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What is the real probability that a person will be murdered tomorrow somewhere in the world? It seems like there should be a right answer to this. In fact, most of us would bet tens of millions of dollars that atleast one person will be murdered tomorrow.

But what is the correct probability then? Is it 1? As in it’s guaranteed to happen? Why would one think it’s guaranteed to happen? It would likely be because someone was murdered every day for many years. But this is nothing more than an inductive inference. Where does probability come here or even needed as a concept to talk about whether there will be a murder somewhere in the world?

Let’s look at a seemingly harder example. What is the probability that there will be a murder tomorrow in your city? As you can imagine, it would depend on how you interpret probability and what class of past events you use. Do you look at the murder rates of your city or the country? What about the time period? What should that be? What makes one interpretation more valid than the other? If there is no clear answer to this, does this mean that the concept of probability is inherently subjective?

I can imagine a retort being that it is subjective but one can make models based on certain classes of past data and see which models predict the future better. But that doesn’t tell you what the “correct” probability is. It just tells you what model works the best. Where is the concept of probability actually needed here?

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  • Another tangential thing: even if you get a precise probability number, that's usually only half the story. You will then want to compare that number to others, interpret the number as "a lot", "not a lot", or "so much we need more police" - and where to set that threshold is also a difficult problem. "Probabilities" are not the entire story, there is an interpretation or decision step that follows and which is separate. Having the raw probability number in itself doesn't say much. Suppose the probability of being killed is "0.7" for this town, but all other towns you can go to have "0.8"?
    – Frank
    Apr 2, 2023 at 15:08
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    Yeah, and that's where it's going to be real subjective. The numbers are one thing, but the decision you take based on those numbers one way or another is going to rely on additional inputs, which are likely going to be "subjective".
    – Frank
    Apr 2, 2023 at 21:59
  • See also here. Apr 3, 2023 at 2:46
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    That doesn't sound different from many other metrics. They are all subject to context. Fuel mileage, BMI, or CPU benchmarks, for example. Subjective? Maybe, but they are also useful.
    – DKNguyen
    Apr 3, 2023 at 14:28
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13 Answers 13

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In fact there is no way to define unique objective probabilities for anything that happens (or not) in the real world. Every definition of probability requires model assumptions that cannot be verified (one can try to falsify some of them, which occasionally works, but even this usually incurs error probabilities, so any falsification won't be 100% reliable). So every definition requires some judgement that will ultimately involve subjective components.

What is usually referred to as "objective" probability is called like this either because it refers to data generating processes in the real world even if there is no objective knowledge how exactly they play out, or ("objective Bayes") because it refers to absence of personal judgement, which in practice is not possible, but rather seen as an ideal to aspire to.

We have argued that the subjective/objective-distinction is often unhelpful in statistics, and also often misunderstood, as the term "objectivity" means different things to different people or in different fields in this paper:

Andrew Gelman, Christian Hennig, Beyond Subjective and Objective in Statistics, Journal of the Royal Statistical Society Series A: Statistics in Society, Volume 180, Issue 4, October 2017, Pages 967–1033, https://doi.org/10.1111/rssa.12276

"Where is the concept of probability actually needed here?" You can find quantifications of uncertainty helpful for orientation even if they are not grounded in "objective truth". Such quantifications can also to some extent be empirically tested.

Your discussion of the murder rate example is known as Reference class problem by the way, and it is clear that compromises need to be struck in practice, i.e., you want a large enough reference class so that there is enough data, but a small enough one so that information is used as precisely as possible. This requires some subjective judgement, however that doesn't make it worthless.

PS: One thing that I think is special about probability (as opposed to other scientific concepts for which mathematical models exist) is that probability is always at least partly about something counterfactual, about what could happen that actually doesn't happen (and how likely that would've been). This makes it much harder to think about probabilities as "objective".

