Throughout the exercise, probabilities will be rounded to $10^{-3}$ if necessary.
A binary datum is data that can only take two values: 0 or 1. Data of this type is transmitted successively from one machine to another. Each machine transmits the received data either faithfully, that is, by transmitting the information as it received it (1 becomes 1 and 0 becomes 0), or in the opposite way (1 becomes 0 and 0 becomes 1). The transmission is faithful in $90\%$ of cases, and therefore opposite in $10\%$ of cases. Throughout the exercise, the first machine always receives the value 1.
For any natural integer $n \geqslant 1$, we denote:

• $V_n$ the event: ``the $n$-th machine holds the value 1'';
• $\overline{V_n}$ the event: ``the $n$-th machine holds the value 0''.

Part A

1. a. Copy and complete the probability tree. b. Prove that $P(V_3) = 0{,}82$ and interpret this result in the context of the exercise. c. Given that the third machine received the value 1, calculate the probability that the second machine also received the value 1.
2. For any natural integer $n \geqslant 1$, we denote $p_n = P(V_n)$. The first machine received the value 1, so $p_1 = 1$. a. Prove that for any natural integer $n \geqslant 1$: $$p_{n+1} = 0{,}8\, p_n + 0{,}1.$$ b. Prove by induction that for any natural integer $n \geqslant 1$, $$p_n = 0{,}5 \times 0{,}8^{n-1} + 0{,}5.$$ c. Calculate the limit of $p_n$ as $n$ tends to infinity. Interpret this result in the context of the exercise.

Part B
To model in Python language the transmission of the binary datum described at the beginning of the exercise, we consider the simulation function which takes as a parameter a natural integer $n$ which represents the number of transmissions carried out from one machine to another, and which returns the list of successive values of the binary datum. The incomplete script of this function is given below. We recall that the instruction rand() returns a random number from the interval $[0; 1[$.
\begin{verbatim} def simulation(n): donnee = 1 liste = [donnee] for k in range(n): if rand() <0.1 donnee = 1 - donnee liste.append(donnee) return liste \end{verbatim}
For example, simulation(3) can return $[1, 0, 0, 1]$.

1. Determine the role of the instructions on lines 5 and 6 of the algorithm above.
2. Calculate the probability that simulation(4) returns the list $[1,1,1,1,1]$ and the probability that simulation(6) returns the list $[1,0,1,0,0,1,1]$.

4. An individual is said to be a "universal donor" when their blood can be transfused to any person without risk of incompatibility. Blood group O with negative rhesus is the only one satisfying this characteristic. [Figure]
Show that the probability that an individual chosen at random from the French population is a universal donor is 0.0714.

Question 154
Uma urna contém 5 bolas vermelhas, 3 bolas azuis e 2 bolas verdes. Retira-se uma bola ao acaso. A probabilidade de que a bola retirada seja vermelha ou verde é
(A) $\dfrac{1}{10}$ (B) $\dfrac{1}{5}$ (C) $\dfrac{7}{10}$ (D) $\dfrac{3}{5}$ (E) $\dfrac{7}{10}$

Uma urna contém 5 bolas vermelhas, 3 bolas azuis e 2 bolas verdes. Retira-se uma bola ao acaso. A probabilidade de que a bola retirada seja azul ou verde é
(A) $\dfrac{1}{10}$ (B) $\dfrac{1}{5}$ (C) $\dfrac{3}{10}$ (D) $\dfrac{1}{2}$ (E) $\dfrac{3}{5}$

José, Paulo, and Antônio are playing fair dice, on which, on each of the six faces, there is a number from 1 to 6. Each of them will roll two dice simultaneously. José believes that, after rolling his dice, the numbers on the faces facing up will give him a sum equal to 7. Paulo believes that his sum will be equal to 4, and Antônio believes that his sum will be equal to 8.
With this choice, who has the greatest probability of getting their respective sum correct is
(A) Antônio, since his sum is the largest of all the chosen ones.
(B) José and Antônio, since there are 6 possibilities for both José's choice and Antônio's choice, and there are only 4 possibilities for Paulo's choice.
(C) José and Antônio, since there are 3 possibilities for both José's choice and Antônio's choice, and there are only 2 possibilities for Paulo's choice.
(D) José, since there are 6 possibilities to form his sum, 5 possibilities to form Antônio's sum, and only 3 possibilities to form Paulo's sum.
(E) Paulo, since his sum is the smallest of all.

