I Why Does Unstable Particle Decay Follow an Exponential Curve?

[Trollfaz]

Given that an unstable particle has a constant probability of decaying per unit time, why is it said that its chance of surviving falls exponentially?

 

[BvU]

It's like froth on beer: the more there is, the more there decays.
Or (in the other direction) like money accumulating on a bank: the more there is, the more interest (in absolute sense) you get, which gains interest again, etc.

Probability $p$ is given, number of decays is proportional to number of particles $N$, so $$ {dN\over dt} = - p N $$.

At an "I" level you can solve this kind of differential equation

 

[mfb]

It has a constant probability to decay per unit time if it still lives. This survival probability goes down over time, therefore the probability to see a decay after time x goes down with increasing x. If you solve the differential equation, you see that it goes down exponentially.