Given two events $A$ and $B$ with $P(A)>0$ and $P(B)>0$, define the conditional probability $P(A \mid B)$.

Show that

$$P(B \mid A)=P(A \mid B) \frac{P(B)}{P(A)}$$

A random number $N$ of fair coins are tossed, and the total number of heads is denoted by $H$. If $P(N=n)=2^{-n}$ for $n=1,2, \ldots$, find $P(N=n \mid H=1)$.