A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we denote by $A_n$ the event: ``the customer buys a melon during week $n$''. Thus $p(A_1) = 1$. a. Reproduce and complete the probability tree below, relating to the first three weeks. b. Prove that $p(A_3) = 0{,}85$. c. Given that the customer buys a melon during week 3, what is the probability that he bought one during week 2? Round to the nearest hundredth.

According to a study, regular users of public transport represent $17\%$ of the French population. Among these regular users, $32\%$ are young people aged 18 to 24 years old.
A person is randomly interviewed and we note:

• $R$ the event: ``The person interviewed regularly uses public transport''.
• $J$ the event: ``The person interviewed is aged 18 to 24 years old''.

Part A:

1. Represent the situation using a probability tree, reporting the data from the problem statement.
2. Calculate the probability $P(R \cap J)$.
3. According to this same study, young people aged 18 to 24 represent $11\%$ of the French population. Show that the probability that the person interviewed is a young person aged 18 to 24 who does not regularly use public transport is 0.056 to $10^{-3}$ precision.
4. Deduce the proportion of young people aged 18 to 24 among non-regular users of public transport.

Part B: During a census of the French population, a census taker randomly interviews 50 people in one day about their use of public transport. The French population is large enough to assimilate this census to sampling with replacement. Let $X$ be the random variable counting the number of people regularly using public transport among the 50 people interviewed.

1. Determine, by justifying, the distribution of $X$ and specify its parameters.
2. Calculate $P(X = 5)$ and interpret the result.
3. The census taker indicates that there is more than a $95\%$ chance that, among the 50 people interviewed, fewer than 13 of them regularly use public transport. Is this statement true? Justify your answer.
4. What is the average number of people regularly using public transport among the 50 people interviewed?

Exercise 2

We have two opaque urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$. Urn $\mathrm { U } _ { 1 }$ contains 4 black balls and 6 white balls. Urn $\mathrm { U } _ { 2 }$ contains 1 black ball and 3 white balls. Consider the following random experiment: We randomly draw a ball from $U _ { 1 }$ which we place in $U _ { 2 }$, then we randomly draw a ball from $\mathrm { U } _ { 2 }$. We denote:

• $N _ { 1 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 1 }$''.
• $N _ { 2 }$ the event ``Drawing a black ball from urn $\mathrm { U } _ { 2 }$''.

For any event $A$, we denote $\bar { A }$ its complementary event.
PART A

1. Consider the probability tree opposite. a. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ given that a white ball was drawn from urn $\mathrm { U } _ { 1 }$ is 0.2. b. Copy and complete the probability tree opposite, showing on each branch the probabilities of the events concerned, in decimal form.
2. Calculate the probability of drawing a black ball from urn $U _ { 1 }$ and a black ball from urn $\mathrm { U } _ { 2 }$.
3. Justify that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28.
4. A black ball was drawn from urn $\mathrm { U } _ { 2 }$. Calculate the probability of having drawn a white ball from urn $\mathrm { U } _ { 1 }$. The result will be given in decimal form rounded to $10 ^ { - 2 }$.

PART B $n$ denotes a non-zero natural number. The previous random experiment is repeated $n$ times in an identical and independent manner, that is, urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$, between each experiment. We denote $X$ the random variable that counts the number of times a black ball is drawn from urn $\mathrm { U } _ { 2 }$. We recall that the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.28 and that of drawing a white ball from urn $\mathrm { U } _ { 2 }$ is equal to 0.72.

1. Determine the probability distribution followed by $X$. Justify your answer.
2. Determine by calculation the smallest natural number $n$ such that: $$1 - 0{,}72 ^ { n } \geqslant 0{,}9$$
3. Interpret the previous result in the context of the experiment.

PART C In this part urns $\mathrm { U } _ { 1 }$ and $\mathrm { U } _ { 2 }$ are returned to their initial configuration, with respectively 4 black balls and 6 white balls in urn $U _ { 1 }$ and 1 black ball and 3 white balls in urn $\mathrm { U } _ { 2 }$.
Consider the following new random experiment: We simultaneously draw two balls from urn $\mathrm { U } _ { 1 }$ which we place in urn $\mathrm { U } _ { 2 }$, then we randomly draw a ball from urn $\mathrm { U } _ { 2 }$.

1. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ are there?
2. How many possible draws of two balls simultaneously from urn $\mathrm { U } _ { 1 }$ containing exactly one white ball and one black ball are there?
3. Is the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with this new experiment greater than the probability of drawing a black ball from urn $\mathrm { U } _ { 2 }$ with the experiment in part A? Justify your answer. You may use a weighted tree diagram modeling this experiment.

We have a bag and two urns A and B.

• The bag contains 4 balls: 1 ball with the letter A and 3 balls with the letter B.
• Urn A contains 5 tickets: 3 tickets of 50 euros and 2 tickets of 10 euros.
• Urn B contains 4 tickets: 1 ticket of 50 euros and 3 tickets of 10 euros.

A player randomly draws a ball from the bag:

• if it is a ball with the letter A, he randomly draws a ticket from urn A.
• if it is a ball with the letter B, he randomly draws a ticket from urn B.

We note the following events:

• $A$: the player obtains a ball with the letter A.
• $C$: the player obtains a 50 euro ticket.

1. Copy and complete the tree opposite representing the situation.
2. What is the probability of the event ``the player obtains a ball with the letter A and a ticket of $50 €$''?
3. Prove that the probability $P(C)$ is equal to 0.3375.
4. The player obtained a 10 euro ticket. Is the statement ``There is more than $80\%$ chance that he previously obtained a ball with the letter B'' true? Justify.
5. We denote $X_1$ the random variable that gives the sum, in euros, obtained by the player. Example: if the player obtains a 50 euro ticket, then $X_1 = 50$. Show that the expectation $E(X_1)$ is equal to 23.50 and that the variance $V(X_1)$ is equal to 357.75.
6. After returning the ball to the bag and the ticket to the urn from which it was taken, the player plays a second game. We denote $X_2$ the random variable that gives the sum obtained by the player in this second game. We denote $Y$ the random variable defined as follows: $Y = X_1 + X_2$. a. Show that $E(Y) = 47$. b. Explain why we have $V(Y) = V(X_1) + V(X_2)$.
7. The player plays likewise a third, fourth, \ldots, hundredth game. We thus define in the same way the random variables $X_3, X_4, \ldots, X_{100}$. We denote $Z$ the random variable defined by $Z = X_1 + X_2 + \ldots + X_{100}$. Prove that the probability that $Z$ belongs to the interval $]1950; 2750[$ is greater than or equal to 0.75.

The Spanish national team will compete in the 2023 FIFA Women's World Cup. In the first two matches of the group stage, which consists of three matches, the probability of winning each one is 80\%. However, due to increased morale among the players, if they win the first two matches, the probability of winning the third rises to 90\%. Otherwise, the probability of winning the third match remains at 80\%. It is requested:
a) ( 0.5 points) Determine the probability that the Spanish national team does not win any match during the group stage.
b) (1 point) Calculate the probability that the national team wins the third match of the group stage.
c) (1 point) If we know that the national team has won the third match, determine the probability that they did not win one of the two previous matches.