Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{\text {th }}$ coin toss, let $X_i=1$ if head appears, and $X_i=0$ if tail appears.

Consider

$$ \hat{p}=\frac{1}{n} \sum_{i=1}^n X_i $$

where $n$ is the total number of independent coin tosses.

Let $\quad f: \mathbb{R} \rightarrow \mathbb{R} \quad$ be such that $|f(x)-f(y)| \leq(x-y)^2$ for all $x, y \in \mathbb{R}$.

Then $\quad f(1)-f(0)=$ ____________

A bag contains 5 white balls and 10 black balls. In a random experiment, $n$ balls are drawn from the bag one at a time with replacement. Let $S_n$ denote the total number of black balls drawn in the experiment.

The expectation of $S_{100}$ denoted by $E\left[S_{100}\right]=$ ___________ (Round off to one decimal place)

Consider a directed graph $G=(V, E)$, where $V=\{0,1,2, \ldots, 100\}$ and $E=\{(i$, $j): 0 < j-i \leq 2$, for all $i, j \in V\}$. Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0 . The number of vertices that will be discovered after vertex 50 is___________