# Calculating Expected Hamming Distance

In my Statistics class, Huffman coding has just been introduced to us. However, I don't believe I have fully grasped the concept behind hamming. In lecture, we were presented with a question:

The hamming distance between two binary strings is the number of coordinates in which they are different. If X and Y are independent random binary strings of length 4, what is the expected value of the hamming distance between X and Y?

I understand this gives me 16 equally likely strings of length 4 to consider here. From 0000, 0001, 0011... 1110, 1111. However, I'm unaware of how to calculate the expected hamming distance here. Originally, I was mistakening this for entropy. I first tried calculating

$$\sum_{n=1}^{16} \frac1{16}\log_2\frac{1}{\frac{1}{16}} = 4$$

Where $\frac{1}{16}$ represents the probability of being any one of the 16 string lengths. However, I was told this is incorrect. How do I calculate the expected hamming distance?

• Shorten the problem. We don't really care what the first string is, we just want to look at how the second string can differ. In how many ways can the strings differ in 0 places (i.e. be identical)?In how many ways can the strings differ in exactly 1 place?Exactly 2 places? 3? 4? 5? 6? 7? 8? I think you'll find combinations to be of some use here. – Sean Henderson May 1 '17 at 21:04

Expected Hamming distance is the expected number of positions where the binary strings $X$ and $Y$ are different.
For example, if you are looking at strings of length $2$, you have 4 possible values for each of $X$ and $Y$.
You will have 16 total entries and are looking for the average value in the Different column. After some observation it should be clear each 4 entries will contain $0,1,1,2$ in some permutation, so the overall average is $$\frac{0+1+1+2}{4} = 1.$$ Hence, one average, in strings of length 2, you get 1 common place. Can you do this for strings of length $4$?