition improbable in the sense that each event precedes an infinite series of causes that could have (theoretically atleast) been different?

Specific events don't have probabilities.

Probabilities express a relationship between measurements of initial conditions and expectations of future measurements. This relationship is based off of past measurements of one or more countable frequencies in one or more corresponding finite samples.

If you ran the experiment many times with the same measurements of initial conditions (and only the same measurements, not a metaphysically identical experiment in which all the things you don't know are also the same), what fraction of the outcomes will fall into the category being evaluated? That fraction is the probability.

We think of someone winning five lotteries as extremely improbable and a rare event. But the probability of five different, specific people winning five lotteries is the same.

Yes, if you remember to mentally translate the ambiguous tense of the present-continuous winning to a future tense. Common use of probability in the past tense to describe past events is misleading; the correct tense is the past progressive would have been.

When it comes to other types of mundane events, such as a person walking across the street, isn’t that also an “absurdly improbable” event? Once you factor in the details and specifics of that event, such as the way in which the person is walking, what clothes he’s wearing, what time it is, it also seems very improbable event.

So when, exactly, is an event probable?

When the "event" is defined as the category of outcomes of a hypothetical future experiment whose initial conditions have the values you (or your sources) have already measured, the event has a probability.

(Note all measurements have limited precision, so all predictable outcomes of measurements are categories of outcomes. If I predict a measurement of length on a meter stick with millimeter precision, I'm predicting a category of outcomes that would be measured within half a mm of the predicted value on a meter stick with greater precision.)

The probability of the category of outcomes is the ratio of how many hypothetical future experiments have outcomes that fall within the designated category, divided by the hypothetical large number of experiments whose initial conditions have the values you (or your sources) have previously measured. Making the designated category smaller or bigger makes the probability smaller or bigger respectively, which is what you seem to be getting at: for any experiment with a given set of initial condition measurements, the probability of a category of outcomes approaches zero as the size of the category becomes arbitrarily small.

If the "event" is not so defined, probability isn't one of the characteristics it can have.

Answer 3

Yes, it's extremely improbable for practically anything to happen in the exact manner that it does. You are correct.

But then what do you do with that information? So it was very improbable for those specific five people to win the lottery, so what? Is that actionable information? Not really.

But if one specific person won the lottery five times, that is actionable. Because we would then suspect that particular person is cheating. The observable results are improbable given the null hypothesis that the person is not cheating, but become far more probable given the hypothesis that the person found some way to cheat the lottery. So, the probability of that hypothesis increases, according to Bayes' rule P(H|O) = P(O|H) P(H)/P(O).

Let's assume we're dealing with 1 chance in a million lotteries here. P(O) is very low because it's so unlikely to win the lottery five times. Crucially, P(H) (probability they're cheating) is low, because most winners of a single lottery aren't cheaters, but with five wins it's not nearly as low as P(O). And P(O|H) is 1. So P(H|O) is quite high, close to 1.

For five different random people winning the lottery, though it is equally unlikely, there's no interesting hypothesis for us to update. The chance of them all cheating is not increased in any remarkable way. Unless, perhaps, the five people share an interesting property, such as being friends and family members of a lottery official.

In the five-winners case, P(O) is about the same as in the single-person five-time winner case. P(H), however, is much lower here, because the chance all five of them cheated is the probability each one cheated (very unlikely, same as in the previous example) to the power of five. P(O|H) is still 1, but in this case that wouldn't be enough; P(H|O) is still much less than 1%.

Anyway, that's the difference. Practically any event can be viewed as unlikely, but few events are "actionable" in the sense that they would cause a Bayesian update to favor an interesting hypothesis.

Answer 4

This kind of problem arises when there is an attempt to apply probability to the real world. The theory of probability is about precise sample spaces and probabilities and events. The real world, on the other hand, isn't like that.

In order to try to apply probability to the real world, it's necessary to describe precisely the state of knowledge about an event before and after it happens. For example, if a fair coin is flipped, most people would agree that it has an equal chance of landing heads or tails. But if we knew precisely which side was up to begin with, and exactly how much force is exerted on exactly which part of the coin, as well as the structure that it lands on, then physics could say with very high probability exactly which side it will land on.

So: Given full knowledge of the universe at time t, the future is virtually determined and not probabilistic at all. Given partial knowledge, that can be quantified well, we can estimate the probability of future events given the present as we understand it.

So there is no god-given probability associated with any event in the real world; its probability depends on how much is known about it before it happens.

It does not make sense to claim that a real-world event has a given probability without referring to some state of prior knowledge.

Answer 5

The questions you need to ask are "improbable given what?", and "improable relative to what?"

