Beta analytic inc radiocarbon dating
Note that after one half-life there are not exactly one-half of the atoms remaining, only approximately, because of the random variation in the process.Nevertheless, when there are many identical atoms decaying (right boxes), the law of large numbers suggests that it is a very good approximation to say that half of the atoms remain after one half-life.
In other words, the probability of a radioactive atom decaying within its half-life is 50%.
For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay.
The number at the top is how many half-lives have elapsed.
Note the consequence of the law of large numbers: with more atoms, the overall decay is more regular and more predictable.
Mathematically, the sum of two exponential functions is not a single exponential function.