I'm struggling to see how to prove that a typical sequence will constitute approximately 0.5n s if it is generated by $i.i.d$ source with P(0)=0.5 and P(1)=0.5. Could we prove this from the definition of a typical set?
When $P(0)=P(1)=1/2$, by definition all binary sequences are in the typical set. Since they may contain any number of successes, I don't believe we can come up with an approximation argument regarding it. However, we can talk about the mean and other relevant statistics on the number of success, which won't differ from discussions made on Binomial RV.