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In a deterministic universe where knowing every possible cause and initial condition leads you to figure out the effects with precision, each event ends up having a “real” probability of 0 or 1. Things would either necessarily happen or not. That is that probability is seen as a function of ignorance and without this ignorance, the concept ceases to exist as envisioned by Laplace’s demon.

The question then is what is the “real” probability of an event if the universe was fundamentally stochastic/random, especially on a macro scale? Even if we can’t fully determine each effect despite knowing everything there is, what are the actual possibilities and how can we mathematize them?

For example, would the probability of a particular coin toss landing on heads still be 0.5 even if we knew every possible thing about the initial conditions and prior causes before the coin toss? Or given that we’d still be in a macro scale, would we have enough knowledge to predict with 100% or near certainty where the coin will land?

What about the impact of the time at which this knowledge is discovered if determinism is false? For example, let’s assume a coin toss occurs at time t. If we knew everything that we could about the universe at time t - 1 seconds, would the probability of the coin landing on heads be vastly different than if all we could know about the universe was at time t - 1000 days?

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  • Coin tosses do not qualify (given complete information about initial state), but double slit experiments on quantum particles might. “True” (metaphysical) probability can only apply to "true" randomness. If there is such a thing, as is supposed in the standard interpretation of quantum mechanics, then there will be laws of nature that determine what the "true" probabilities are in a given state of affairs. In quantum mechanics, they are given by the Born rule for collapse into eigenstates.
    – Conifold
    Jun 15, 2023 at 23:33
  • @Conifold: It should perhaps also be noted that, under QM, probability is only meaningful at the moment the wave function collapses (or whatever your favorite interpretation says that "wave function collapse" physically means). If you compute a probability, and then the wave function doesn't collapse, then that probability is just an arbitrary number you pulled out of a formula.
    – Kevin
    Jun 16, 2023 at 0:02
  • This is the most sensible thing I have seen you say. I was just thinking about this before breakfast. Yes, if something will either happen or not, then the probability must be 0.5.
    – Scott Rowe
    Jun 16, 2023 at 1:54
  • "That is that probability is seen as a function of ignorance..." That is still the case with indeterminism. This question includes it's answer.
    – tkruse
    Jun 16, 2023 at 5:31
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    We should not speak of "true" probabilities. If you know the outcome of an event, then for you the probability is 0 or 1. All probabilities between 0 and 1 are a description of what some subject doesn't know about the event. Even the perspective of "what a person could predict if they had all starting variables before the coin toss and could calculate perfectly" is still only a certain person's lack of knowledge, and should not be called a "true" probability as if it were independent of the subject.
    – causative
    Jun 16, 2023 at 5:56

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Probabilities are attributes of models, not of the modeled world itself. Talking about the probability of an event is meaningless without a model which allows looking at the possible outcomes.

Look at an idealized "model" die. It has an equal probability of 1/6 for showing each of its sides on top after a throw. Like all models, it serves as a good predictor for the actual behavior of dice, so it can be helpful in calculating winning chances at games etc. However, it does not model deviations of the real object from the model - for example, leaded dice could have distorted distributions, or a sufficiently well-balanced die thrown by someone with extreme accuracy could land on its edge in an unstable equilibrium, never deciding on one or the other side to face up, or you could even have a bomb explode under the gaming table before the die settles into a final position, destroying the die together with all observers around the table. All of this is not part of the model, and it is not meaningful to talk about probabilities of these "special" events.

If you change your model to include special cases, distortions in the probability distributions, tricks performed by the coin or dice thrower etc., you may get different probabilities, however, it's much harder to get a good agreement between model and reality. Leaded dice could probably be modeled quite accurately using a physics simulation.

If the coin thrower is a trained table magician, they may be able to make the coin show a desired outcome 95% of the time, so by knowing in advance that the coin will be thrown by such a magician, and that he intends the coin to show heads, you might modify your coin throw model to have probabilities of 0.95 for heads and 0.05 for tails, and predict (or bet on) heads accordingly. The actual outcome might still be tails, that's the nature of probabilistic models, but that doesn't affect the validity of your prediction.

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  • Propensity interpretations of probability do treat them as objective properties of the world, including as properties of single events. For this reason, they are popular in quantum mechanics. These are the "true probabilities" OP is asking about. Epistemic probabilities that are assigned by models are not relevant to this question.
    – Conifold
    Jun 16, 2023 at 20:21
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That is that probability is seen as a function of ignorance...

