# How to count the number of independent parameters in a Bayesian network?

I'm currently going through Prof. Daphne Koller's probabilistic graphical models course on Coursera and had a question regarding an exercise problem.

The problem is as follows:

How many independent parameters are required to uniquely define the conditional probability distribution of $$C$$ in the graphical model below, if $$A$$, $$B$$, and $$D$$ are binary, and $$C$$ and $$E$$ have three values each?

My approach is that since $$C$$ is dependent on $$A$$ and $$B$$ (i.e., the CPD of $$C$$ is $$P(C\ \vert\ A, B)$$) and $$A$$, $$B$$, and $$C$$ each have 2, 2, and 3, values, respectively, we can write $$2 \times 2 \times 3$$. But we have to subtract $$1$$ since knowing all but the last values of $$P(C\ \vert\ A, B)$$ automatically determines the last. Therefore, we get $$(2 \times 2 \times 3) - 1 = 11$$ independent parameters.

Where am I going wrong? Any tips are appreciated, thanks.

Given the values of $$A$$ and $$B$$, there are $$2\times2=4$$ different conditional probability functions of $$C$$. For example, if $$A=1,B=1$$: $$P(C=c_1|A=1,B=1)+P(C=c_2|A=1,B=1)+P(C=c_3|A=1,B=1)=1$$
So, if you happen to know any two of these probabilities, you can find the third because they sum up to $$1$$. For each of these four conditional probabilities, you need to know $$2$$ parameters, ending up with a total of $$8$$ parameters.