If the time at which a single atom decays is random, why do groups of atoms behave in predictable ways?

Question

Why do groups of atoms decay at predictable rates even though a single atom’s decay point is completely unpredictable? I’m having trouble wrapping my head around this.

From my reading, it seems that the law of large numbers can explain this. For example, even though it is unknown where a particular coin will land, we can say with reasonable certainty that when tossing tons of coins, about half will land on heads and the other half tails.

But in the case of a coin, there seems to be physical reasons explaining why about half will land on tails and heads. For starters, the coin is balanced on each side and given the rest of our knowledge about nature, there is no reason to prefer one side over the other given how objects behave when thrown in the air.

If one had never seen or tossed a coin before, assuming there was nothing biasing one side, one could still guess that about half the times you land it, it will land on heads. But in the case of atoms, one cannot seem to predict this. We observe frequencies of atoms decaying and then after the fact determine their probabilities of decay.

The question, then, is twofold. First, why is the decay rate or half life of a particular group of atoms X instead of Y? What influences this if we’ve found (and apparently proved) no possible local influence of decay on a single atom?

Second, why is there a constant decay rate for these groups of atoms in the first place? Why not complete and utter chaos (I.e. pure chaos or complete indeterminism)?

Answer 1

This can be answered from statistics only.Lets assume that a atom A has a probability of decaying before 1 min 10% between 1-4 min 80% and after 4 mins 10%.Since the decay of a atom A doesnt depend on the decay of another atom B the probability $P(A\cap B) = P(A)*P(B)$. If you do the multiplication for a sample with n elements you will see that the probability of the sample to decay between 1 and 4 min is almost certain.

Answer 2

An atom, which is initially in the state $\psi_i$, must decay into a state $\psi_f$. This is described by the transition probability using quantum mechanics. To visualise it think of a barrier, through which a particle has to tunnel. The height and width of the barrier determines the transition probability. This answers your first question.

Your second question, why the decay rate is constant, is in fact the result of many random and independent processes. You can check this yourself.

1. Either take one dice and roll it 100 times or take $N=100$ dices and roll them once. Next, count how many times you rolled a six. With 95% probability you will count $[10, 24]$. These numbers correspond to the probabilities $[0.1, 0.24]$.
2. Next, take $N=500$ dices and roll them once. Counting the number six, you will (probability) find $[67, 100]$. This corresponds to the probabilities $[0.134, 0.2]$. Thus, the interval around $1/6=1.\bar6$ has decreased.
3. Finally, take $N=10^{23}$ dices ... I know you won't, but this is approximately the number of atoms in your radio active sample. If we take such a large sample and count how often we find the number six, we will get $\frac{10^{23}}{6} \pm 230\,982\,418\,432$ with 95% probability. If we calculate he corresponding probabilitie we get $\frac{1}{6}\pm 2.3098\cdot 10^{-12}$. Thus, although we know that rolling each dice yields a random result, we are able to predict the cumulated outcome extremely precisely.