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    @ChristianHennig But, if I count the number of defective beer bottles out of my assembly line compared to the number of beer bottles I produce, can I not calculate an objective probability of a defect?
    – Frank
    Apr 2, 2023 at 14:04
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    @Frank The first thing is, what is your definition of "objective"? Anyway, the standard frequentist idea is that counting gives you a relative frequency; a probability is a limit of relative frequencies for overall number of bottles converging to infinity, which is an idealisation. You can calculate an estimate of the "true probability" but not the probability itself. Apr 2, 2023 at 14:12
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    @Frank An alternative idea is that all the produced bottles are the complete population, however then a probability statement would not regard the population as it is (as this is deterministic), but rather "random" draws from the population, and then standard frequentist probabilities would concern a number of draws converging to infinity (there is no way to objectively secure that a draw is "really random"). There's no way around idealisation, and probabilities are not directly observable. Apr 2, 2023 at 14:14
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    @ChristianHennig I see. I guess I am too accustomed to using concrete "probabilities" in a frequentist sense, ie confusing the relative frequency with a "probability" for practical purposes :-)
    – Frank
    Apr 2, 2023 at 14:17
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    @Frank As I said, the term "objective" is ambiguous, however I'd agree with you that it can make sense to call the observed relative frequency "objective". But the observed relative frequency isn't the probability. The probability is defined by idealisation, which among other things involves idealising away differences between different cap-actions of the machine. You need judgement to decide what you see as "repetition of the same experiment", and then, as said before, probability involves the idea that this could be repeated infinitely often. Apr 2, 2023 at 14:27
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What is the real probability that a person will be murdered tomorrow somewhere in the world? It seems like there should be a right answer to this.

I would accept that there is a true, correct, mathematical probability for that, which is very close to (but less than) 1. This is an hypothesis that I cannot prove, but I believe it anyway.

Why would one think it’s guaranteed to happen?

One might think so because one has a flawed understanding of probability.

Or one might deploy the word "guaranteed" in a probabilistic sense, meaning that they believe the probability to be so close to 1 that the alternative is not pragmatically worth considering.

It would likely be because someone was murdered every day for many years. But this is nothing more than an inductive inference.

I think that's a straw man. Yes, if indeed one has that statistic to draw on (do we?) then by inference one could predict that there will also be at least one murder tomorrow. But that's no basis for assigning probability 1 to such an event, nor any other specific probability. Anyone coming to such a conclusion on that basis would fall into the "flawed understanding of probability" group.

Where does probability come here or even needed as a concept to talk about whether there will be a murder somewhere in the world?

Quantitative probability is not particularly useful in this area, because for most practical purposes, the exact mathematical probability of real-world events is not computable. On the other hand, qualitative probability is foundational. We can often estimate fairly well how likely or unlikely one event is either on its own or relative to an alternative. We make such estimates all the time, and rely on them in making decisions.

What is the probability that there will be a murder tomorrow in your city? As you can imagine, it would depend on how you interpret probability

Why?

and what class of past events you use.

You seem to be assuming an inferrential approach. This is certainly the statistician's method for estimating the probability of an event, but it gives you only an estimate, often with an associated confidence estimate, not a true, exact probability.

Do you look at the murder rates of your city or the country? What about the time period? What should that be? What makes one interpretation more valid than the other? If there is no clear answer to this, does this mean that the concept of probability is inherently subjective?

No, it means that statistical inference is qualified by a set of subjective assumptions. The methodologies of statistical inference are related to probability, but they do not compute true probabilites in the sense you seem to mean.

I can imagine a retort being that it is subjective but one can make models based on certain classes of past data and see which models predict the future better.

Yes. That's more or less a description of science.

But that doesn’t tell you what the “correct” probability is.

True. So what?

Where is the concept of probability actually needed here?

There in particular, mathematical probability is the framework by which we understand statistical inference. However, exact numeric probabilities of real-world events indeed are not needed for most purposes.

No doubt one can go a very long way in philosophy without involving the concept of probability. Eventually, however, one has to come to grips with the fact that in the real world, we often* need to make decisions without sure, advance knowledge of all the outcomes of each choice. A rational decision under these circumstances must be based on some evaluation of the likelihood of some of the outcomes of each alternative.

It does not matter whether the exact mathematical probability of various outcomes under various conditions is known or even hypothetically computable. I use probability if I judge, for example, that because I am plastered, the risk of being in an automobile accident or being arrested for DUI is is too great, and the consequences too severe, for me to accept driving myself home as a rational decision. (And I hope that I retain so much reason in that diminished state.)


