Problem 1 (yau-contest 2017)
Section: Probability and Statistics

A box contains 750 red balls and 250 blue balls. Repeatedly pick a ball uniformly at random from the box and remove it until all remaining balls have a single color. (Note: no replacement).

Please find integer $m$ such that the expectation value for the total number of the remaining balls $\in [m, m + 1]$.

Problem 2 (yau-contest 2017)
Section: Probability and Statistics

Suppose a number $X_0 \in \{1,-1\}$ at the root of a binary tree is propagated away from the root as follows. The root is the node at level $0$. After obtaining the $2^h$ numbers at the nodes at level $h$, each number at level $h+1$ is obtained from the number adjacent to it (at level $h$) by flipping its sign with probability $p \in (0, 1/2)$ independently.

Let $X_h$ be the average of the $2^h$ values received at the nodes at level $h$. Define the signal-to-noise ratio at level $h$ to be
$$
R_h := \frac{\left(\mathbb{E}[X_h \mid X_0 = 1] - \mathbb{E}[X_h \mid X_0 = -1]\right)^2}{\operatorname{Var}[X_h \mid X_0 = 1]}.
$$
Find the threshold number $p_c$ such that $R_h$ converges to $0$ if $p \in (0, p_c)$, as $h \to \infty$ and diverges if $p \in (0, p_c)$, as $h \to \infty$.

Problem 3 (yau-contest 2017)
Section: Probability and Statistics

Consider the space representing an infinite sequence of coin flips, namely $\Omega := \{H, T\}^\infty$, (H: head, T: tail) with the associated $\sigma$-field $\mathcal{F}$ generated by finite dimensional rectangles. For $0 \le p \le 1$, denote by $\mathbb{P}_p$ the probability measure on $(\Omega,\mathcal{F})$ corresponding to flipping a coin an infinite number of times with probability of $H$ being $p$ and probability of $T$ being $q = 1 - p$ at each flip.

Show that for each $p \in [0,1]$, there exists $A_p$ such that
$$\mathbb{P}_p(A_p) > 1/2$$
and for any $p' \ne p$, $p' \in [0,1]$
$$\mathbb{P}_{p'}(A_p) < 1/2.$$

Problem 4 (yau-contest 2017)
Section: Probability and Statistics

Let $G := G(n,p)$ be a random graph with $n$ vertices where each possible edge has probability $p$ of existing. The existence of the edges are independent to each other. With $G$, we say $A \subset \{1,2,\cdots,n\}$ is a fully connected set if and only if
$$i,j \in A \Longrightarrow i\text{–th and }j\text{–th vertices are (directly) connected with an edge in }G.$$
Define $T$ as the size of the largest fully connected set
$$T := \max\{|A| : A \text{ is a fully connected set}\}.$$
Let’s fix $p \in (0, 1)$, please prove that
$$\lim_{n \to \infty} \mathbb{P}\left( \frac{T}{2 \log_{1/p} n} \le 1 + \epsilon \right) = 1, \quad \forall \epsilon > 0,$$
and
$$\lim_{n \to \infty} \mathbb{P}\left( \frac{T}{\sqrt{2 \log_{1/p} n}} \ge 1 - \epsilon \right) = 1, \quad \forall \epsilon > 0,$$
Hint:
$$\mathbb{P}(T = n) = p^{\binom{n}{2}} = p^{n(n-1)/2}.$$

Problem 5 (yau-contest 2017)
Section: Probability and Statistics

Consider a population of constant size $N + 1$ that is suffering from an infectious disease. We can model that spread of the disease as Markov process. Let $X(t)$ be the number of healthy individuals at time $t$ and suppose that $X(0) = N$. We assume that if $X(t)$
$$\lim_{h \to 0} \frac{1}{h} \mathbb{P}(X(t + h) = n - 1 \mid X(t) = n) = \lambda n (N + 1 - n)$$
For $0 \le s \le 1$, $0 \le t$, define
$$G(s,t) := \mathbb{E}(s^{X(t)}).$$
Please find a non-trivial partial differential equation for $G(s,t)$, which involves $\partial_t G.$