A store monitored the number of buyers of two products, A and B, during the months of January, February and March 2012. With this, it obtained this graph.
The store will draw a prize among the buyers of product A and another prize among the buyers of product B.
What is the probability that both winners made their purchases in February 2012?
(A) $\frac{1}{20}$ (B) $\frac{3}{242}$ (C) $\frac{5}{22}$ (D) $\frac{6}{25}$ (E) $\frac{7}{15}$

QUESTION 151
A bag contains 5 red balls, 3 blue balls, and 2 green balls. The probability of randomly drawing a blue ball is
(A) $\frac{1}{5}$
(B) $\frac{3}{10}$
(C) $\frac{2}{5}$
(D) $\frac{1}{2}$
(E) $\frac{3}{5}$

A box contains a $\mathrm{R}\$ 5.00$ bill, a $\mathrm{R}\$ 20.00$ bill, and two $\mathrm{R}\$ 50.00$ bills of different designs. A bill is randomly drawn from this box, its value is noted, and the bill is returned to the box. Then, the previous procedure is repeated.
The probability that the sum of the noted values is at least equal to $\mathrm{R}\$ 55.00$ is
(A) $\frac{1}{2}$
(B) $\frac{1}{4}$
(C) $\frac{3}{4}$
(D) $\frac{2}{9}$
(E) $\frac{5}{9}$

A couple, both 30 years old, intends to take out a private pension plan. The insurance company researched, to define the value of the monthly contribution, estimates the probability that at least one of them will be alive in 50 years, based on population data, which indicate that 20\% of men and 30\% of women today will reach the age of 80.
What is this probability?
(A) $50\%$
(B) $44\%$
(C) $38\%$
(D) $25\%$
(E) $6\%$

A box contains 5 red balls, 3 blue balls, and 2 green balls. One ball is drawn at random from the box.
What is the probability that the drawn ball is blue?
(A) $\dfrac{1}{10}$
(B) $\dfrac{1}{5}$
(C) $\dfrac{3}{10}$
(D) $\dfrac{2}{5}$
(E) $\dfrac{1}{2}$

Four candidates presented themselves to take the exam of a competition. Before starting the exam, the cell phones of the four candidates were collected by the proctor, who stored them, each one, inside a black envelope. At the end of the exam, the proctor returned the four envelopes with the cell phones to the four candidates, in a random manner, since he had not identified the envelopes.
The probability that all candidates received back the envelopes with their respective cell phones is
(A) $\dfrac{1}{2}$
(B) $\dfrac{1}{10}$
(C) $\dfrac{1}{16}$
(D) $\dfrac{1}{24}$
(E) $\dfrac{1}{256}$

A magazine report addressed the use of social networks by Brazilian internet users. Some of the data collected by the report are presented in the infographic.
According to the infographic data, when randomly selecting a Brazilian internet user in the period to which the report refers, the probability that he is a man who accesses some social network is
(A) $\dfrac{30}{90}$
(B) $\dfrac{36}{100}$
(C) $\dfrac{40}{100}$
(D) $\dfrac{40}{90}$
(E) $\dfrac{46}{90}$

(a) $n$ identical chocolates are to be distributed among the $k$ students in Tinku's class. Find the probability that Tinku gets at least one chocolate, assuming that the $n$ chocolates are handed out one by one in $n$ independent steps. At each step, one chocolate is given to a randomly chosen student, with each student having equal chance to receive it.
(b) Solve the same problem assuming instead that all distributions are equally likely. You are given that the number of such distributions is $\binom { n + k - 1 } { k - 1 }$. (Here all chocolates are considered interchangeable but students are considered different.)

Ten people sitting around a circular table decide to donate some money for charity. You are told that the amount donated by each person was the average of the money donated by the two persons sitting adjacent to him/her. One person donated Rs. 500. Choose the correct option for each of the following two questions. Write your answers as a sequence of two letters (a/b/c/d).
What is the total amount donated by the 10 people?
(a) exactly Rs. 5000
(b) less than Rs. 5000
(c) more than Rs. 5000
(d) not possible to decide among the above three options.
What is the maximum amount donated by an individual?
(a) exactly Rs. 500
(b) less than Rs. 500
(c) more than Rs. 500
(d) not possible to decide among the above three options.

There are four distinct balls labelled $1, 2, 3, 4$ and four distinct bins labelled A, B, C, D. The balls are picked up in order and placed into one of the four bins at random. Let $E_i$ denote the event that the first $i$ balls go into distinct bins. Calculate the following probabilities.
(i) $\Pr[E_4]$
(ii) $\Pr[E_4 \mid E_3]$
(iii) $\Pr[E_4 \mid E_2]$
(iv) $\Pr[E_3 \mid E_4]$.
Notation: $\Pr[X] =$ the probability of event $X$ taking place. $\Pr[X \mid Y] =$ the probability of event $X$ taking place, given that event $Y$ has taken place.

Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questions correctly and a poorly prepared student can answer only 3 questions correctly.
For each probability below, write your final answer as a rational number in lowest form.
(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly?
Note: The student and the questions are chosen independently of each other. "Random" means that each individual student/each pair of questions is equally likely to be chosen.
(b) Now suppose that a student was chosen at random and asked two randomly chosen questions from the exam, and moreover did answer both questions correctly. Find the probability that the chosen student was well prepared.

Positive integers $a$ and $b$, possibly equal, are chosen randomly from among the divisors of 400. The numbers $a, b$ are chosen independently, each divisor being equally likely to be chosen. Find the probability that $\gcd(a, b) = 1$ and $\text{lcm}(a, b) = 400$.

You are given an $8 \times 8$ chessboard. If two distinct squares are chosen uniformly at random find the probability that two rooks placed on these squares attack each other. Recall that a rook can move either horizontally or vertically, in a straight line.

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).

Solve the following two independent problems.
(i) A mother and her two daughters participate in a game show. At first, the mother tosses a fair coin.
Case 1: If the result is heads, then all three win individual prizes and the game ends. Case 2: If the result is tails, then each daughter separately throws a fair die and wins a prize if the result of her die is 5 or 6. (Note that in case 2 there are two independent throws involved and whether each daughter gets a prize or not is unaffected by the other daughter's throw.)
(a) Suppose the first daughter did not win a prize. What is the probability that the second daughter also did not win a prize?
(b) Suppose the first daughter won a prize. What is the probability that the second daughter also won a prize?
(ii) Prove or disprove each of the following statements.
(a) $2 ^ { 40 } > 20!$
(b) $1 - \frac { 1 } { x } \leq \ln x \leq x - 1$ for all $x > 0$.

Suppose $A$, $B$ and $C$ are three events and $P ( A ) = a , P ( B ) = b , P ( C ) = c$ are known. Let $P ( A \cup B \cup C ) = p$. The statements below are about whether we can find the value of $p$ if we know some additional information. (Note: $\cup$ is the same as OR. Similarly $\cap$ is the same as AND.)
Statements
(33) We can find the value of $p$ if we know that at least one of $a , b , c$ is 1. (34) We can find the value of $p$ if we know that at least one of $a , b , c$ is 0. (35) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are mutually exclusive. (36) We can find the value of $p$ if we know that any two of $A , B$ and $C$ are independent and we know the value of $P ( A \cap B \cap C )$.

You play the following game with three fair dice. (When each one is rolled, any one of the outcomes $1,2,3,4,5,6$ is equally likely.) In the first round, you roll all three dice. You remove every die that shows 6. If any dice remain, you roll all the remaining dice again in the second round. Again you remove all dice showing 6 and continue.
Questions
(29) Let the probability that you are able to play the second round be $\frac { a } { b }$, where $a$ and $b$ are integers with $\gcd = 1$. Write the numbers $a$ and $b$ separated by a comma. (30) Let the probability that you are able to play the second round but not the third round be $\frac { c } { d }$ where $c$ and $d$ are integers with $\gcd = 1$. Write only the integer $c$ as your answer.

17. One day, Captain Haddock receives a mysterious letter with a confusing paragraph. Captain Haddock and Tintin are investigating the matter. There are two possible suspects: Professor Calculus and Thomson $\&$ Thompson. Based on their past experience:

• The probability that Professor Calculus sends a letter is $60 \%$, while the probability that Thomson $\&$ Thompson send a letter is $40 \%$.
• When Professor Calculus sends letters, there is an $80 \%$ probability that the letter contains a confusing paragraph.
• When Thomson \& Thompson send letters, there is a $5 \%$ probability that the letter contains a confusing paragraph.

What is the probability that the letter was sent by Professor Calculus?
(a) 0.96
(b) 0.80
(c) 0.50
(d) 0.48

15. A game being offered in a casino consists of guessing the outcomes of two tosses of a fair coin. The gambler wins if she/he has correctly guessed at least one of the two tosses. To play a game, the gambler has to pay a fee of Rs. 80, and the winner gets a reward of Rs. 100 on winning the game (and nothing otherwise). Which of the following statements are correct?
(a) In the first 10 minutes on a given day exactly three gamblers play the game, one after the other. The probability that the casino owner makes a profit in the first 10 minutes equals $1 / 4$.
(b) One gambler plays the game three times. The probability that she wins exactly two of the three games is $27 / 64$.
(c) Three friends go together and play the game with each playing once. The probability that all three win equals $27 / 64$.
(d) If 1200 players play the game on a given day, the expected profit of the casino owner for the day equals Rs. 6000.

There are four people of different heights. When they stand in a line, what is the probability that the third person from the front is shorter than both of their neighbors? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$