[fsckboy]

>purely a result of more likely things happening more often on average

according to your wording, no. if you have a perfect six sided die (or perfect two sided coin), none/neither of the outcomes are more likely at any point in time... yet something approximating entropy occurs after many repeated trials. what's expected to happen is the average thing even though it's never the most likely thing to happen.

you want to look at how repeated re-convolution of a function with itself always converges on the same gaussian function, no matter the shape of the starting function is (as long as it's not some pathological case, such as an impulse function... but even then, consider the convolution of the impulse function with the gaussian)