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{ I am explaining things as I know it. Please feel free to correct as necessary. }

As I have understood it, Gaussian randomness forms a predictable pattern when sample size is very high. If we take an ideal Gaussian coin and keep tossing it, there is no way to predict the outcome of each toss, but if the sample size is 1000 tosses, there will be near equal number of heads and tails.

However, it is entirely theoretically possible that all 1000 tosses will yield head, even though the probability of that happening is 1/2^1000 which is absurdly low.

Now lets move from Gaussian randomness to quantum randomness.

We can see that despite the underlying randomness, nature seems to follow very concrete laws on a macro scale. Nothing observable ever goes totally out of whack. But is it theoretically possible for something totally unreal to happen on macro scale no matter how low the probability of such an event be?

I know that due to quantum superposition I cannot calculate quantum probability the way I calculate normal probability, but I don't know enough to be able to tell whether quantum randomness yields non-zero probability for anomalies for a high sample size.

Suppose we have 1 mg pure Uranium 238. Can we calculate the probability of the uranium not releasing a single alpha particle for one whole second? Or is this a meaningless question?

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    $\begingroup$ As far as I get the axiomatics of QM, randomness there has the same properties. Go to tunneling effect, for instance! That is the classical example of allying the pure probability interpretation. In essence, all the outcomes of the experiment are possible as long as they obey Heisenberg principle. $\endgroup$ – MsTais Mar 29 '17 at 22:16
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Yes, the stochastic nature of quantum mechanics implies that extremely strange effects like no decays of U-238 for a long time can happen, albeit with astronomically low probability. One can also invoke here that such effects explain why radioactive decay can happen at all. Alpha decay involves the tunneling of an alpha particle through a potential barrier. Now, at the relevant nuclear scale where this happens, this is usually an extremely rare event. So, almost never does the alpha particle escape. Even if the half life is one second, this still means that the alpha particle got out on one of a huge number of escape attempts and then consider that the half life of 180-W is $1.8\times 10^{18}\text{ years}$ .

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