*maybe always

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  • IMHO @John Bollinger has made a good point: there is no value in knowing the probability as a means of guiding action. If I am driving along and another motorist does it does not signal a turn, what is the probability that they will turn? My estimate of that probability (and estimates are the best we can do) should determine what I do. Apr 5, 2023 at 1:46
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All probability models are subjective, but some models are more useful than others

There are various ways you can come up with a probability for an event. Bayesian vs frequentists are the 2 main approaches to statistical inference, and within each of those, one can look at different variables or use different formulas. We can also map a continuous variable onto one of many probability distributions to predict the future value of that variable.

For each of these methods, we can see how reliable it is historically, and the more reliable it has been in the past, the more reliable it's likely to be in the future, and the more sense it would make to base any decisions we make on that.

So the method that's proven to be the most reliable is the most useful and is the one we should use (there may be some cases where we'd pick one model above another may be more useful, depending on the factors at play, but "most reliable = most useful" is a good rule of thumb).

This is inductive inference, yes, but not "just" inductive inference. Induction is universally considered to be a fact (there is some philosophical discussion questioning our reasons for believing this, but we still consider it to be a fact).

It is, however, not always easy to come up with a reliable measure, and there are plenty of ways you can go wrong. This is why statistics degrees and statistician occupations exist, and why there is additional knowledge sharing and why results are often checked by others (especially if it's used in science).

What is the "correct" probability?

That's arguably either 1 or 0 for any instance of an event, depending on whether or not the event happens.

But we don't know the future, so the best we can do is stick to our models.

Will someone die tomorrow?

For this problem, I might consider the number of people who die per day, model this as a probability distribution and then determine the probability of getting 0 on that distribution.

This shouldn't be too hard, although we'll need to take into account the hard cutoff at 0, which could cause problems if not careful. Also, the death rate changes over time, but we could limit the results to somewhat recently.

But this isn't a maths forum, so I wouldn't try to actually model this here.

There are most definitely other ways to solve the problem too, but you should end up with something very close to 1 (I'd say the probability shouldn't be equal to 1, but some reasonable methods may end up with that result).

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Is all of probability fundamentally subjective and unneeded as a term outright?

Of a murder somewhere in the world tomorrow, "…what is the correct probability then? Is it 1?"

No, it is less than 1, because the question asks about the real world. To give another example, the probability of the sun rising tomorrow is less than one. Granted, the prediction of tomorrow’s sunrise is a safe bet, but the probability of the event is still some sliver less than one.

In a hypothetical world, governed only by assumptions run through formulas, the solution might in fact be exactly 1. It just depends on the assumptions.

When you mention inductive inference, you are on the right track. In an inductive analysis, probability is all that one ever gets, never certainty.

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  • Related: stats.stackexchange.com/questions/590861/…
    – Galen
    Apr 2, 2023 at 23:35
  • When you say that it is less than 1, what is the value then? And what interpretation of probability are you following? What past series of data are you looking at to answer this question? If there is no clear answer to these, then your interpretation remains subjective
    – user62907
    Apr 3, 2023 at 0:02
  • A probability of 1 does not mean "certain", there can still be an exceptional event of probability zero. But according to ourworldindata.org/homicides the annual number of homicides in the world is around 400000, so I would estimate the probability of some murder somewhere tomorrow as 1. Apr 3, 2023 at 5:14
  • @thinkingman. I do not know the value < 1. As I say, induction is always an estimate. Here, we would have to know the exact age of the sun and the exact moment that it would cease to exist. Apr 3, 2023 at 17:03
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    @MarkAndrews If you're choosing a random number over the interval [0,1], the probability that it's irrational is 1. But it's still in the event space that you could choose a rational number.
    – Idran
    Apr 3, 2023 at 21:50
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There are two parts to an analysis using probabilities:

  1. Create an experiment to determine the probabilities of an event (coin flip). These experiments determine the probability that a coin will land heads or tails. The experiment can take the form of data gathering (The number of murders per day in the US). This part should be as objective as possible and in most cases is objective. The only subjective part in the post would be the definition of "murder"

  2. Using the results of part one to make a prediction about a future event. This is the part that is nearly always subjective. If the coin in the experiment is 50/50, what makes one person choose heads and another choose tails? That decision is subjective.

Suppose that experiments and data collection can't be done. For example, determining the probability of intelligent life somewhere other than Earth. This is where subjective probability is useful in making predictions.

So not all of probability is subjective. It's only subjective in its use to predict future events and the future is notoriously unpredictable.