• Suppose I flip a coin. The probability that it will be heads, is 0.5.
• Suppose I flip 10 coins. The probability that they will all be the sequence HTTTHTTHH is 2^-9.
• Finally, suppose that I flipped 9 coins, and they came up HTTTHTTH. The probability that my next coin flip, ignoring what came before, will be heads is P(x_9 = H) = 0.5. Further, the probability that my total sequence will be HTTTHTHTHH, given that I already flipped HTTTHTTH, is P(HTTTHTTHH | HTTTHTTH) = 0.5.

When asking the probability of a specific event happening, where that event depends on billions and billions of other things happening first, don't disregard the fact that those billions and billions of things already happened¹. You don't want P(universe reaches the point where this particular 1997 quarter is minted and ends up in my pocket AND I flip it AND it comes up heads)². You want P(heads | all that). Yes, all prior distributions for things in the universe are extremely small, but once you condition your distributions on all the prior events we don't care about, the giant numbers mostly cancel out and we end up with useful probabilities.

We think of someone winning five lotteries as extremely improbable and a rare event. But the probability of five different, specific people winning five lotteries is the same.

"Specific" is the key word there. If the probability of winning is 10^-6, and you have 10^6 people with 5 draws each, the probability that each draw will have at least one winner is about 10%. If you want your intuition to match the math, suppose that you named 5 specific people in advance, and they all won. You'd consider that just as suspicious as one person winning 5 times in a row (and it is exactly as improbable if you specify which person wins which draw. A bit (~3125x) more likely if you don't specify which draw each person wins)).

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¹ And were observed, but more on that later.

² Note that in this case, we don't need to have observed which specific sequence of events happened so as to lead to that quarter being in my pocket, just that some such sequence happened. Actually computing the probability of the end result would require knowing the probability of the various sequences summing over them, so it's really convenient that we won't need that number. But to head off related questions, it's probably worth noting that the result of that sum will be astronomically higher than the prior probability of any specific sequence in it.

Answer 6

Maybe you should read about logical operators like AND in the context of statistics: there is absolutely nothing paradoxical about what you claim.

when is an event probable

let's suppose when its chance is 50% or more.

Any two coin tosses will result in an event that is "improbable". Indeed, I am less likely to reply to a question than not. Yet no-one is very impressed when I do.

Answer 7

The field of algorithmic information theory gives an answer to this question: we expect "random" events to be complex. On the other hand, simple events are not likely to arise from random chance, so they are surprising.

Here "complex" is measured by compressibility: how concisely we can describe the event. For example, a string of 100 heads can be described very succinctly. However, a more typical result of 100 fair coin flips will not be compressible at all, and we will have list out the entire sequence of results manually. If the coin flips are actually random, we do not expect to get any simple-to-describe result such as "alternate heads and tails", "heads at each Fibonacci number and tails otherwise", etc. Any such result would be "surprising", or, we can say, "improbable".

Answer 8

I would dispute the notion that all events are equally improbable or absurd in probability. While in an absolute sense, any precisely specified event may be seen as improbable, that is not how our minds apprehend events in the real world.

Our intuitions about probability are contextual - we perceive events against a backdrop of patterns, regularities, and hierarchical abstractions. Walking across the street seems mundane because it aligns with the common abstractions of 'human behaviour' and 'everyday activities'. It is embedded within a known context.

Winning the lottery five times disrupts our established patterns massively. It deviates from what we expect, grabbing our attention. The greater the deviation, the more improbable and meaningful an event becomes within its context.

So I would argue that our minds are attuned to hierarchical thinking about probability based on conceptual patterns, not just specific details. We do not treat all events as equally improbable, because we inherently recognize that some occurrences align with intelligible contexts and regularities, while others are radical departures demanding explanation. It is these improbable disruptions that reveal the gaps in our knowledge

Answer 9

I don't think you can discuss probability in the real world without at least a passing reference to entropy.

Imagine a sealed airtight container filled with a gas. If you could take a snapshot of all the states of all the molecules, the likelihood that they would return to exactly that configuration again in the next five minutes, is the same as that they would all simultaneously end up in a particular corner of the box in the next five minutes.

However there are astronomically more ways that molecules can be evenly spread out in the box than that they can all be in one corner.

Entropy in one sense is a measure of how things can be distributed throughout space.

The human aspect is that we classify things. We can classify many different arrangements of molecules as being the same when in fact they are all completely different. E.g. We can have "the molecules are evenly spread out" as a category and "the molecules are all in one corner" as another. Naïvely these are equal categories whereas in truth they are vastly different.

The probability that a person crosses a street is contingent on a person existing, and a street existing for them to cross. Those are contingent on there being a planet that supports human life orbiting a sun that provides energy, and so on.

In terms of entropy, if you have enough time and enough matter that interacts and combines according to certain rules then most things that are possible within the rules will happen somewhere and somewhen.

Conclusion

Paradoxically, within a suitable dynamic system, any event, no matter how improbable, is nevertheless inevitable given sufficient time.

