What is the compressibility of this simple “book”?

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?

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• Can you add in the question an accessible reference where your compressibility is introduced? – Apass.Jack Jan 14 at 5:43