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]

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.

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 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.''

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 $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.

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?

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 $\_\_\_\_$

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

A company has carried out a personnel selection process.
a) ( 1.25 points) It is known that $40 \%$ of the total number of applicants have been selected in the process. If among the applicants there was a group of 8 friends, calculate the probability that at least 2 of them have been selected.
b) (1.25 points) The scores obtained by the applicants in the selection process follow a normal distribution, X, with mean 5.6 and standard deviation $\sigma$. Knowing that the probability of obtaining a score $\mathrm { X } \leq 8.2$ is 0.67, calculate $\sigma$.

The probability that a fish of a certain species survives more than 5 years is 10\%. Find: a) (1 point) If in an aquarium we have 10 fish of this species born this year, find the probability that at least two of them are still alive in 5 years. b) ( 1.5 points) If in a tank of a fish farm there are 200 fish of this species born this same year, using an approximation by the corresponding normal distribution, find the probability that after 5 years at least 10 of them have survived.

In an autonomous community, three out of every five second-year high school students are enrolled in Mathematics II. Six students are randomly selected from all second-year high school students. It is requested: a) ( 0.75 points) Calculate the probability that exactly four of them are enrolled in Mathematics II. b) (0.75 points) Calculate the probability that at least one of them is enrolled in Mathematics II. c) (1 point) If an institute has 120 students enrolled in second-year high school, calculate, approximating the binomial distribution by a normal distribution, the probability that more than 60 of these students are enrolled in Mathematics II.

65\% of 18-year-old university students who attempt the practical driving exam pass it on the first try. 10 randomly selected 18-year-old university students who have already passed the practical driving exam are chosen.\ Find:\ a) (0.75 points) Calculate the probability that exactly 3 of them needed more than one attempt to pass the practical driving exam.\ b) (0.75 points) Calculate the probability that at least one of them needed more than one attempt to pass the practical driving exam.\ c) (1 point) Using a normal distribution approximation, determine the probability that, given 60 of these university students, at least half passed the practical driving exam on the first try.

According to data from the Community of Madrid, in the 2021-2022 season the coverage of the flu vaccine among people over 65 years old was 73.2%. a) ( 1.5 points) In the face of an epidemic outbreak situation, the authorities decide to restrict those gatherings in which the probability that there is more than one unvaccinated person is greater than 0.5. Assuming that attendees at a gathering constitute a random sample, should gatherings of 5 people over 65 years old be restricted? And gatherings of 7 people over 65 years old? b) ( 1 point) A random sample of 500 people over 65 years old is taken. Calculate, approximating by the appropriate normal distribution, the probability that at least 350 of them are vaccinated against the flu.