Answer 10

Of course the reality we inhabit is fantastically unlikely. I thought that's obvious after a casual look around. ;-)

Answer 11

The probability of getting 100 straight heads is the same as any other sequence, yet the other sequences aren’t

Other sequences are seen as improbable. At least they are for anyone who has studied probability.

Something that you haven't taken into account is the number of coin flips.

If I make 100 tosses, the probability of getting 100 straight heads is approximately 0.0000000000000000000000000000008 - For practical purposes it is zero.

If I make an infinite number of tosses, the probability of getting 100 straight heads is 1, i.e. it is certain.

In order for your question to be answerable, you must define your sample space and specify the number of trials.

Answer 12

Given the premise "there is a universe", the probability of a mammal calling itself Joe Smith walking on two legs across the bit of space I know as "6th Avenue" at the moment I reckon as "12:33pm on July 31st" is ludicrously, incalculably small.

Conditioned on the premise "there is a universe, and it has matter as we know it, and planets, and cities, and streets, and human beings, and calendars, and Joe Smith — and Joe Smith works at 6th Ave and 33rd Street from Monday to Friday, and has a lunch break at 12:30, and likes Japanese-Mexican fusion", the probability of Joe Smith walking across 6th Avenue at 12:33pm on any given weekday, on the way to Takumi Taco, is not so small after all. The "unlikelihood" of the premise and the conclusion do a good job of canceling one another out.

Probabilities are only usefully considered in light of information we already have. Thinking about a probability conditioned on nothing doesn't get you much of anywhere.

Answer 13

For any event or definition of an event where you're adding on successive conditions, where those conditions are either mutually exclusive (at a given time in a given place, only one object can occupy that space, and someone cannot be on a flight to Melbourne and floating through the Horsehead Nebula at the same time) or the negation of that condition is true for at least one thing in your potential sample, the probability of all of those conditions being true will always go down, because you're multiplying probabilities that are less than 1 together or creating a set of conditions that cannot happen together.

When observing a real world event, the observer can describe that event using as many or as few of the actual preceding conditions as they like, generally removing all of the conditions for which we do not observe a causal link (we observe that gravity is 9.81m/s^2 on every day where at least one person wakes up, but most other things also happen when that's true, and we have no known laws implying one should follow the other).

A "probable" event then is, in general, one with as few conditions as possible, where each individual condition is itself "probable", and where those conditions are not mutually exclusive with each other. This is why abstracting away conditions makes something more likely, i.e. what are the odds that my left foot will be directly over the 52th sidewalk pad of North Avenue Chicago at 5:12pm on March 19th 2034, vs. what are the odds that anyone walks down any street at any time in the future. So there is at least one probable event in the universe: Somewhere, sometime, something will happen!

Answer 14

You throw a coin. Before it lands, you describe three results: It ends up "heads", it ends up "tails", or something entirely unforeseen happens (just in case to cover all possibilities). The first two have probabilities about 49.999999999%.

Now you throw the coin, and look for the event "coin falls on the table and comes to rest after between 1.3746 and 1.3747 seconds". That is much much less likely. There are also a few thousand similar events, all with very small probabilities. But the add up to a probability very very close to 1. You know have a very unlikely event.

Now you throw the coin ten times in a row, and look for the event "coin rests after 1.3746 to 1.3747 seconds on the first throw, 1.0853 to 1.0854 seconds on the second throw, ..., xxx on the tenth throw". This is incredibly unlikely to happen. But there are gazillions of similar events with a total probability of close to 1. (You might have a heart attack after the seventh throw, that's why the total probability is not 1).

So you can describe events with arbitrary small probabilities, but events that have no interesting meaning. You need to look at interesting events.

Answer 15

I suck at statistics. But perhaps my simplistic explanation will be useful if correct. Please comment if I need to fix this.

Suppose you flip a coin five times, and the coin has two faces, tails (0) and heads (1).

For example, I could flip the coin five times and end up with this sequence:

01101

So now let's ask two different questions about this experiment.

1. How likely is it that we achieve the sequence 11111 versus all other sequences?
2. How likely is it that we flip 5 heads?

This sounds like the same question, but I suggest to you that it is not. The underlying question to ask in both cases is how many ways there are to achieve that outcome versus the other competing outcomes?

For 1, it's simple. Every possible sequence of 5 flips is unique, so they are all equally probable. That is to say there is only one way to flip 01101 - one tail, then one head, then another head, then a tail, then a head. It's an identity.

For 2 it's different. How many ways are there to flip exactly one head? five: 10000 01000 00100 00010 00001

But how many ways are there to flip five heads? Exactly one: 11111

We can definitively say, having written out all applicable outcomes, that flipping one head is 5x more likely than flipping all five heads.