Why does radioactive decay follow an exponential pattern?

Radioactive decay follows an exponential pattern because it is a statistical process where the probability of decay for any individual atom is constant over time.12

In physics, this is expressed by the fact that the number of atoms likely to decay in any given infinitesimal time interval is directly proportional to the total number of radioactive atoms currently present. Mathematically, this relationship is described by the first-order differential equation:1

$$\frac{dN}{dt}=-\lambda N$$

In this equation, $N$ represents the population of radioactive atoms, $t$ is time, and $\lambda$ is the decay constant, which represents the probability of decay per unit time. Integrating this equation results in the exponential decay law:31

$$N=N_0{e}^{-\lambda t}$$

Because each atom acts independently and has a constant probability of decaying—a property known as being "memoryless"—the overall rate of decay decreases as the population of parent nuclei diminishes. This results in the characteristic exponential curve rather than a linear one.123