Question: I wondered this since my teacher told us about half life of radioactive materials back in school. It seems intuitive to me to think this way, but I wonder if there's a deeper explanation which proves me wrong.

When many atoms are involved, half life can statistically hold, but since decaying of an individual atom is completely random and stateless, can't all the atoms in a 1 kg of matter just decide to decay in the next minute, even if the probability of this event occurring is extremely small?

Answer: 
TLDR: statistical models are models, and thus by definition not a perfect reflection of reality.

Nihar's answer is good but I'm going to tackle it from a different direction.

First off, if we only look at statistical mechanics you can run through the math and of course you will find an extremely small probability. You might stop there. But statistical mechanics uses statistical models, and all models are wrong. They make assumptions and necessarily simplify reality to solve complicated problems. There could very well be some physical processes unaccounted for in statistical mechanics that negate any possibility of such a rapid decay.

A classic example is having a room and figuring out the probability that all the oxygen all of a sudden is only in one half of the room. From a stat mechanics standpoint, it's basically the probability of flipping a fair coin an unimaginably large number of times and having them all land the same way. But in reality, the unimaginably small number you would calculate wouldn't actually be correct, because the assumptions made by your model wouldn't perfectly reflect reality (particles interact with each other, for one). Much like the ideal gas law, these things are useful but can completely fail if you deviate too far from the assumptions made. This is true of all statistical models, of course.

**So if we assume that the stat model of half life is a completely accurate representation of reality, the answer to your question is technically yes. Of course we know it's not, so that leads me to my final point.**

There's also a heavy philosophical component to these sorts of questions since we are dealing with probabilities that are so small they are effectively 0. If someone flips a coin a billion times and it lands tails every time no one is going to think it's a fair coin, because it's obviously not\*. You could also consider state of the art cryptography. The odds of successfully randomly guessing a key is so low that for all intents and purposes it is 0. Or imagine watching a video of a bunch of shattered glass forming into a vase. Your conclusion wouldn't be 'see ya thermodynamics, wouldn't want to be ya', it would be 'I'm watching a video of a vase shattering in reverse'. Yes, there are technically tiny probabilities associated with these events but it is so small that saying they are technically possible is more of a philosophical statement than anything else.

\* The idea of a fair coin is a rabbit hole on its own. How do you determine that a coin is fair? By tossing it a bunch of times and observing a nearly equal number of tails and heads. If it deviates too much from 50/50, we declare it to be biased. But of course no matter what outcome we observe, there's always a chance it was a fair coin, so technically we can never know for sure. In order to make use of statistics then, we must arbitrarily pick a cut off point for random chance. Usually this is 2 sigma, maybe 3. CERN uses 5 sigma for new particle detection but again, this is arbitrary. Applied statistics is very much an art as much as it a branch of math.
To understand this, you need to see what triggers a nuclear decay. The answer is a beautiful example of quantum mechanical behavior. Nothing triggers it. It is just that the world is fundamentally quantum mechanical, and probabilistic.

> 
> All the other answers that "no, there is no triggering event, it just happens, quantum mechanics is like that" are perfectly right.
> 
> 
> 

[What happens before a radioactive element decays?](https://physics.stackexchange.com/questions/164665/what-happens-before-a-radioactive-element-decays/164679#164679)

All you can do is calculate the probabilities.

So the answer to your question is, that yes, there is a non-zero probability for the material to decay in the next minute.

But your question is more about if there is a chance that all the atoms in the material decay simultaneously in the next minute. And the answer is again yes, there is a non-zero probability for that to happen, but it just happens so that the probability is so little, that even on giant timescales like the age of our universe, there is very little probability for us to observe that to happen.