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: 
The answer is 'no'. This 'no' is on the same level like:

* Can it happen that you float for 15 minutes in the middle of your room. (Statistical mechanics tells technically yes, but again with a for all practical purposes zero probability)
* Can you put a monkey in front of a typewriter and get Shakespeare novels out of it?
* Can you walk through a solid wall (non zero tunnel probability due to quantum mechanics)
@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.