There are 6 weights of 1 unit, 3 weights of 2 units, and 1 empty bag. Using one die, the following trial is performed. (Here, the unit of weight is g.)
Roll the die once. If the number shown is 2 or less, put one weight of 1 unit into the bag. If the number shown is 3 or more, put one weight of 2 units into the bag.
Repeat this trial until the total weight of the weights in the bag is first greater than or equal to 6. Let $X$ be the random variable representing the number of weights in the bag. The following is the process of finding the probability mass function $\mathrm { P } ( X = x ) ( x = 3,4,5,6 )$ of $X$.
(i) The event $X = 3$ is the case where 3 weights of 2 units are in the bag, so $$\mathrm { P } ( X = 3 ) = \text{ (a) }$$ (ii) The event $X = 4$ can be divided into the case where the total weight of weights put in by the third trial is 4 and a weight of 2 units is put in on the fourth trial, and the case where the total weight of weights put in by the third trial is 5. Therefore, $$\mathrm { P } ( X = 4 ) = \left( \text{ (b) } + { } _ { 3 } \mathrm { C } _ { 1 } \left( \frac { 1 } { 3 } \right) ^ { 1 } \left( \frac { 2 } { 3 } \right) ^ { 2 } \right) \times \frac { 2 } { 3 }$$ (iii) The event $X = 5$ can be divided into the case where the total weight of weights put in by the fourth trial is 4 and a weight of 2 units is put in on the fifth trial, and the case where the total weight of weights put in by the fourth trial is 5. Therefore, $$\mathrm { P } ( X = 5 ) = { } _ { 4 } \mathrm { C } _ { 4 } \left( \frac { 1 } { 3 } \right) ^ { 4 } \left( \frac { 2 } { 3 } \right) ^ { 0 } \times \frac { 2 } { 3 } + \text{ (c) }$$ (iv) The event $X = 6$ is the case where the total weight of weights put in by the fifth trial is 5, so $$\mathrm { P } ( X = 6 ) = \left( \frac { 1 } { 3 } \right) ^ { 5 }$$ If the values corresponding to (a), (b), (c) are $a , b , c$ respectively, what is the value of $\frac { a b } { c }$? [4 points]
(1) $\frac { 4 } { 9 }$
(2) $\frac { 7 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 13 } { 9 }$
(5) $\frac { 16 } { 9 }$

16. A bank stipulates that if a bank card has 3 incorrect password attempts in one day, the card will be locked. Xiaowang went to the bank to withdraw money and found that he forgot his bank card password, but he is certain that the correct password is one of his 6 commonly used passwords. Xiaowang decides to randomly select one without replacement to try. If the password is correct, he stops trying; otherwise, he continues trying until the card is locked.
(1) Find the probability that Xiaowang's bank card is locked that day;
(2) Let $X$ denote the number of password attempts Xiaowang makes that day. Find the probability distribution of $X$ and its mathematical expectation.

To promote the development of table tennis, a certain table tennis competition allows athletes from different associations to form teams. There are 3 athletes from Association A, of which 2 are seeded players, and 5 athletes from Association B, of which 3 are seeded players. Randomly select 4 people from these 8 athletes to participate in the competition.
(I) Let A be the event ``exactly 2 seeded players are selected, and these 2 seeded players are from the same association''. Find the probability of this event.
(II) Let X be the number of seeded players among the 4 selected people. Find the probability distribution and mathematical expectation of the random variable X.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. Give the distribution of the random variable $X_{i,j} = X_i \times X_j$.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Give the distribution, expectation and variance of the random variables $\operatorname{tr}(M)$ and $\operatorname{rg}(M)$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ have the same parity, $\mathbb{P}\left(S_{n} = k\right) = \binom{n}{(k+n)/2} \frac{1}{2^{n}}$.

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. For every $i \in \llbracket 0, n \rrbracket$, show $\mathbb{P}(M_n > i) = p_i$.

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws.
Identify the distribution of $X_{n}$ and give its expectation.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn.
By listing all possible outcomes, give the distribution of $X_{3}$.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Verify that $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.

We consider a general balanced urn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$. We denote by $g_{n}$ the generating function of $X_{n}$ (the number of white balls after $n$ draws).
Justify the equalities $$\begin{aligned} & g_{n}(t) = \frac{1}{\operatorname{card}(\Omega_{n})} P_{n}(t, 1) \\ & E(X_{n}) = \frac{1}{\operatorname{card}(\Omega_{n})} \frac{\partial P_{n}}{\partial u}(1,1) \end{aligned}$$

In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), fix an integer $n \geqslant 2$.
Using the Taylor expansion of $g_{n}$ to order $n$ at 0, deduce that, for all $m$ in $\llbracket 1, n \rrbracket$, $$P(X_{n} = m) = \frac{1}{n!} \sum_{k=0}^{m-1} (-1)^{k} \binom{n+1}{k} (m-k)^{n}.$$

In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), an algorithm constructs a permutation of $S_{n}$ from an outcome of $n$ draws. Using question 33, compare, for any outcome, the number of white balls in the final composition of the urn with the number of ascents of the permutation associated with it by the algorithm above.

In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ For $n \in \mathbb{N}$, determine $P \left( S _ { 2 n + 1 } = 0 \right)$ and justify the equality: $$P \left( S _ { 2 n } = 0 \right) = \binom { 2 n } { n } ( p q ) ^ { n }$$

Let $n$ be a non-zero natural number.

