Somewhere the connection is not being made. I have seen all the analogies (flipping pennies, popcorn, etc) and know all the equations. What is the simplest self-contained explanation for why radioactive (ie random) decay is exponential, rather than linear, for example? How do you translate saying that "something has an equal probability of occurring at any given time" into an exponential curve on a graph, and why is that curve exponential rather than linear? I understand that "the amount of decay events will reduce", as will "the amount of material that's left to decay", but how do those phrases translate into a non-linear decay over time. Thanks.