Probability with powers of $2$ There are $2^H$ tickets. Let $k<H$ (and so $2^k$ is a submultiple of $2^H$). Let $D$ be a random integer number taken uniformly from $1,2,..k$. Then $2^D$ is in turn a submultiple of $2^k$ and then also of $2^H$. Once we select $D$ at random, we pick at random one ticket every $2^D$ tickets. That is, we divide the $2^H$ tickets in $\frac{2^H}{2^D}$ distinct groups and we pick at random one ticket from each group (indeed we will then pick $\frac{2^H}{2^D}$ tickets). What is in the end the probability of being picked for each ticket in the set of $2^H$ tickets?

If $D=1$, which has probability $\frac 1k$, the chance a ticket is picked is $\frac 12$. If $D=2$, also probability $\frac 1k$, the chance a ticket is picked is $\frac 1{2^2}$ The total probability a ticket is picked is then $$\frac 1k\sum_{i=1}^k \frac 1{2^i}=\frac 1k\left(1-\frac 1{2^k}\right)$$