1. Let $p$ be a vector projection of rank $r \in \mathbb { N }$.
   1. [1.1.] Give, as a function of $r$, a matrix $W$ of $p$ in an adapted basis.
   2. [1.2.] Give the possible spectra of $W$.
   3. [1.3.] Compare $\boldsymbol { \operatorname { rg } } ( W )$ and $\boldsymbol { \operatorname { tr } } ( W )$.
   4. [1.4.] Calculate $\boldsymbol { \operatorname { det } } ( W )$.

We consider the family $X _ { 1 } , \ldots , X _ { n }$ of independent random variables defined on the same probability space $( \Omega , \mathscr { A } , \mathbb { P } )$ all following the Bernoulli distribution with parameter $p \in ]0,1[$.
Let $M$ be a discrete random variable from $\Omega$ to $\mathscr { M } _ { n } ( \mathbb { R } )$ such that for all $\omega$ in $\Omega , M ( \omega )$ is diagonalisable and similar to $\Delta ( \omega ) = \operatorname { diag } \left( X _ { 1 } ( \omega ) , \ldots , X _ { n } ( \omega ) \right)$.

1. We denote by $T$ the random variable $\mathbf { tr } ( M )$.
   1. [2.1.] Determine $T ( \Omega )$, that is the set of values taken by the random variable $T$.
   2. [2.2.] Give the probability distribution of $T$ and the expectation of the random variable $T$.
2. Deduce the probability distribution of the random variable $R = \mathbf { rg } ( M )$.
3. We denote by $D$ the random variable $\boldsymbol { \operatorname { det } } ( M )$.
   1. [4.1.] Determine $D ( \Omega )$.
   2. [4.2.] Give the probability distribution of $D$ and calculate the expectation of the random variable $D$.
4. We propose to determine the probability of the event $Z$: ``the eigenspaces of the matrix $M$ all have the same dimension''.
   1. [5.1.] We denote by $V$ the event: ``$M$ has only one eigenvalue''. Calculate $\mathbb { P } ( V )$.
   2. [5.2.] Suppose $n$ is odd. Determine $\mathbb { P } ( Z )$.
   3. [5.3.] Suppose $n$ is even and set $n = 2 r$. Calculate $\mathbb { P } ( T = r )$. Deduce $\mathbb { P } ( Z )$.
5. For all $\omega \in \Omega$, we denote $U ( \omega ) = \left( \begin{array} { c } X _ { 1 } ( \omega ) \\ \vdots \\ X _ { n } ( \omega ) \end{array} \right) \in \mathscr { M } _ { n , 1 } ( \mathbb { R } )$ and $A ( \omega ) = U ( \omega ) \times ( U ( \omega ) ) ^ { \top } = \left( a _ { i j } ( \omega ) \right) _ { ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } }$.
   1. [6.1.] Let $\omega \in \Omega$. Determine, for all pairs $( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , a _ { i j } ( \omega )$.
   2. [6.2.] Give the probability distribution of each random variable $a _ { i j }$.
   3. [6.3.] Show that $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } X _ { i }$.
   4. [6.4.] Determine the values taken by the random variable $\boldsymbol { \operatorname { rg } } ( A )$.
   5. [6.5.] For all $\omega$ in $\Omega$, give the eigenvalues of the matrix $A ( \omega )$.
   6. [6.6.] Determine the probability distribution of the random variable $\mathbf { rg } ( A )$.

Show that the distribution of the random variable $X_n$ is given by $$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability. We define a random variable $Z_{n}$ by $Z_{n}(\sigma) = \nu(\sigma)$.
Explicitly state the distribution of $Z_{n}$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Specify the distribution of $Z_n$.

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 $$Z_n(h) = \sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{\beta x_i x_{i+1} + h x_i}$$

Let $z _ { n }$ and $w _ { n } ( n = 0,1,2 , \ldots )$ be complex numbers. Consider a bag that contains two red cards and one white card. First, take one card from the bag and return it to the bag. $z _ { k + 1 } ( k = 0,1,2 , \ldots )$ is generated in the following manner based on the color of the card taken.
$$z _ { k + 1 } = \begin{cases} i z _ { k } & \text { if a red card was taken, } \\ - i z _ { k } & \text { if a white card was taken. } \end{cases}$$
Next, take one card from the bag again and return it to the bag. $w _ { k + 1 }$ is generated in the following manner based on the color of the card taken.
$$w _ { k + 1 } = \begin{cases} - i w _ { k } & \text { if a red card was taken, } \\ i w _ { k } & \text { if a white card was taken. } \end{cases}$$
Here, each card is independently taken with equal probability. The initial state is $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Thus, $z _ { n } , w _ { n }$ are the values after repeating the series of the above two operations $n$ times starting from the state of $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Here, $i$ is the imaginary unit.
Answer the following questions.
(1) Show that $\operatorname { Re } \left( z _ { n } \right) = 0$ if $n$ is odd, and that $\operatorname { Im } \left( z _ { n } \right) = 0$ if $n$ is even. Here, $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$ represent the real part and the imaginary part of $z$ respectively.
(2) Let $P _ { n }$ be the probability of $z _ { n } = 1$, and $Q _ { n }$ be the probability of $z _ { n } = i$. Find recurrence equations for $P _ { n }$ and $Q _ { n }$.
(3) Find the probabilities of $z _ { n } = 1 , z _ { n } = i , z _ { n } = - 1$, and $z _ { n } = - i$ respectively.
(4) Show that the expected value of $z _ { n }$ is $( i / 3 ) ^ { n }$.
(5) Find the probability of $z _ { n } = w _ { n }$.
(6) Find the expected value of $z _ { n } + w _ { n }$.
(7) Find the expected value of $z _ { n } w _ { n }$.