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  • Your choice to use frequentist statistics would be subjective. How you conduct the experiment would be subjective. How you analyse the results would be subjective (you observe it with your subjective senses and process it with your subjective mind). "as objective as possible", sure, "seems to be objective", fair enough, but not "is objective".
    – NotThatGuy
    Apr 3, 2023 at 17:53
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Joseph lives in New York. What is the probability of Joseph being murdered tomorrow? It is the number of murders in New York divided by the total population. This is the definition.

We cannot know this number in advance. We can only estimate it based on past data.

It doesn't mean anything to talk about probability of once-off events. What is the probability the Sun will rise tomorrow? It is the number of times the Sun rose in the morning divided by the number of mornings. That's just 1. What's the probability of the Earth blowing up tomorrow? Same problem.

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    Not everyone in New York has an equal probability of being murdered tomorrow.
    – Davislor
    Apr 3, 2023 at 19:07
  • @Davislor It doesn't mean anything to talk about the probability of a single person. Only probabilities among subgroups of the population.
    – Daron
    Apr 3, 2023 at 20:04
  • @Daron One can only verify statistical accuracy for large enough subgroups. Simultaneously, there is more than one subgroup and the probability distribution is not uniform.
    – Davislor
    Apr 3, 2023 at 22:59
  • @Davislor I don't know what is a probability distribution today.
    – Daron
    Apr 4, 2023 at 13:58
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I concur with the answer by user @Christian Hennig:, there is no "objective" way of ascertaining probabilities in the real world, so in practice the so called objective and subjective viewpoints are intertwined. We use the most objective random situations we can think about, such as coin-tossing or card-shuffling (often called classical probability) as measuring sticks, so less clear situations are assigned probabilities by comparison to these.

I have written more about this at our sister site: https://stats.stackexchange.com/questions/332026/can-we-think-of-a-probability-in-both-the-classical-and-subjective-sense-simulta/332218#332218

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  • As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.
    – Community Bot
    Apr 3, 2023 at 21:18
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There is no such thing as subjective probability. All probabilities are based on objective data.

You cannot assign, calculate or estimate a probability for a single human action. A larger data set is required, either about earlier behaviour of the same person or a larger group of people in a similar situation.

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    "You cannot assign, calculate or estimate a probability for a single human action." You'd better not drive a car, then. Consider the following situation. I am approaching a roundabout. Traffic is heavy, and I face a significant delay getting through. But, wait: there is a car signalling that they are about to exit the roundabout, which would let me get in. A Bayesian driver might estimate the probability that the signal is accurate, and act accordingly. A frequentist would need to wait until N identical cars with identical drivers had signaled.... Apr 5, 2023 at 1:52
  • @SimonCrase That scenario is not about probability. Mathematics is not involved. You cannot do math without any parameters. Apr 5, 2023 at 3:50
  • "That scenario is not about probability". Which scenario? Do you man my driving example? I'll accept that motorists in your country always follow the rules; they signal perfectly, and there are no probabilistic judgements involved. That is not the case where I live; many drivers signal randomly (maybe they got their license in a raffle), so the choices are: don't drive; or estimate probabilites. Apr 6, 2023 at 3:19
  • @SimonCrase The traffic scenario is not about probability. You are not doing any math or statistics. This is a human interaction, you are making a judgement about whether you can trust the signal or not. You have no numerical data to calculate any probabilities. Apr 6, 2023 at 7:19
  • OK,so that's where you are coming from: your guess that I'm not carrying a notebook while I drive to perform statistics calculations is correct. But I'm afraid you are missing the point. I'm thinking of the Bayesian brain, which considers the brain as a statistical organ of hierarchical inference that predicts current and future events on the basis of past experience. Possibly you have your own (no doubt charming) little idea of the workings of the brain: this one appears to me to account for the evidence. Apr 7, 2023 at 4:53
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You are just asking for things that are part of a basic college probability course.

The only philosophical aspect of this is that yes, future probabilities cannot be definitively stated without a complete mechanistic understanding of the process.

Invariably, it is not possible to have full mechanistic understanding of any real world process, because there may always be hidden variables. Coin tosses may be influenced by little gust of air or fluctuations in the Earth's magnetic sphere or invisible little trickster demons... Therefore, probabilities are typically calculated from an idealized model (imaginary coin in a room with perfectly still air, perfectly stable magnetic field and no demons) which is then assumed to be "close enough" to the real-world situation.

