This applet demonstrates an interesting probability problem that I learned about some few years ago. I do not know it's name, but I'll try to explain what's it about.

Given a grid graph (What is a graph?) and knowing that each edge has three possible states: unset, active and inactive, we assume, that every edge has a given probability of being active. Starting from the "borders" of the graph (== from the edges that are connected to only one vertice) we check each edge whether it is active or not by generating a random number in range of <0;1) and comparing it to the given probability. Once the border edges are set, we iterate over every other edge applying given algorithm: if the edge hasn't been set and it is directly connected to at least one active edge we try to activate it in the same way as the border edges. If the edge hasn't been set and it is connected to inactive edges only we deactivate it.

The algorythm processes the graph from "outside" - the outermost edges - to "inside" - the center of the graph. The result is, as I call it, a flow graph.

The interesting part is the way the graph behaves for different probability values. For low numbers the whole centre of the graph almost always stays deactivated, only the borders are (and event them only partially) active. For large numbers, almost the whole graph is active, with no big inactive areas. The interesting part is that the function of activated edges rises very quickly around 0.6 probability.

Of course the grid should be infinite (or at least very, bery big) for proper results. What you can see below is only a small approximation of the phenomenon - and that's the reason (or at least I think it is) why there are some interesting effects to see for numbers around 0.6 value - as I call them, the "branch effect". See it for yourself in the applet below.

A quick explaination of the applet:
    it shows a 300/300 grid. Red colour is deactivated, blue is activated.
    You can modify the probability value in the input field to the right, to recompute the grid click "compute".
    The computation can take a second or two, so be patient and don't click the button several times. It won't help;)
    The possible values for the probability are <0;1)