That is still the case with indeterminism. Quantum indeterminacy just added a little more ignorance about the future. And of course the farther ahead we try to predict, the worse any indeterminancy would make our prior ignorance.

How much probabilities change compared to a deterministic world depends on how much indeterminism affects the macro scale. Indeterminism on the macro scale can be likened to a permanent earthquake. Earthquakes can be very light, almost imperceptible, or strong. Probabilities of events change depending on the "strength" of indeterminancy.

In practice, indeterminism as a source of ignorance about the future has not become relevant to mankind. All other sources of ignorance about initial conditions are much more strongly impacting prediction models that quantum events, for most practical applications.

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  • We are ignorant, about a great many things."
    – Scott Rowe
    Jun 16, 2023 at 11:31
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Re: paragraph 3

Coin flips are almost entirely predictable given even very limited knowledge of initial conditions. Nothing close to 'every possible thing about the initial conditions' is needed to guarantee the outcome of a coin flip to close to 100%, just a simple machine manufactured to reasonably tight tolerances. See SIAM Review Vol. 49, No. 2, pp. 211–235 (pdf via berkeley.edu) for Newtonian model and experimental verification.

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  • this seems like a (good) comment rather than an answer.
    – tkruse
    Jun 16, 2023 at 8:47
  • So we could improve our lives by controlling initial condions better, like avoiding shootings or car accidents. But people feel that this limits their free will. One wrong idea crashes into another. We could control what people learn...
    – Scott Rowe
    Jun 16, 2023 at 11:28
  • @tkruse IMO direct answers of questions should be answers, even if there are multiple questions in the Question.
    – g s
    Jun 16, 2023 at 15:34
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True probability of a certain event can only be calculated, if we have enough knowledge about how the system works or statistical data about similar events.

  • We know the properties of a deck of cards well enough to calculate the probability of a certain card being drawn: 1/52.
  • We have enough statistical data of radioactive decay to calculate the probability of a single atom decay: 50% during the timeframe called half-life.
  • We know how the weather works and have lots of statistics about weather in the past. Therefore we can calculate the probabilities of future weather events.

The very idea of probability is that future events are not knowable before they can be observed. Knowledge about future events does not exist before the events actually occur.

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  • There's no "true probability". Any probability is true: if you predict that the probability of falsehood is 99.999% POSITIVE, and the event results NEGATIVE, this does not make the probability "false".
    – RodolfoAP
    Jun 28, 2023 at 4:25
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You are mixing multiple notions here.

Determinism is the idea that our decisions are determined by external causes which are physical. Determinism, therefore, is moreover a metaphysical doctrine. Determinism has no relationship with probability.

Probability is a prediction of the future, independently of how it is calculated. The definition of a probability implies a lack of knowledge of the future. Probability might be similar or different to the actual fact.

The "true" value of probability is not the value of the fact. Probability is just probability.

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  • What do you mean by "decisions determined by causes"? How can they be determined when they are not physical events? How can they be called "decisions" if the results are already determined? Jun 16, 2023 at 18:29
  • @PerttiRuismäki 'How can they be called "decisions" if the results are already determined?' Well duh, ask that to determinists...
    – RodolfoAP
    Jun 27, 2023 at 18:48
  • I have tried to ask them, but their answers are equally illogical as their beliefs. Jun 28, 2023 at 3:24
  • @PerttiRuismäki please avoid generating noise on the discussion. Just read on the Internet. Your question is elementary to determinism, this is not the place to start learning about it.
    – RodolfoAP
    Jun 28, 2023 at 4:17
  • What noise? The very thing that distinguishes reality from determinism is the noise. There is no noise in determinism. I know everything there is to know about actual determinism. I am only curious about the so called "determinists" and their beliefs. Jun 28, 2023 at 5:09
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If determinism is false, what is the “true” probability of each event?

What about the impact of the time at which this knowledge is discovered if determinism is false?

Probability requires a history of prior events to be useful and the lack of determinism means that it's not possible to know all the states of a system in advance. For a coin that has never been flipped, the probability is assumed or calculated by determining all the possible states the coin can have: Heads or Tails. Two states that can have an equal chance if occurring.

But the "true" probability can only be determined through experiment and observation over multiple events.

Otherwise you miss that a nickel has a third state: On edge, which does occur very infrequently. Or unfair coins.

If the the very beginning universe was hot dense and too hostile of an environment to support any life. No life at the beginning of the universe: Zero probability. At what time in the early universe did this change to a non-zero probability? This is the temporal nature of probabilities and there is no easy solution.

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