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.
@Nihar has an excellent answer: It's possible but with a chance of 1 in $10^{1.94\times10^{25}}$

That is a truly large number. When you use exponents that need to be represented with their own exponents, it can sometimes be difficult to think about what they actually mean. for some perspective:

* There are about $5\times10^{19}$ atoms in a grain of sand
* There are about $8\times10^{18}$ grains of sand in the world
* That's about $4\times10^{38}$ atoms in all the sand in the world
* There are about $1.33\times10^{50}$ atoms of all kinds in the world
* There are about $10^{56}$ atoms in the solar system
* There are between $10^{78}$ and $10^{82}$ atoms in the universe

Using the largest estimate of $1\times10^{82}$ atoms in the universe, we've gone only from an exponent of 19 to 82 comparing a grain of sand and the entire universe. This *exponent* is 1,940,000,000,000,000,000,000,000,000.

How many trials would we have to do to get a reasonable chance of this happening? The formula for figuring out the odds of a random event happening *at least* once is $1-(1-P)^y$ where P is the probability $1/{10^{1.94\times10^{25}}}$. I couldn't find any app that would give sensical results given large values for y, but if y = P then the odds approach ${-(1-e)}/e$ as P gets large. That's about 63.2%. So if we do $10^{1.94\times10^{25}}$ trials, there's about a 63.2% chance of it happening at least once and about a 37.8% chance of it not happening at all.

So how can we imagine doing $10^{1.94\times10^{25}}$ trials?

If we take all the atoms in the universe and change them all into separate 1kg bundles of iodine-131, we would have about $2.2\times10^{57}$ of them. Spread out over the volume of the visible universe ($3.57\times10^{80} m^3$), that's one bundle every $1.6\times10^{23}$ cubic meters, that's a cube that is 57,000 kilometers per side with a 1kg bundle of iodine-133 in the center. The age of the universe is estimated at 13.772 billion years, that's about $7.24\times10^{15}$ minutes. If we took all those bundles of iodine-133 and re-ran our experiment every minute (converting the decayed atoms back to iodine-131 for each trial) from the big bang until now, that's about $1.6\times10^{73}$ individual trials.

That exponent of 73 is no-where near the exponent we need to achieve a 63.2% chance of it happening. There would need to be about $2.66\times10^{23}$ universes of atoms converted to iodine-131 re-running the experiment each minute for 13.777 billion years to have a 63.2% chance of it happening at least once.