To conduct a survey, an employee interviews people chosen at random in a shopping mall. He wonders whether at least three people will agree to answer.

1. In this question, we assume that the probability that a person chosen at random agrees to answer is 0.1. The employee interviews 50 people independently. We consider the events: $A$: ``at least one person agrees to answer'' $B$: ``fewer than three people agree to answer'' $C$: ``three or more people agree to answer''. Calculate the probabilities of events $A$, $B$ and $C$. Round to the nearest thousandth.
2. Let $n$ be a natural integer greater than or equal to 3. In this question, we assume that the random variable $X$ which, to any group of $n$ people interviewed independently, associates the number of people who agreed to answer, follows the probability distribution defined by: $$\left\{\begin{array}{l}\text{For every integer } k \text{ such that } 0 \leqslant k \leqslant n-1,\; P(X = k) = \frac{\mathrm{e}^{-a} a^k}{k!}\\\text{and } P(X = n) = 1 - \sum_{k=0}^{n-1} P(X=k)\end{array}\right.$$

On a tennis court, a ball launcher allows a player to train alone. This device sends balls one by one at a regular rate. The player then hits the ball and the next ball arrives. According to the manufacturer's manual, the ball launcher sends the ball randomly to the right or to the left with equal probability.
Throughout the exercise, results will be rounded to $10 ^ { - 3 }$ near.

Part A

The player is about to receive a series of 20 balls.

1. What is the probability that the ball launcher sends 10 balls to the right?
2. What is the probability that the ball launcher sends between 5 and 10 balls to the right?

Part B

The ball launcher is equipped with a reservoir that can hold 100 balls. Over a sequence of 100 launches, 42 balls were launched to the right. The player then doubts the proper functioning of the device. Are his doubts justified?

Part C

To increase the difficulty, the player configures the ball launcher to give spin to the balls launched. They can be either ``topspin'' or ``slice''. The probability that the ball launcher sends a ball to the right is still equal to the probability that the ball launcher sends a ball to the left. The device settings allow us to state that:

• the probability that the ball launcher sends a topspin ball to the right is 0.24;
• the probability that the ball launcher sends a slice ball to the left is 0.235.

If the ball launcher sends a slice ball, what is the probability that it is sent to the right?

In 2016, in France, law enforcement carried out 9.8 million alcohol screening tests with motorists, and $3.1\%$ of these tests were positive. In a given region, on 15 June 2016, a gendarmerie unit conducted screening on 200 motorists. Statement 2: rounding to the nearest hundredth, the probability that, out of the 200 tests, there were strictly more than 5 positive tests, is equal to 0.59. Indicate whether Statement 2 is true or false, justifying your answer.

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required.
A wrong answer, multiple answers or the absence of an answer to a question neither awards nor deducts points. The five questions are independent.
A production line produces mechanical parts. It is estimated that $4 \%$ of the parts produced by this line are defective. We randomly choose $n$ parts produced by the production line. The number of parts produced is large enough that this choice can be treated as a draw with replacement. We denote $X$ the random variable equal to the number of defective parts drawn. In the following three questions, we take $n = 50$.

1. What is the probability, rounded to the nearest thousandth, of drawing at least one defective part? a. 1 b. 0,870 c. 0,600 d. 0,599
2. The probability $p ( 3 < X \leqslant 7 )$ is equal to : a. $p ( X \leqslant 7 ) - p ( X > 3 )$ b. $p ( X \leqslant 7 ) - p ( X \leqslant 3 )$ c. $p ( X < 7 ) - p ( X > 3 )$ d. $p ( X < 7 ) - p ( X \geqslant 3 )$
3. What is the smallest natural integer $k$ such that the probability of drawing at most $k$ defective parts is greater than or equal to $95 \%$ ? a. 2 b. 3 c. 4 d. 5

In the following questions, $n$ no longer necessarily equals 50.
4. What is the probability of drawing only defective parts? a. $0,04 ^ { n }$ b. $0,96 ^ { n }$ c. $1 - 0,04 ^ { n }$ d. $1 - 0,96 ^ { n }$
5. Consider the Python function below. What does it return?
\begin{verbatim} def seuil (x) : n=1 while 1-0.96**n a. The smallest number $n$ such that the probability of drawing at least one defective part is greater than or equal to x . b. The smallest number $n$ such that the probability of drawing no defective parts is greater than or equal to x. c. The largest number $n$ such that the probability of drawing only defective parts is greater than or equal to x. d. The largest number $n$ such that the probability of drawing no defective parts is greater than or equal to x.

A fair die is thrown 100 times in succession. Find probabilities of the following events.
(i) 4 is the outcome of one or more of the first three throws.
(ii) Exactly 2 of the last 4 throws give an outcome divisible by 3 (i.e., outcome 3 or 6).

When a coin is tossed 6 times, the probability that the number of heads is greater than the number of tails is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

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) $\frac { 256 } { 729 }$

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 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 $\_\_\_\_$