Grade 12 / Annals on probability 15 exercises (100% corrected)

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1. Probability and sequences E.3188 Peter and Claude play tennis. Both players have the same chance of winning the first game. Thereafter, when Peter wins a game, the probability that he wins the next is 0.7 . And if he loses one game, the probability that he will lose the next is 0.8 . Throughout the exercise, n is a non-zero natural number. Consider the events : G n : ˇ Peter wins the n -th partie ı. P n : ˇ Pierre loses the n -th partie ı. We pose : p n = P ( G n ) et q n = P ( P n ) . 1 Finding a recurrence relation. a Determine p 1 and then the conditional probabilities P G 1 ( G 2 ) and P P 1 ( G 2 ) . b Justify the equality: p n + q n =1 . c Demonstrate that for any non-zero natural number n : p n +1 = 0.5 · p n + 0.2 . 2 Study of the sequence ( p n ) . We pose, for any non-zero natural number n , v n = p n − 2 5 . a Prove that the sequence ( v n ) is a geometric sequence and express v n as a function of n . b Deduce the expression of p n as a function of n . c Determine the limit of the sequence ( p n ) when n tends to + ∞ . E.4326 A player starts a video game and plays several successive games. It is assumed that : the probability of him winning the first game is 0.1 ; if he wins one game, the probability of winning the next is equal to 0.8 ; if he loses one game, the probability of winning the next is equal to 0.6 . We note, for any non-zero natural number n : G n the event ˇ the player wins the n -th partie ı ; p n the probability of the event G n . We therefore have : p 1 =0.1 1 Show that p 2 =0.62 . A weighted tree may be helpful. 2 The player has won the second game. Calculate the prob-ability that he lost the first. (an approximate value will be given to the nearest 10 − 2 ) . 3 Calculate the probability of the player winning at least one of the first three games. 4 Show that for any non-zero natural number n : p n +1 = 1 5 · p n + 3 5 . 5 Show by recurrence that for any non-zero natural num-ber n : p n = 3 4 − 13 4 · 1 5 n 6 Determine the limit of the sequence p n when n tends to + ∞ . 7 For what values of the natural integer n does one have : 3 4 − p n < 10 − 7 ? https://chingmath.fr chapExoCorrec/3188 sacados/3188 Asie Juin 2006 4 points chapExoCorrec/4326 sacados/4326 Polynesie Juin 2011
3epassageboucle Pour2epassageboucle Pour1erpassageboucle Pourinitialisationvariablesidabs E.3155 We study the random movement of a flea. This flea moves across three squares labeled A , B , and C . At time 0 , the flea is at A . For any natural number n : if at time n the chip is at A , then at time ( n +1) , it is : Either in B with a probability equal to 1 3 ; Or at C with a probability equal to 2 3 . If at time n the chip is at B , then at time ( n +1) , it is : either in C or in A with equal probability. If at time n the chip is at C , then it stays there. We note A n (respectively B n , C n ) the event ˇ at time n the chip is in A ı (respectively in B , in C ) . We note a n (respectively b n , c n ) the probability of event A n , (respectivement B n , C n ) . We therefore have : a 0 =1 ; b 0 = c 0 =0 To solve the exercise, we can use weighted trees. 1 Calculate a k , b k and c k for k ∈ N such that 1 k 3 . 2 a Show that, for any natural number n : a n + b n + c n = 1 ; a n +1 = 1 2 · b n b n +1 = 1 3 · a n b Show that, for any natural number n : a n +2 = 1 6 a n c Deduce that, for any natural number p : a 2 p = 1 6 p et a 2 p +1 = 0 b 2 p = 0 et b 2 p +1 = 1 3 · 1 6 p 3 Show that : lim n ↦→ + ∞ a n =0 . We assume that lim n ↦→ + ∞ b n =0 . What is the limit of c n when n tends towards + ∞ ? E.6800 There is a balanced die with 6 numbered sides from 1 to 6 and 2 coins A and B each with a heads and tails side. A game consists of rolling the die one or more times. After each throw of the die, if we get 1 or 2 , then we turn over the piece A , if we get 3 or 4 , then we return part B and if we get 5 or 6 , then we return neither part. At the start of the game, the 2 pieces are on the face side. 1 In the algorithm below, 0 encodes the headside of a coin and 1 encodes the tailside. If a encodes the side of the coin A at a given