# Total Probability and Bayes's Theorem

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.

x