How do particles "know" when to decay? So, as I understand it, in a substance that is made of radioactive elements, the half-life tells us how long until the half of those atoms decay into their next atom [is there a name for that: the element or isotope that is the result of a prior radioactive decay?]. My question is, is there some sort of pattern to which atoms decay at which time, or is it some miraculous property of quantum mechanics that somehow each atom knows when to decay? or do I just understand radioactive decay incorrectly? 
Are the particles entangled or in some otherway attached (aside from their molecular bonds)? If there is no known answers, I would prefer any serious, leading theories. 
 A: There are no patterns. When a particle decays, the moment when it does so is absolutely random, chosen from the distribution
$$  P_{\rm decay}(t\lt T\lt t+dt) = \frac{dt}{t_0}\cdot \exp(-t/t_0)  $$
For $t=t_0$, the beginning of time when we knew that the particle still existed, the exponential is equal to one and we see that the "probability of the decay per unit time" is $1/t_0$. As the probability that the particle still exists exponentially decreases, so does the probability that it decays at a later moment.
The randomness of the decay time is just another example of the randomness that quantum mechanics, the basic framework for all the laws of physics since 1925, predicts for every phenomenon in Nature.
In the most widespread description of quantum mechanics, the decaying particle is described by a wave function. And that wave function evolves into a superposition of the undecayed and  (various) decayed components, and the probability amplitude (value of the wave function) associated with the undecayed particle decreases as $\exp(-t/2t_0)$. This probability amplitude has to be squared and the result, $\exp(-t/t_0)$, gives us the probability that the particle hasn't decayed yet.
Theories that would try to find some "internal" reason why the particle decayed at the given moment are called "hidden variable theories" and they may be shown incorrect – either incompatible with the experiments about the decay in this case, or with experiments backing the special theory of relativity. So physicists have to embrace the intrinsic randomness of Nature as a fact. The randomness of the decay time is a Nature's perfect random generator, one that can't be fooled or cheated.
A: 
Is there some sort of pattern to which atoms decay at which time, or is it some miraculous property of quantum mechanics that somehow each atom knows when to decay?

Atoms are dumb. They don't know anything. Radioactive decay is a memoryless process, a process that  doesn't depend on history. Consider three atoms of radon 222. One was created a month ago (8 half-lives), another four days ago (~1 half-life), and the third, 12 hours ago (~ 1/8 half-life). Which will be the next to decay? The atoms don't know. Nobody knows; in fact, nobody can know. Each of the three atoms has the exact same small chance of decaying in the next minute.
Radioactive decay is the canonical example of a Poisson process. Knowledge (or state) is not required in a Poisson process. State gets in the way. An ideal Poisson process is stateless and memoryless. While there are lots and lots of examples of processes whose probability distribution is close to Poisson, nothing comes closer to it than does radioactive decay.
A: You have just read reasonable answers of knowledgeable people, so now you know that "radioactive decay...- it's completely stateless (@Luaan)", "There are no patterns" (@Luboš Motl), and "Atoms are dumb" (@David Hammen). However, there is a bit more to it. Atoms may be dumb, but they happen to know quantum mechanics much better than we, mere mortals, do. So there may be randomness, but there cannot be perfect randomness. Someone Khalfin showed many years ago that strictly exponential decay is not compatible with quantum theory, there must be some tiny deviations, both at very short and very long times. Please see references to theoretical and experimental work in my answer at Does average lifetime even mean anything?
