Difficulty with Poisson Processes

The wiki page is pretty good.

But basically: Poisson processes are simply a recognition that everything is memoryless.

Throw a dice a thousand times. How many sixes do you expect? 166ish, right? Right, because the odds of getting one are one in six.

Now throw a dice three times. You get no six. How many more throws do you expect until you see a six? Three, because there's a six on average every six throws and we've thrown three, right? Wrong. The dice doesn't know it's been thrown three times before. The probability of the next throw being a six hasn't changed, so the expected number of throws until a six is still six.

You see these sorts of Poisson processes everywhere once you look for them. How long until the next random bus arrives? How many car accidents should we expect per year? Etc.

Hope that helps (and hope my understanding is right :-))

if you're looking for how the poisson probability distribution arises, I did a brief (and quite rubbish) video about it: https://www.youtube.com/watch?v=iT7nxSL1gNQ