What is the real probability that a person will be murdered tomorrow somewhere in the world?

Impossible to know for sure, but can be estimated with various straightforward techniques. The simplest one is to simply calculate the murder rate for each past day, average them, and declare that tomorrow will be no different.

One thing you can say is that it will be almost 1. Because each person has probability p of being murdered tomorrow, and the probability of someone being murdered is equal to 1-q where q is the probability nobody will be murdered. Simple math shows (I won't) that q = (1-p)^n where n is the number of people in the world. Because n is so large and p for most people is not that small, q is going to be tiny, therefore 1-q will be almost 1.

But what is the correct probability then? Is it 1?

Of course not. Under any reasonable model there is a nonzero chance nobody will be murdered. Even a pre-planned murder has some small chance to unexpectedly fail.

When calculating the probability of a future event which is not fully understood (99.9% of real life cases) the two common approaches are frequentist and Bayesian.

To greatly simplify, a frequentist will look at how often someone was murdered in the past under the same conditions of note. Which conditions are of note is up to the analyst to decide and convince his audience of. If he chooses them poorly, his probability predictions will be off by more.

A Bayesian will select some probabilities for relevant independent events, which he thinks influence the event of interest (eg. murder rate). He will then use well established mathematical theorems to combine these probabilities and calculate a conditional probability for the event given those probabilities of other relevant events. The decision of which events are relevant and what their probabilities are is up to the analyst to decide and convince his audience of. If he chooses poorly, his probability predictions will be off by more, but it is often believed that Bayesian approaches are not very sensitive to slight inaccuracies in the priors. Sometimes frequentist approaches may be used, so that Bayesian probability becomes a sort of meta-technique for combining less reliable frequentist estimates into a more accurate Bayesian one.

This is of course very superficial, and there is a wealth of specialized techniques to either approach. I cannot possibly cover them here and they would be irrelevant to the question. Luckily, just about any college textbook on probability should cover them in detail.

As for subjectivity, when using models of probability, just like any other model, you must choose some assumptions and be confident they correspond to reality closely enough. The choice of these assumptions and what counts as "close enough" is subjective. Everything else (the calculations from those) is objective. In this sense probability is subjective, but in this sense no study of the natural world can be objective.

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This question basically boils down to the question of determinism. Do we live in a deterministic universe? If so, then with enough information, anything can be predicted. The way a die will land can be determined by calculating the forces acting on it. Probability just functions as a tool to guess what will probably happen when not all information is known or there is so much information that needs to be taken into account that making a prediction is unfeasible. And yes, the more information you have, the more accurate your probability estimations can be.

However, the physicists currently believe that we in fact do not live in a deterministic universe. The current consensus of physics is that on the most microscopic level of quantum mechanics, there are indeed things that happen truly random. And those effects do occasionally affect our macroscopic world to some extent.

And then there is the question if human behavior is perfectly predictable. Which boils down to the age-old question "Do humans actually have free will?". This is an open question philosophers constantly debate about since the antique and which we will never be able to answer conclusively. Probably.

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  • "the physicists currently believe that we in fact do not live in a deterministic universe" - not all physicists. It's hard to differentiate unpredictability due to randomness vs due to insufficient understanding. Also, the more we learn about reality and brains, the less likely free will seems: we have plenty of evidence of behaviour being heavily influenced by biology and external factors, and there is no conceivable mechanism that would enable free will as we imagine it (even if quantum mechanics is "truly random", randomness is not freedom).
    – NotThatGuy
    Apr 3, 2023 at 9:30
  • This question has nothing to do with determinism. In a deterministic system everything happens with 100% certainty and accuracy, there is no concept of probability. Nor is there any concept of belief either. Therefore it is illogical to believe that we live in a deterministic universe. Apr 3, 2023 at 11:17
  • Probability "functions as a tool to guess what will probably happen when not all information is known" regardless of whether or not determinism is true. If there are "truly random" or otherwise non-deterministic events, we might need probability in any case, but knowing that something is (likely) deterministic doesn't affect whether applying probability to it is useful, unless you also know what has been determined.
    – NotThatGuy
    Apr 3, 2023 at 11:24
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The mathematical theory of probability is logically as sound as anything we have. The issue is, what is its significance to the real world?

