Given is a Bernoulli chain with length $n$ and success probability $p$. Explain that for all $k \in \{ 0 ; 1 ; 2 ; \ldots ; n \}$ the relationship $B ( n ; p ; k ) = B ( n ; 1 - p ; n - k )$ holds.
A company organizes trips with an excursion ship that has space for 60 passengers.
(1) [3 marks] Consider a trip where the ship is fully booked. Among the passengers are adults, teenagers and children. Half of the passengers eat ice cream during the trip, of the adults only one in three, of the teenagers and children 75\%. Calculate how many adults participate in the trip.
To participate in a trip, one must make a reservation in advance without having to pay the fare yet. Based on experience, some of the people with reservations do not appear for the trip. For the 60 available seats, the company therefore allows up to 64 reservations. It should be assumed that 64 reservations are actually made for each trip. If more than 60 people with reservations appear for the trip, only 60 of them can participate; the rest must be turned away. The random variable $X$ describes the number of people with reservations who do not appear for the trip. For simplicity, it should be assumed that $X$ is binomially distributed, where the probability that a randomly selected person with a reservation does not appear for the trip is 10\%. The table shown supplements the approved reference material.
Binomial distribution cumulative; $k \mapsto \sum _ { i = 0 } ^ { k } B ( n ; p ; i )$

n	k	$\mathrm { p } = 0.10$	$\mathrm { p } = 0.11$	$\mathrm { p } = 0.12$	$\mathrm { p } = 0.13$	$\mathrm { p } = 0.14$	$\mathrm { p } = 0.15$	$\mathrm { p } = 0.16$	p = 0.17
\multirow[t]{7}{*}{64}	0	0.00118	0.00058	0.00028	0.00013	0.00006	0.00003	0.00001	0.00001
	1	0.00956	0.00514	0.00272	0.00142	0.00073	0.00037	0.00019	0.00009
	2	0.03891	0.02290	0.01321	0.00748	0.00417	0.00228	0.00123	0.00065
	3	0.10629	0.06827	0.04277	0.02620	0.01572	0.00924	0.00533	0.00302
	4	0.22047	0.15377	0.10425	0.06886	0.04439	0.02797	0.01725	0.01043
	5	0.37271	0.28059	0.20485	0.14534	0.10040	0.06763	0.04450	0.02863
	...	...	...	...	...	...	...	...	...

(2a) [1 marks] Give a reason why the assumption that the random variable $X$ is binomially distributed is a simplification in the context of the problem.
(2b) [3 marks] Determine the probability that no person with a reservation needs to be turned away.
(2c) [3 marks] For the company, it would be helpful if the probability of having to turn away at least one person with a reservation were at most one percent. For this, the probability that a randomly selected person with a reservation does not appear for the trip would need to be at least a certain value. Determine this value to the nearest whole percent.
The company sets up an online portal for reservations and suspects that this could increase the proportion of people with reservations who do not appear for the respective trip. As a basis for deciding whether more than 64 reservations should be allowed per trip in the future, the null hypothesis "The probability that a randomly selected person with a reservation does not appear for the trip is at most 10\%." is to be tested using a sample of 200 people with reservations at a significance level of 5\%. Before the test is conducted, it is determined that the number of possible reservations per trip will only be increased if the null hypothesis would need to be rejected based on the test result.
(2d) [5 marks] Determine the associated decision rule.
(2e) [3 marks] Decide whether the choice of the null hypothesis was primarily motivated by the interest in having fewer empty seats or the interest in not having to turn away more people with reservations. Justify your decision.
(2f) [2 marks] Describe the associated Type II error and the resulting consequence in the context of the problem.

Determine the probability for each case that among 10 contacted households

• at least two do not yet have a fast internet connection.
• exactly eight already have a fast internet connection.

Describe in the context of the problem an event whose probability is given by the term $0,2 ^ { 10 } + ( 1 - 0,2 ) ^ { 10 }$.

Determine how many households the company would need to contact at minimum so that with a probability of more than $99 \%$ at least one contacted household that does not yet have a fast internet connection would decide to set one up. Assume that every hundredth contacted household that does not yet have a fast internet connection decides to set one up.

15 plants are treated with the plant protection product and then sprayed with fungal spores. Determine the probability of each of the following events:\n$E _ { 1 }$ : ``None of the plants become infested with fungi.''\n$E _ { 2 }$ : ``At most two plants become infested with fungi.''\n$E _ { 3 }$ : ``12 or 13 plants remain free of fungal infestation.''

Determine the smallest value of $n$ for which the probability that at least one plant becomes infested with fungi is at least $99 \%$.

