I recently built an interactive visualisation [1] to explore dependent probabilities in terms of area and side length of rectangles.
But P(B) just sat there. Puzzling. Then I realized that it starts at P(B|A) = P(B| not A) which is a special case.
An excellent philosophical toy :-)
One big square represents the whole result space having a probability of 1, ie an area of 1 (side length 1x1). The four colors partiotion the whole space in all 4 combinations: 1. A happens and B happens, 2. A happens but not B, 3. A does not happen but B, 4. neither A nor B happen.
Each of the four colors has an area representing the corresponding probability. Naturally all 4 areas sum to 1.
The left and right big squres are simply the same areas just arranged differently to create a straight vertical or horizontal split.
This leads to the side lenghts of the areas corresponding to the probabilities of A happening, B happensing and the dependent probsbilities (eg A happening given that B happens, or B happening given that A happens, etc).
Given that two areas of the same color are the same size you can derive bayes theorem by converting the side length of one rectangle to that of the other same colored reactangle.
For example: The pink rectangle has the area equal to the probability Pr(B∩A). the left pink rectangle is factorized into the side lengths of width=Pr(A) and height=Pr(B|A). So the area is widthheight = Pr(A)Pr(B|A) = Pr(B∩A). The right pink rectangle is factorized into Pr(A|B) and Pr(B) instead, but has the same area Pr(B∩A).
Also you can see that the aligned side of two rectangle sum to 1. For example on the left side the height of the pink plus the height of the orange always sum to one.
Now by changing the probabilities by dragging the sliders you can see that there are configurations in which the height of a rectangle on the left side differs very stark from the width of the rectangle on the right side. For example when Pr(B|A) is totally different than Pr(A|B). But there are also configurations when they match.