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 $a = \frac{p\ln(p)}{p-1}$. Let $V$ and $W$ be the Lambert functions defined in Part I. When it exists, express the largest natural integer $r$ satisfying condition (II.2) as a function of $p$, $\alpha$ and $a$ using one of the functions $V$ or $W$.

We consider two urns each containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously, in an equiprobable manner, $n$ balls from the first urn. We denote $Y$ the number of white balls obtained. We also draw, in an equiprobable manner, $n$ balls from the second urn, but successively and with replacement. We denote $Z$ the number of white balls obtained. What is the distribution of the variable $Z$? Give the expectation and variance of $Z$.

On the finite probability space $\left(\mathcal{S}_n, P_n\right)$, we define, for all $i \in \llbracket 1, n \rrbracket$, the random variable $U_i$ such that, for all $\sigma \in \mathcal{S}_n$, we have $U_i(\sigma) = 1$ if $\sigma(i) = i$, and $U_i(\sigma) = 0$ otherwise.
Show that $U_i$ follows a Bernoulli distribution with parameter $\frac{1}{n}$.
Show that, if $i \neq j$, the variable $U_i U_j$ follows a Bernoulli distribution whose parameter you will specify.

Let $X$ and $Y$ be two Bernoulli random variables, having parameters $\lambda \in ]0,1[$ and $\mu \in ]0,1[$, respectively. Calculate $d_{VT}\left(p_X, p_Y\right)$.

Exercise V

A six-sided die is rolled five times. Check TRUE if the proposed random variable follows a binomial distribution and FALSE otherwise. V-A- The random variable corresponding to the number of rolls where an even number appears. V-B- The random variable corresponding to the sum of the results of all rolls.

What is the distribution followed by the random variable $A _ { n }$ representing the number of edges of a graph of $\Omega _ { n }$?

144. We toss a fair coin at least a few times so that we are more than 99\% certain that the result of three heads has occurred at least once. How many times at minimum must we toss?
(1) $12$ (2) $12$ (3) $18$ (4) $19$

There are 30 True or False questions in an examination. A student knows the answer to 20 questions and guesses the answers to the remaining 10 questions at random. What is the probability that the student gets exactly 24 answers correct?
(A) $\frac{105}{2^9}$
(B) $\frac{105}{2^8}$
(C) $\frac{105}{2^{10}}$
(D) $\frac{4}{2^{10}}$

8. A die has the shape of a regular dodecahedron with faces numbered from 1 to 12. The die is loaded so that the face marked with the number 3 appears with a probability $p$ double that of each other face. Determine the value of $p$ as a percentage and calculate the probability that in 5 rolls of the die the face number 3 comes up at least 2 times.

8. In a two-player game, each game won earns 1 point and the winner is the first to reach 10 points. Two players who in each game have the same probability of winning challenge each other. What is the probability that one of the two players wins in a number of games less than or equal to 12?

2. A biased coin is tossed 5 times, giving heads with probability $p$. －What is the probability of obtaining heads exactly 2 times? －For which value of $p$ is the probability of obtaining heads exactly 2 times maximum?

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is
(1) $1 / 729$
(2) $8 / 9$
(3) $8 / 729$
(4) $8 / 243$

A multiple choice examination has 5 questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is:
(1) $\frac{11}{3^5}$
(2) $\frac{10}{3^5}$
(3) $\frac{17}{3^5}$
(4) $\frac{13}{3^5}$

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is
(1) $\frac { 496 } { 729 }$
(2) $\frac { 192 } { 729 }$
(3) $\frac { 240 } { 729 }$
(4) $\frac { 256 } { 729 }$

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90 \%$ is:
(1) 2
(2) 4
(3) 5
(4) 3

Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is: (1) $\frac { 5 } { 8 }$ (2) $\frac { 1 } { 8 }$ (3) $\frac { 5 } { 16 }$ (4) 1

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is:
(1) $\frac { 33 } { 2 ^ { 32 } }$
(2) $\frac { 33 } { 2 ^ { 29 } }$
(3) $\frac { 33 } { 2 ^ { 28 } }$
(4) $\frac { 33 } { 2 ^ { 27 } }$

The mean and variance of a binomial distribution are $\alpha$ and $\frac { \alpha } { 3 }$ respectively. If $P ( X = 1 ) = \frac { 4 } { 243 }$, then $P ( X = 4$ or $5 )$ is equal to:
(1) $\frac { 5 } { 9 }$
(2) $\frac { 64 } { 81 }$
(3) $\frac { 16 } { 27 }$
(4) $\frac { 145 } { 243 }$

Let $X$ have a binomial distribution $B ( n , p )$ such that the sum and the product of the mean and variance of $X$ are 24 and 128 respectively. If $P ( X > n - 3 ) = \frac { k } { 2 ^ { n } }$, then $k$ is equal to
(1) 528
(2) 529
(3) 629
(4) 630

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is
(1) $\frac { 46 } { 6 ^ { 4 } }$
(2) $\frac { 275 } { 6 ^ { 5 } }$
(3) $\frac { 41 } { 5 ^ { 5 } }$
(4) $\frac { 36 } { 5 ^ { 4 } }$

In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then $6(n + p - q)$ is equal to:
(1) 51
(2) 52
(3) 53
(4) 50

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac { 1 } { 3 }$ and $\frac { 2 } { 3 }$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm { P } ( | x - y | \leq 2 )$ is $p$, then $3 ^ { 9 } p$ equals $\_\_\_\_$

Q90. In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac { 1 } { 3 }$ and $\frac { 2 } { 3 }$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm { P } ( | x - y | \leq 2 )$ is $p$, then $3 ^ { 9 } p$ equals $\_\_\_\_$

ANSWER KEYS

\begin{tabular}{|l|l|l|} \hline 1. (3) & 2. (1) & 3. (4) \hline 9. (2) & 10. (1) & 11. (1) \hline 17. (3) & 18. (4) & 19. (4) \hline 25. (750) & 26. (32) & 27. (5) \hline 33. (4) & 34. (2) & 35. (4) \hline

Q1 In a box, there are $n$ red balls and $(20 - n)$ white balls, where $0 < n < 20$. In each trial, a ball is taken out of the box, its color is examined, and it is returned to the box.
(1) Let $x$ be the probability that the ball taken out in one trial is red. Then, $x = \frac{n}{\mathbf{AB}}$.
(2) Let $p$ be the probability that in two trials a white ball is taken out at least once. Then $p$ can be expressed as $p = \mathbf{C} - x^{\mathbf{D}}$, where $x$ is the $x$ of (1).
(3) Let $q$ be the probability that in four trials a white ball is taken out at least twice. Then $q$ can be expressed as
$$q = \mathbf{E} - \mathbf{H}x^{\mathbf{G}} + \mathbf{H}x^{\mathbf{I}},$$
where $x$ is the $x$ of (1).
(4) For $p$ and $q$ of (2) and (3), we are to find the maximum value of $n$ such that $p < q$.
From the inequality $p < q$, we obtain the inequality
$$\mathbf{J}x^2 - \mathbf{K} \square \text{ } \square$$
When we solve this, we have
$$x < \frac{1}{\mathbf{L}}$$
Thus the maximum value of $n$ is $\square\mathbf{M}$.