Probability of Sets I need some help on this one:
We have sets $X$ and $Y$ chosen independently and uniformly at random from among all subsets of $\{1,2,\ldots,100\}$. Determine the probability that $X$ is a subset of $Y$.

Let $D_n = \{i| i \in \Bbb N \land i \le n\}$.
For an empty set $D_0$ all its subsets $X,Y \subseteq D_0$ are empty: $X = Y = \emptyset$, so $P_0(X\subseteq Y) = 1.$
Now let's consider $D_n$ for $n\in \Bbb N$. Then any subset $X\subseteq D_n$ either is a subset of $D_{n-1}$ or it contains the number $n$, both with probability equal $\tfrac 12$. For subsets $X,Y$ there are four equally possible cases of $n$ inclusion, which results in $$P_n=\tfrac 14P_{n-1} + 0 + \tfrac 14P_{n-1} + \tfrac 14P_{n-1} = \tfrac 34P_{n-1} $$
Since $P_0 = 1 = \left(\tfrac 34\right)^0$, the above makes a general answer $$P_n=\left(\tfrac 34\right)^n$$