Determine the probabilities of the following events:\ $D$ : ``Among the selected cars there are seven or eight combustion engine cars with diesel motors.''\ $E$ : ``Among the selected cars there are more than 135 with purely electric drive.''

From the newly registered cars with electric motors, 40 vehicles are randomly selected. Determine the probability that exactly ten plug-in hybrids are among them.

The game is played five times. Give an event in the context of the problem whose probability can be calculated using the term $\left( \frac { 1 } { 4 } \right) ^ { 2 } \cdot \left( \frac { 3 } { 4 } \right) ^ { 3 }$.

Describe in the context of the problem a random experiment in which the probability of an event can be calculated using the term $1 - \sum _ { i = 0 } ^ { 8 } \binom { 30 } { i } \cdot 0,2 ^ { i } \cdot 0,8 ^ { 30 - i }$, and give this event.

Determine how many packages must be randomly selected at minimum so that the probability that at least one return is among them is greater than $90 \%$.

Determine the probability that two or three returns are removed.

In this part of the task, Machine $A$ is examined more closely. 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfills'' in a sample is assumed to be binomially distributed with $p = 0.3$.
(1) Determine the probability of the event ``Fewer than 30 underfills occur''.
(2) Determine the probability of the event ``At least 40 underfills occur''.
(3) Give an event in the given context whose probability together with the probabilities from (2) and (3) sums to 1.
(4) Give an event in the given context whose probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.

A total of 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfilled bottles'' in a sample is assumed to be binomially distributed with $p = 0.3$.
(1) Determine the probability of the event ``Fewer than 30 underfilled bottles occur.''
(2) Give an event in the given context for which the probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.

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 $p$.
Calculate the probability that $X_1, \ldots, X_n$ are all equal.

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 $p$.
What is the distribution of $S = X_1 + \ldots + X_n$? A proof of the stated result is expected.

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, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$.
If $\omega \in \Omega$, we introduce the column matrix $$U(\omega) = \begin{pmatrix} X_1(\omega) \\ \vdots \\ X_n(\omega) \end{pmatrix}$$ and the matrix $M(\omega) = U(\omega)\, {}^t(U(\omega))$. The application $M : \left\{\begin{array}{l} \Omega \rightarrow \mathcal{M}_n(\mathbb{R}) \\ \omega \mapsto M(\omega) \end{array}\right.$ is thus a random variable.
a) If $\omega \in \Omega$, justify that $M(\omega) \in \mathcal{X}_n$.
b) If $\omega \in \Omega$, justify that $\operatorname{tr}(M(\omega)) \in \{0, \ldots, n\}$, that $M(\omega)$ is diagonalizable over $\mathbb{R}$ and that $\operatorname{rg}(M(\omega)) \leqslant 1$.
c) If $\omega \in \Omega$, justify that $M(\omega)$ is an orthogonal projection matrix if and only if $S(\omega) \in \{0,1\}$.

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Let $r \geqslant 1$ be an integer and $S_r = N_1 + \cdots + N_r$. What does $S_r$ represent? Give its distribution (you may use the previous question).

Let $X$ be a random variable following the binomial distribution $\mathcal{B}(n, p)$ where $n \geqslant 1$ and $p \in ]0,1[$. Show that $X$ is decomposable if and only if $n \geqslant 2$.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. Determine the distribution of $X$, its expectation and its variance.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits, and $X$ follows the distribution determined in Q16. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ Using Markov's inequality, prove that if $r \leqslant 2 \frac{1-\alpha}{1-p}$, then condition (II.2) is satisfied.

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ We set $a = \frac{p \ln(p)}{p-1}$ and $x = r\ln(p) - a$. Prove that condition (II.2) is equivalent to the condition $$x \mathrm{e}^{x} \leqslant -\alpha a \mathrm{e}^{-a}.$$

A message is transmitted over a channel where each bit is inverted independently with probability $1-p \in ]0,1[$. Blocks of $r$ bits are transmitted. $X$ denotes the number of inversions during the transmission of a block of $r$ bits. We consider $\alpha \in ]0,1[$ and the condition $$\mathbb{P}(X \geqslant 2) \leqslant 1 - \alpha. \tag{II.2}$$ With $a = \frac{p\ln(p)}{p-1}$ and $x = r\ln(p) - a$, condition (II.2) is equivalent to $xe^x \leqslant -\alpha a e^{-a}$. Let $V$ and $W$ be the Lambert functions defined in Part I. Using one of the functions $V$ and $W$ and Question 10, study the existence of a largest natural integer $r$ satisfying condition (II.2).