# Probabilistic prediction (quantum mechanics) - what is the meaning of such a prediction and how do you falsify it?

Suppose there's a hypothetical quantum physics experiment. There are 2 possible outcomes to this experiment A or B. QM predicts that the probability of each is 50%.

Firstly, what is the meaning of this prediction? The only meaning I can come up with is that the limit of (number of A's/number of experiments) approaches 0.5 as the number of experiments approaches infinity.

So practically... what would constitute a falsification of such a prediction by QM? Suppose we conduct the experiment 2 times. We get 2 A's. So our probability distribution violates our prediction. But we decide that we conducted the experiment too few times. So we conduct the experiment a million times. We again get all A's. Are we justified in thinking the prediction has been falsified? What changes in the case of 2 experiments vs a million experiments?

• Probabilities are a modality for which standard logic does not apply (Aristotle knew that), so there is no straight 'falsification'. Conditional probability would show that P(H|T(n)) decreases with a larger n, that is, the hypothesis 0.5/0.5 gets less credible with the accumulation of repetitions. Jun 14 at 7:31
• I'm not well versed in statistics. But in this case I would say the first null hypothesis would be that the outcome is 50/50 because it's random. You have to construct the experiment so that complete randomness is ruled out. Otherwise it's indeed unfalsifiable. So here's my theory: "if I throw a coin million times it will land approx. 50% tails because the devil organizes it so". You can't (un)falsify that just by throwing coins. Jun 15 at 8:42

In science, nothing is ever exactly 100 percent certain. Quantum mechanics has actual randomness (strictly speaking, the interpetation of this depends on weather you take the Copenhagen-interpreation or the many-worlds-interpretation. But thats not really relevant to this question I think). But also all other parts of science (except pure math I guess) have measurement errors, which are effectively random as well.

What you do in science is Hypthesis testing. For this example it goes like this: Your hypothesis (often called "null hypothesis" in this case) is "event A and B are each 50% likely". Then you do the experiment (multiple times). Suppose you do it two times and get two times A. Then you ask "how likely was that outcome if my hypothesis was right?". In this case, the answer is 25%. This is called a "p-value". In this case p=0.25. If you instead repeat the experiment twenty times and get A every time, then p=0.00000095...

If that p-value is very small it means that the outcome of the experiment is very unlikely if the hypothesis were true. Therefore the hypothesis is probably wrong ("you reject the null-hypothesis"). The smaller p is, the more certain you are. Most scientists only consider results with p<0.05 to be meaningful at all. But for interesting questions you should have a (much) smaller cutoff to be more certain.

Note in particular:

• If the p-value is small, you can conclude that the null-hypothesis is (probably) wrong. If the p-value is not small, you cant conclude anything. Your experiment simply wasnt precise enough.
• Strictly speaking this means that it is impossible to completely falsify any scientific theory. But you can get as close to certainty as you want by repeating the experiment often enough.
• Its still true that hypothesis testing can only reject a hypothesis/theory. You can never really verify it. Thats what "Occam's razor" is for.
• Obligatory xkcd Jun 15 at 17:15
• The hidden variables interpretation states that the randomness actually is not truly random. You base your view on science on a very narrow platform. Science isn't done by a method. The method of falsification, however sophisticated, simply halts the progress of science, like adherence to any method.
– user52804
Jun 16 at 9:13
• @Methadont Hidden-Variable-Interpretation was experimentally falsified (see en.wikipedia.org/wiki/Hidden-variable_theory#Bell's_theorem ). Still you are right, there are fundamentally non-random interpreations of quantum mechanics (most notably "many-worlds"). But the method of hypothesis testing still works for anything that can reasonably modelled as random. Like if you throw a die, the result is practically random. Even though strictly speaking it is deterministic if one could know the state of all air particles and the like. Jun 16 at 11:07
• Hi there! I don't think that the hidden variables are falsified by experiment. Local hv's yes, but non-local ones not. I'm not sure if the MWI is non-random. It's not known in advance how states are distributed over the branches. It is known that they separate.
– user52804
Jun 16 at 11:22
• @Methadont Yes, only local HV was disproved. In MWI, there will be two 'worlds' (more precisely "projections of the global wave function"), one with event B and one with event B. I would call that non-random, even though it is of course impossible to predict the outcome, because "both happen". Anyway, this discussion is getting into the semantics of of the word "random" actually means. And thats beside the point here. Hypothesis testing and falsification works. And pretty much all of science nowadays is based on it. Maybe in some future we will come up with a different method. I dont know. Jun 16 at 11:30

Each time you conduct the experiment, the probability of outcome A or B is always going to be 50/50 so its entirely possible to do the experiment a million times and get the outcome A every time. The probability is not determined by previous outcomes, its determined by number of possible outcomes. So just because you get outcome A a million times, it doesn't change the probability the next time you do the experiment, its still 50/50.

• So the prediction is unfalsifiable? Doesn't that make it unscientific? Jun 14 at 8:26
• Yes I would say its unfalsifiable. Not sure what you mean about being unscientific. Jun 14 at 8:32
• @NetServOps: Falsifiability of hypthesees is Popper's criteria for demarcating science from non-science en.wikipedia.org/wiki/Falsifiability Jun 14 at 16:58

Modern philosophers tend to agree that naïve falsification is really the wrong way to look at science. Theories don't really get falsified, because they always summarize (accurately!) what we have experienced in reality. Instead, they just develop more and more exceptions until they violate Ockham's razor.

