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Suppose I wanted to calculate the “true” probability of me tossing a coin tomorrow and it landing on heads.

Now, even though we often say that this is 50%, correct me if I’m wrong, but this can’t be the “true” probability, correct? The true probability, as far as I’m aware, is how many quantum branches of reality starting from the initial conditions of the universe result in this coin toss landing on heads, weighted by perhaps the amplitude of those branches’ wave functions? Forgive me if I’m using the terminology incorrectly but that is atleast how I vaguely pictured it.

Now, correct me if I’m wrong, but on a macro scale, certain quantum effects may be washed out such that the probability may still be 0 or 1 resulting in atleast pseudo determinism. But the more time elapses, the more the chance for a quantum effect to have an impact, yes?

For example, if my decision to toss the coin was influenced by a Geiger counter, we would now have a quantum effect percolating up and affecting the macro scale thus now affecting the true probability of the macro event of my coin toss landing on heads.

So in summary, I’m trying to understand how one would atleast in principle calculate the probability of a certain event occurring in the macro scale.

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    $\begingroup$ The outcome of a coin toss is affected by the force of your thumb and such. If you know all those things accurately enough, you could calculate the outcome to nearly $100$%. If you don't, the odds are $50$%. $\endgroup$
    – mmesser314
    Apr 3 at 3:58
  • $\begingroup$ Yes but that depends on the probability of those initial conditions then and this can keep going back to the Big Bang, yes? $\endgroup$
    – user353810
    Apr 3 at 4:05
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    $\begingroup$ So in summary, I’m trying to understand how one would atleast in principle calculate the probability of a certain event occurring in the macro scale. ---how to go from micro (quantum) scale to macro (everyday) scale is, in general, an unsolved problem and extremely non-trivial. Other than that, the rest of the question is not clear to me. $\endgroup$ Apr 3 at 5:55
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    $\begingroup$ Voting to reopen. A perfectly clear question with a straightforward answer in principle - path integrals. $\endgroup$
    – gandalf61
    Apr 3 at 8:33
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    $\begingroup$ Voted to reopen. In view of your last question, I would ask you to keep in mind that many people here do know physics well and are doing their best to explain it. $\endgroup$
    – mmesser314
    Apr 3 at 13:18

3 Answers 3

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Suppose I wanted to calculate the “true” probability of me tossing a coin tomorrow and it landing on heads.

This is the joint probability of two events, $ P(A,B) $, with $$ A= \text{You tossing a coin tomorrow}.$$ $$ B= \text{The coin landing on heads}. $$ By the structure of the two assertions, it makes sense to compute $ P(A,B) $ using conditional probabilities, $$ P(A,B) = P(B|A) P(A). $$

Having no information on the coins that you have, it is a good guess (or a null model) to state $P(B|A)=1/2$. Note that this probability does not depend on the big-bang or anything like that, as it is conditioned to the fact that you will effectively throw the coin.

The probability that you will throw the coin, $P(A)$, is more difficult to estimate for me, but you might have more information about this ;P.

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In classical probability theory we can in principle calculate the probability of a state B occurring at some time after a state A by considering all of the possible routes from A to B and adding up the probability of each route.

In quantum mechanics there are infinite number of routes from A to B, and the sum becomes an integral known as a path integral. There is another significant difference from classical probability theory. Whereas the probabilities in classical theory are always positive numbers, the quantities in a path integral are wave function amplitudes, which take complex values. This means that some paths can cancel out other paths.

I emphasise that this is how probabilities are calculated in principle because this calculation is only feasible in practice in the very simplest scenarios involving quantum systems that do not interact with the surrounding environment.

