Having two binary numbers of length 6, what is the probability that they match exactly? What is the probability that they have hamming distance of exactly 1? or of 2?
For the first part, the number of possible variants of the binary number is 2^6 I believe. What is the probability of the second binary number matching the first? It seems to me that this probability would be 1/2 ^ 6?
If this probability is found is it a simple matter to then find the probability of when they mismatch by exactly one number?
Your $\frac{1}{2^6}$ is correct if we can assume that all the binary sequences of length $6$ are equally likely. That is not necessarily a safe assumption in the real world.
For Hamming distance $1$, let $A$ be a given sequence. We find the probability that a randomly chosen sequence $B$ matches $A$ at all but one place.
There are $2^6$ possibilities for $B$, by assumption equally likely, If $B$ is to match $A$ in all but $1$ place, the place where $B$ does not match $A$ can be chosen in $\binom{6}{1}$ ways. So our probability is $\frac{\binom{6}{1}}{2^6}$.
Similar reasoning can be used for Hamming distance $2$.