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# Total Probability and Bayes's Theorem

This Demonstration provides examples of total probability and Bayes's theorem. In the given world a figure X is randomly chosen. What is the probability of the given statement S? Suppose the statement is true. What is the probability that X = A? What is the probability that X = B?
If the probability of S is 0, the conditional probability P(X=A|S) is undefined (or undecided, denoted by U).
A simple two-dimensional area is occupied by white or gray triangles, squares, and pentagons. A disk means that the shape of the element is not known; in such a case a proposition of type Shape() has probability 1/3. A gray-white figure means that the color of the figure is not known; in such a case a proposition of type Color(x) has probability 1/2.

### DETAILS

The conditional probability of an event A assuming that B has occurred, denoted P(A|B), equals
P(A|B)=(P(A ⋂ B))/(P(B)). If P(B)=0, P(A|B) is undefined.

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