We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. A path is any $2n$-tuple $\gamma = (\varepsilon_1, \cdots, \varepsilon_{2n})$ whose components $\varepsilon_k$ equal $-1$ or $1$. An equality index of a path is any integer $k \in \llbracket 1, 2n \rrbracket$ such that $\sum_{i=1}^k \varepsilon_i = 0$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by: $$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$ Calculate the probability $\mathbf{P}(A_i)$, for every integer $i$ between $1$ and $n$.

We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. Let $\ell \in \mathbf{Z}$ be an integer and $n \geqslant 1$ be another integer. By distinguishing the case where the integer $\ell - n$ is even or odd, calculate $\mathbf{P}(S_n = \ell)$.

For $N = 1$, among random variables with usual distributions, give without justification one example of a random variable satisfying (8) and two examples of random variables not satisfying (8), where (8) states $\mathbb{P}(|X_n| \leq K) = 1$ for some constant $K \geq 1$, with $\mathbb{E}[X_n] = 0$ and $\operatorname{Var}(X_n) \leq 1$.

For all $N \geq 1$, give an example of random variables satisfying hypotheses $$\mathbb{P}(|X_n| \leq K) = 1, \quad \mathbb{E}[X_n] = 0, \quad \operatorname{Var}(X_n) \leq 1$$ and such that $\mathbb{P}(|S_N| \geq N) \geq 1/2$, where $S_N := X_1 + \cdots + X_N$.

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
Verify that $$Z_n(h) = \sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{\beta x_i x_{i+1} + h x_i}$$

Establish the following identity:
$$\mathbf { E } \left[ \Delta \tilde { R } _ { n } \right] = \rho \mathbf { E } \left[ \tilde { I } _ { n } \right]$$
where each infected person recovers at the end of the day with probability $\rho \in ]0,1[$, independently of others, and $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$.

Show that
$$\mathbf { E } \left[ \Delta \tilde { S } _ { n } \right] = - \mathbf { E } \left[ \tilde { S } _ { n } p \left( \tilde { I } _ { n } \right) \right]$$
then deduce the equation satisfied by $\mathbf { E } \left[ \Delta \tilde { I } _ { n } \right]$.
Here $\Delta \tilde{S}_n = \tilde{S}_{n+1} - \tilde{S}_n$, $\Delta \tilde{I}_n = \tilde{I}_{n+1} - \tilde{I}_n$, $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$, and $\tilde{S}_n + \tilde{I}_n + \tilde{R}_n = M$ for all $n$.

Let $0 < x < \frac { 1 } { 6 }$ be a real number. When a certain biased dice is rolled, a particular face $F$ occurs with probability $\frac { 1 } { 6 } - x$ and and its opposite face occurs with probability $\frac { 1 } { 6 } + x$; the other four faces occur with probability $\frac { 1 } { 6 }$. Recall that opposite faces sum to 7 in any dice. Assume that the probability of obtaining the sum 7 when two such dice are rolled is $\frac { 13 } { 96 }$. Then, the value of $x$ is:
(A) $\frac { 1 } { 8 }$
(B) $\frac { 1 } { 12 }$
(C) $\frac { 1 } { 24 }$
(D) $\frac { 1 } { 27 }$.

In a room with $n \geqslant 2$ people, each pair shakes hands between themselves with probability $\frac{2}{n^2}$ and independently of other pairs. If $p_n$ is the probability that the total number of handshakes is at most 1, then $\lim_{n \rightarrow \infty} p_n$ is equal to
(A) 0
(B) 1
(C) $e^{-1}$
(D) $2e^{-1}$

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X > Y )$ is
(A) $\frac { 1 } { 4 }$
(B) $\frac { 5 } { 12 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 7 } { 12 }$

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X = Y )$ is
(A) $\frac { 11 } { 36 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 13 } { 36 }$
(D) $\frac { 1 } { 2 }$

Consider the $6 \times 6$ square in the figure. Let $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$ be the points of intersections (dots in the picture) in some order. We say that $A _ { i }$ and $A _ { j }$ are friends if they are adjacent along a row or along a column. Assume that each point $A _ { i }$ has an equal chance of being chosen.
Let $p _ { i }$ be the probability that a randomly chosen point has $i$ many friends, $i = 0,1,2,3,4$. Let $X$ be a random variable such that for $i = 0,1,2,3,4$, the probability $P ( X = i ) = p _ { i }$. Then the value of $7 E ( X )$ is

Let $X$ be a random variable, and let $P ( X = x )$ denote the probability that $X$ takes the value $x$. Suppose that the points $( x , P ( X = x ) ) , x = 0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P ( X = x ) = 0$ for all $x \in \mathbb { R } - \{ 0,1,2,3,4 \}$. If the mean of $X$ is $\frac { 5 } { 2 }$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_$ .

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is
(1) $22 \left( \frac { 1 } { 3 } \right) ^ { 11 }$
(2) $\frac { 5 } { 19 }$
(3) $55 \left( \frac { 2 } { 3 } \right) ^ { 10 }$
(4) $220 \left( \frac { 1 } { 3 } \right) ^ { 12 }$

An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is
(1) $\frac { 3 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $- \frac { 3 } { 16 }$
(4) $- \frac { 1 } { 8 }$

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is:
(1) $\frac { 5 } { 31 }$
(2) $\frac { 31 } { 61 }$
(3) $\frac { 5 } { 6 }$
(4) $\frac { 30 } { 61 }$

The probability distribution of random variable $X$ is given by:

$X$	1	2	3	4	5
$P ( X )$	$K$	$2 K$	$2 K$	$3 K$	$K$

Let $p = P ( 1 < X < 4 \mid X < 3 )$. If $5 p = \lambda K$, then $\lambda$ is equal to

A six faced die is biased such that $3 \times P ($ a prime number $) = 6 \times P ($ a composite number $) = 2 \times P ( 1 )$. Let $X$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $X$ is
(1) $\frac { 3 } { 11 }$
(2) $\frac { 5 } { 11 }$
(3) $\frac { 7 } { 11 }$
(4) $\frac { 8 } { 11 }$

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m - 3n$ is equal to
(1) 6
(2) 12
(3) 10
(4) 8

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p : q = m : n$, where $m$ and $n$ are co-prime, then $m + n$ is equal to

If the mean of the following probability distribution of a random variable $X$ :

X	0	2	4	6	8
$\mathrm { P } ( \mathrm { X } )$	$a$	$2a$	$a + b$	$2b$	$3b$

is $\frac { 46 } { 9 }$, then the variance of the distribution is
(1) $\frac { 173 } { 27 }$
(2) $\frac { 566 } { 81 }$
(3) $\frac { 151 } { 27 }$
(4) $\frac { 581 } { 81 }$

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is
(1) $\frac { 5 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 5 } { 11 }$
(4) $\frac { 6 } { 11 }$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $\mathrm { a } = \mathrm { P } ( \mathrm { X } = 3 ) , \mathrm { b } = \mathrm { P } ( \mathrm { X } \geq 3 )$ and $\mathrm { c } = \mathrm { P } ( \mathrm { X } \geq 6 \mid \mathrm { X } > 3 )$. Then $\frac { \mathrm { b } + \mathrm { c } } { \mathrm { a } }$ is equal to

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar { X }$ and $\bar { Y }$ are the means of $X$ and $Y$ respectively, then $7 \bar { X } + 4 \bar { Y }$ is equal to $\_\_\_\_$

One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 4 } { 9 }$
(4) $\frac { 3 } { 5 }$