Simple information theory question: where is this equation coming from?

Question

I am reading a simple example of a joint distribution that looks like this:

                C
      _____________________
       p        t        k

 |  a  1/16     3/8      1/16 
V|  i  1/16     3/16     0
 |  u   0       3/16     1/16

The book says "we can work out the entropy of the joint distribution in more than one way. Let's use the chain rule:

$$ H(C) = 2 * (1/8) * 3 + (3/4)(2 - log 3) \\ H(C) = \frac{9}{4} - \frac{3}{4} * log 3 $$

Where is that coming from? I get that the marginal probability of p and k is 1/8 and the marginal probability of t is 3/4. I also know the formula for entropy is

$$ \Sigma(x) = p(x) * log \frac{1}{p(x)} $$

So applying the formula I would get

$$ 2 * 1/8 * log \frac{1}{8} + 3/4 * log \frac{1}{\frac{3}{4}} \\ 2 * 1/8 * 3 + 3/4 * log \frac{1}{\frac{3}{4}} $$

But how do I get from that last line to their final step? Have I set up the equation wrong? It seems like they are skipping some algebraic steps that I do not follow.

Answer 1

You are almost there. The second term of your last formula can be writen as

$$ \frac{3}{4}\log{\frac{4}{3}}= \frac{3}{4}(\log 4 -\log3)=\frac{3}{4}(2 -\log3)$$