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 $Y$.

For every element $a$ of initial set one of four options is true:  


*

*$\quad a \in X,\quad a \in Y$

*$\quad a \not\in X,\quad a \in Y$

*$\quad a \in X,\quad a \not\in Y$

*$\quad a \not\in X,\quad a \not\in Y$


The probability of each option is $1/4$ (due to independence and uniformity).
$X\subseteq Y$ if for all $a \in \{1,\ldots,100\}$ options number $1$, $2$ or $4$ are true. For every element probability is $\dfrac{3}{4}$ and for whole set it is $\left(\dfrac{3}{4}\right)^{100}$.
