Usually, Venn diagrams are not very useful for illustrating independence, as the sizes of the circles and their intersections have no meaning. It can help to illustrate independence if we force the area of each region to be equal to its probability of occurring. Independence is even easier to see if we represent the events with rectangles instead of circles. I illustrate these concepts in this video.

All plots were created in R. The appropriate diameters of the circles and distance between the centres of the circles were calculated in R using the package venneuler.

Not all plots in this video are officially called Venn diagrams.