Exercise 6.4 Suppose we want to diagnose the errors school students make when adding multidigit binary numbers. Suppose we are only considering adding two two-digit numbers to form a three-digit number.

That is, the problem is of the form:
A1 A0 + B1 B0 C2 C1 C0 , where Ai, Bi, and Ci are all binary digits.

(a) Suppose we want to model whether students know binary addition and whether they know how to carry. If they know how, they usually get the correct answer, but sometimes they make mistakes. If they don’t know how to do the appropriate task, they simply guess.
What variables are necessary to model binary addition and the errors students could make? You must specify, in words, what each of the variables represents. Give a DAG that specifies the dependence of these variables.

(b) What are reasonable conditional probabilities for this domain?

(c) Implement this, perhaps by using the AIspace.org belief-network tool. Test your representation on a number of different cases.
You must give the graph, explain what each variable means, give the probability tables, and show how it works on a number of examples.