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  • $\begingroup$ So what is the true probability on average of a coin toss landing on heads then? The sum of the probabilities of the paths leading to heads vs. tails on average would be? $\endgroup$
    – user353810
    Apr 3 at 17:26
  • $\begingroup$ @TruthSeeker Impossible to say in practice as there are too many variables. For example, how far ahead are you trying to predict the coin toss ? If you start before the coin has been tossed than there is a chance it will never even be tossed. On the other hand, if you start a microsecond before it lands than in most cases you can predict heads or tails with near certainty. $\endgroup$
    – gandalf61
    Apr 3 at 17:31
  • $\begingroup$ What about a second after it’s been tossed? In most cases, would basically all branches lead to either heads or tails with 100% certainty and function as determinism? Or not? $\endgroup$
    – user353810
    Apr 3 at 17:43
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Suppose I wanted to calculate the “true” probability of me tossing a coin tomorrow and it landing on heads.

Now, even though we often say that this is 50%, correct me if I’m wrong, but this can’t be the “true” probability, correct?

50% could be the correct probability. For a macroscopic system you're not going to calculate it from quantum theory: the calculation would be intractable even if you knew the initial conditions perfectly, which is impossible since it requires far too much information storage.

The true probability, as far as I’m aware, is how many quantum branches of reality starting from the initial conditions of the universe result in this coin toss landing on heads, weighted by perhaps the amplitude of those branches’ wave functions? Forgive me if I’m using the terminology incorrectly but that is at least how I vaguely pictured it.

No. The probability is not the number of branches. For a start, branches are an approximation because information being copied out of a system reduces interference, but never eliminates it so you have to make a choice about the level of interference you will tolerate in your definition of branching:

https://arxiv.org/abs/1111.2189

Also, branch counting is an inconsistent rule anyway

https://arxiv.org/abs/0906.2718

Suppose you have two radioactive atoms A1 and A2. You measure A1 and record whether it has decayed or not. There are two branches as a result of this event, so the probability would be 1/2 that A1 has decayed. But then suppose that if A1 has decayed you measure and record whether A2 has decayed. Then there are three branches A1-undecayed, A1-decayed+A2-decayed, A1-decayed+A2-undecayed. So then the probability that A1 has decayed would be 1/3 by the branch counting rule, not 1/2.

The probability of a branch is the weight given by the standard formula for the probability of an outcome: the Born rule.

Now, correct me if I’m wrong, but on a macro scale, certain quantum effects may be washed out such that the probability may still be 0 or 1 resulting in atleast pseudo determinism.

On the macroscopic scale interference washes out to a good approximation and this may make the result of some process deterministic to a good approximation. But in general all the lack of interference actually buys you is that the Born rule amplitudes actually act like probabilities, which they don't when there is interference. So the probability could be 0, 1 or anything in between in general.

But the more time elapses, the more the chance for a quantum effect to have an impact, yes? For example, if my decision to toss the coin was influenced by a Geiger counter, we would now have a quantum effect percolating up and affecting the macro scale thus now affecting the true probability of the macro event of my coin toss landing on heads.

What matters for the probability is the motion of the specific system you're looking at.

So in summary, I’m trying to understand how one would atleast in principle calculate the probability of a certain event occurring in the macro scale.

Your best option is usually to repeat the same kind of event many times and measure the relative frequency of different outcomes, not to use quantum mechanics.

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  • $\begingroup$ I’m a bit confused by this answer. Suppose I am about to toss a coin. Either it is determined by physical laws that it will land on heads, or it is not determined. If it is not determined, presumably there is a probability. If there is a probability, how do I calculate this in principle? If there isn’t a probability, then how is it not determined? Or can something be not determined without a probability? $\endgroup$
    – user353810
    Apr 3 at 18:06
  • $\begingroup$ In principle, you would know the initial conditions relevant to the final state of the coin to some finite level of accuracy: stuff like the state of tension in your muscles when you toss it, variations in air pressure and density in the air, the shape and material properties of the surface it will land on, the hardness of your nails, variations in the density of the ground that could change the gravitational acceleration slightly and so on. You would then come up with a model of the coin throwing, simulate it and count trajectories leading to heads or tails. $\endgroup$
    – alanf
    Apr 4 at 7:33
  • $\begingroup$ Quantum theory isn't very relevant since coin tossing is decoherent. $\endgroup$
    – alanf
    Apr 4 at 7:33

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