In your specific example, Quantum Mechanics predicts event A occurs under conditions B with probability 0.5. Now suppose a scientist C using this theory sets up n independent repetitions of B. Then event A will occur (n/2+E)-many times, where E is an error term of size roughly sqrt(n)/4.

With 2 trials, two occurrences of A is well-within the space of expected results described above. With 1 million trials, 1 million occurrences of A is not. That is, our theory no longer accurately summarizes what we know about reality. On the other hand, the theory can be rectified by changing quantum mechanics to the statements

1. Event A occurs in conditions B with probability 0.5.
2. The 1 million trials of B performed by C on such-and-such a date always turn up A. This is fantastically unlikely, but we happen to live in that universe.

As we add more and more trials with more and more dates/places/scientists, we need to keep adding caveats (unless reality starts producing the expected number of As). But then the theory

Event A always occurs in conditions B.

starts looking a lot more simple and convincing, so more and more scientists start using it in their work.

The fact that you have thrown heads a zillion times would prove that the dice is biased or that the probability is small indeed but that it can be thrown a zillion times. You have to examine the dice for that (or the quantum system). If it's a pure (non-biased) dice then it's just a coincidence. How do you know it's pure? By throwing it a zillion times!

The prediction of the possibility that an event can occur a zillion times while using a 50/50 chance system is confirmed in your example.

So the experiment doesn't falsify the prediction (the outcome can appear according to the prediction). It falsifies the premise that the chance distribution is a 50/50 one. It confirms that it is a 100/0 distribution.

So, the prediction of 50/50 is not falsified by throwing it two times. The prediction is falsified by throwing it a zillion times. So now you have a new prediction, based on this zillion times throwing. Namely that the probability distribution is 0/100. This means that if you perform the experiment again you will get zero whatever and 100 whatever elses. you have to readjust your image of the system (the quantum system) though. It isn't the state you thought it was before (superposition of two equally probable outcomes).

• So it's impossible to falsify the QM prediction? So no matter what the results of the experiment are, we're still going to say QM is true and the probability is 0.5? Jun 14 at 9:53
• @AmeetSharma Yes. The experiment merely confirms that it is possible to get the same result a zillion times. The opposite is not falsified (if it is a real chance event). It will be a big hint though that it's not a 50/50 chance event though.
– user52804
Jun 14 at 9:55
• Ok. Just to clarify the original post. Assume the experiment is conducted perfectly, and QM predicts unambiguously a 50/50 probability. So any hint that it's not a 50/50 chance event would be a hint that QM is false. Jun 14 at 10:01
• No. It would show that the state is not one that has a 50/50 chance distribution. It would show that the state is one for which you always measure the same outcome. That is, the 50/50 chance is falsified.
– user52804
Jun 14 at 10:03
• But QM predicts a 50/50 distribution for this experiment. Jun 14 at 12:16

Quantum probabilities are not Bayesian, determined from expectations adjusted by experience. Instead they come from the wavefunctions of particles, by the Born rule. The Stern-Gerlach experiment is perhaps the clearest example of the distinct impact of their non-classical qualities.

The Schroedinger's cat arrangement produces 50% likelihood based on a single atom of a radionuclide, and a timespan passing equal to the atoms half-life. Half-lives were initially determined by experiment, but can now (nearly always) be predicted accurately from theory. They decay truly randomly, 'god playing dice', but with a specific distribution related to available decay modes, barrier potential etc. The decay of an individual atom is 50% likely after one half-life, and approaches certainty as time goes to infinity, but might never decay.

We don't know if protons decay, just over a very long time. That's the closest example I can think of to your question. Neutrino observatories additionally look for evidence of proton decays, progressively constraining the minimum proton half-life as they get bigger, and time passes.

As for how many cases lead to a given conclusion, this is well established in science by statistical hypothesis testing. You compare the model with the results, and state something has been discovered when an appropriate level of confidence is reached that the prediction could not have been chance. That level depends a bit: 3 sigma means 99.7% sure of the result, which is fine for most regular physics results; the Higgs-boson was announced at only 2.3 sigma, 98% confidence, but it fitted the models, and as CERN runs the confidence gets higher, over 5 now; gravity waves were all about minimising noise, & they were measured at 5. 1sigma, the 'gold standard' of 5 sigma is already 99.99994% sure; a faster-than-light neutrino result was announced with 6 sigma confidence, but considered so unlikely it was still written off as experimental error.

You get Type 1 & Type 2 errors (and some other higher number errors more rarely used), which specifically relate to the equivalent of 'unlikely coinflips', you might phrase it. Numbers of particle collisions at CERN per second, or atoms in a visible piece of nuclide, are very large - a mole of atoms is enough for a coin flip every second for longer than the current age of the universe. The chances of getting all heads by chance over those timescales, is vastly different to our ordinary experience of coins.

That still leaves examples like the tachyon result: systematic error, bad models, confused thinking. So we like replication of results. Detailed methodologies. The team did in fact, find flaws in their equipment set-up that caused errors far outside their original confidence interval, a fiber optic cable attached improperly, and a clock oscillator ticking too fast.