Compressibility is defined as

$ $ C=\frac{2^{HN}}{2^{H_{max}N}}$ $

The book is made up of a simple alphabet of only {a,b,c,d} which occur with probabilities $ $ P(a)=0.2, P(b)=0.4, P(c)=0.1, P(d)=0.3$ $

In class we were given the example of calculating the entropy of a simple string of coin tosses, where $ N$ is the number of coin tosses.

So, for example, we could say that $ P(heads)=0.1$ and $ P(tails)=0.9$ . The entropy of each coin toss is therefore $ H(p=0.9)=0.469$ If we toss the coin 4 times then we end up with a compressibility of: $ $ C=\frac{2^{HN}}{2^{H_{max}N}}=\frac{2^{0.469\cdot4}}{2^{1\cdot 4}}=22.3\%$ $ How do we extend this to the book case?

The entropy of the letter occurance is $ $ H(p=0.2,p=0.4,p=0.1,p=0.3)=1.85$ $

and the maximum entropy is just 2 bits (i.e. we require 2 bits to descibe four characters)

Since this is all the information given in the question, I presume that we just set $ N=4$ and plug in the values for $ H$ and $ H_{max}$ to get the entropy. However, I am struggeling to see why. Why is $ N$ not the number of letters that make up this book?