My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood. Eg. if an event has a 1/10 chance of

My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood. Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials? Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is: = 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%) This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!