A lot of philosophers here seem to be saying that we can't define the real-world probability of, say, a particle experiencing radioactive decay, which would come as a surprise to most physicists. If the real world has infinite branching universes, we can define probability objectively using physics: something with probability p happens in universes whose probability mass adds up to p.

Without that, the best we can do is: if we have a large number of events with probability x%, approximately x% of them will happen.

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  • Practically every physicist or other scientist should agree that scientific knowledge is not completely certain, because we have limited and subjective knowledge of reality. We may have lots and lots of knowledge about some things, and we may know things with very high confidence, but there's always some possibility of being incorrect. This is a very core part of doing science. All knowledge is open to revision or falsification, but some knowledge would require a whole lot of contrary evidence to outweigh the supporting evidence.
    – NotThatGuy
    Apr 4, 2023 at 13:25
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I think your use of the word subjective is somewhat loaded- as if you considered any assessment of probability to be purely a personal matter. Probabilities can be clear-cut in certain circumstances, but in others you must fall back on estimates. In some respects it is no different to making any other form of estimate.

If I am laying an irregular patch of concrete on uneven ground I cannot know exactly how much concrete I need, so I make an estimate. I get an idea of the average length, width and depth and I calculate a volume. I only know how close my estimate was when I lay the concrete. If you were to say that my estimate of the volume of concrete was 'subjective', you would be overlooking the fact that most other people would approach the question in the same sort of way and it could be relied upon to give broadly the right answer, notwithstanding the fact that some degree of personal judgement pays a role.

Now take the question of whether a murder is likely to happen in my town tomorrow. I can estimate the probability in a number of ways, some more sophisticated than others. I could look at the average number of murders in the UK over some period, assume the number is related to overall population, then scale it to the population of my town, and so on. I might see if the murder statistics vary by time of year, by socio-economic conditions, etc, and take them into account in my estimates. There are many approaches I might adopt, and whether I picked one over another might be a matter of personal preference (or ignorance, laziness etc), but I think you are going too far by writing off the whole exercise as purely subjective.

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  • The subjective/objective knowledge distinction in philosophy refers to what we believe about reality vs what reality actually is (similarly for subjective/objective morality), so I wouldn't consider "subjective" itself to be loaded. However, "subjective" is often falsely treated as little more than whims and personal preference, usually by people who are arguing for objective knowledge or especially objective morality (despite them having neither), or by people who use it as a "justification" for their unjustified belief.
    – NotThatGuy
    Apr 4, 2023 at 11:20
  • @NotThatGuy agreed. I suppose that the I tension of my example- of estimating the volume of an irregular patch of concrete- was to highlight the overlap, rather than the distinction, between what we believe and what is. Apr 4, 2023 at 13:00
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Probabilities in quantum mechanics don't face these issues.

The problem with classical probabilities is that they are born out of ignorance, and there is no limit to ignorance. There can be infinite number of things that you are not considering. When you try to predict the probability that a coin will land heads, you usually reject the possibility that the coin is rigged.

If you don't reject that possibility, you are faced with an infinite number of sub-possibilities : The coin could have probability p for heads and q for tails, where p and q are any real numbers between zero and one.

Even after you consider the above, you need to assume another probability distribution over the space of possible coins, indicating the probability density that you have any particular coin. You can be even further ignorant about this ignorance.

This process can keep on exploding forever. There is no limit to ignorance. In practice, we stop this process somewhere and then apply Bayes' theorem over our thus obtained sample space. Classical probability works in practice

The only way to escape this process without an artificial full stop is when you have zero ignorance : i.e. when you already know the definite state of the coin to be, say, Heads. Then the probability of Heads is 1, and Tails is zero.

Now we come to quantum mechanics

Quantum probabilities are not born out of ignorance. If you have collapsed a particle into a (non-degenerate) eigenstate, you already know the definite state of your system. So you no longer need to consider ignorance.

Does this mean that all the probabilities become 1 or 0 just like what happens classically when you have zero ignorance?

No. In quantum mechanics, a state can be definite with respect to one observable but probabilistic with respect to others. This probability is completely given by the Born rule, and you do not need to consider an endless explosion of ignorance when you calculate this probability.

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