Entropy contribution from variable length segment of a sequence.

If I have a sequence which is comprised of one of $$10$$ prefixes, one of $$5$$ suffixes and a variable length middle, how do I compute the entropy of the sequence?

Using Shannon-Entropy $$H= -\sum_{i=1}^{m} p_i \ln(p_i)$$

I can compute the entropy contributions from the $$10$$ starting and $$5$$ ending sequences using this sum, but I am unsure how to computer the variable length section. If the distribution of length is:

• $$0=0.5$$
• $$1=0.25$$
• $$2=0.2$$
• $$3=0.05$$

And each character can be A-F say, for $$16$$ possibility with equal probability how do I calculate the number of bits of entropy? If I simply use the sum above, I am underestimating the contribution from the longer sequences, which have more possibilities. Do I also need to ignore the $$p_0$$ term, as $$0.5\ln(0.5)$$ isn't actually adding any additional characters to the sequence.

If all $16$ possibilities has an equal probability, that entropy must $\ln 16\approx 2.77$ its; $H\sim \max= \sum[ \frac1n \ln(\frac1n) ]$. Adding new information with "equal" probability is not add extra information into entropy.