# How to prove that there will be 0.5n 1s and 0s in the typical sequence for a $i.i.d$ source with 0.5 probability?

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?

• You're going to have to define what you intend by 'typical' here, because typically there will be an unequal number of 0's and 1's, and the counts will diverge as n increases. – Glen_b Nov 28 '19 at 14:00

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.