instant, then 1 − a encodes the side of the coin A after flipping it. i , n are integers greater than or equal to 1 Function f(n) a ← 0 b ← 0 For i from 1 to n do d ← of a random integer between 1 and 6 If d 2 Then a ← 1 − a Otherwise If d 4 Then b ← 1 − b End Si End If s ← a+b End To Return s a We call the function f with for argument n =3 and as-suming that the successively generated random values for d are 1 ; 6 and 4 . Copy and complete the table given below containing the state of the variables during the execution of the algorithm: b Does calling the function f decide whether both coins are on the stack side at the end? 2 For any natural number n , note : X n the event : ˇ At the end of n rolls of the dice, both pieces are on the face ı side Y n the event : ˇ At the end of n throws of the dice, one coin is on the tails side and the other is on the face ı side Z n the event : ˇ At the end of n throws of the dice, both coins are on the pile ı side. Furthermore we note, x n = P X n , y n = P Y n et z n = P Z n the respective probabilities of the events X n , Y n and Z n : https://chingmath.fr chapExoCorrec/3155 sacados/3155 chapExoCorrec/6800 sacados/6800 3epassageboucle Pour2epassageboucle Pour1erpassageboucle Pourinitialisationvariablesidabs
xnXnYnZnXnynXnYnZnYnznXnYnZnZn a Give the respective probabilities x 0 , y 0 and z 0 that at the start of the game there are 0 , 1 or 2 coins on the stacked side. b Justify that : P X n X n +1 = 1 3 c Copy the tree below and complete the probabilities on its branches, some of which may be zero: d For any natural number n , express z n as a function of x n and y n . e Deduce that, for any natural number n ,: y n +1 = − 1 3 · y n + 2 3 f We pose, for any natural number n : b n = y n − 1 2 Show that the sequence b n is geometric. Deduce that, for any natural number n : y n = 1 2 − 1 2 × − 1 3 n g Calculate lim n ↦→ + ∞ y n . Interpret the result. 2. Conditional probabilities E.1449 A player has an urn containing 3 red balls, 4 white balls and n green balls ( 0 n 10 ) . The balls are indistinguishable to the touch. 1 The player randomly draws a ball from the urn. Calcu-late the probability of each of the following events : a R : ˇ the ball drawn is rouge ı. b B : ˇ the ball drawn is blanche ı. c V : ˇ the ball drawn is verte ı. 2 The player decides to play a game. This takes place as shown below. The player draws a ball from the urn : if it’s red, he wins 16 F ; if it’s white, he loses 12 F ; if it’s green, he puts the ball back in the urn, then draws a ball from the urn ; if this ball is red, he wins 8 F ; if this ball is white, he loses 2 F ; if this ball is green, he neither loses nor gains any-thing. The draws are equiprobable and two successive draws are independent. At the start of the game, the player has 12 F . Let X be the random variable that takes as its value the sum the player owns at the end of the game (one draw or two draws as the case may be) . a Determine the values taken by X . b Determine the probability distribution of X . c Show that the mathematical expectation of X is : E ( X ) = 12 + 16 · n ( n + 7) 2 . 3 Consider the function f defined on the interval 0 ; 10 by: f ( x ) = x ( x + 7) 2 Study the variations of f . 4 Deduce the value of n for which the mathematical expec-tation X is maximum. Calculate this maximum value (the result will be given as an irreducible fraction) . https://chingmath.fr xnXnYnZnXnynXnYnZnYnznXnYnZnZn chapExoCorrec/1449 sacados/1449 Sportifs de haut-niveau Octobre 1998
RS0S1S2S3S4S5S6S7S8 E.3143 A gardener has two lots 1 and 2, each containing a large number of bulbs producing tulips of various colors. The probability of a bulb in lot 1 producing a yellow tulip is 1 4 . The probability of a bulb from lot 2 giving a yellow tulip is equal to 1 2 . This gardener randomly selects a lot and plants 50 tulip bulbs. Let n be a natural number verifying 0 n 50 . The following events are defined : A : ˇ the gardener has chosen the 1 ı lot; B : ˇ the gardener has chosen lot 2 ı ; J n : ˇ the gardener gets n tulips jaunes ı. 1 In this question, it is assumed that the gardener chooses lot 1. a What probability distribution does the number of yel- low tulips obtained from 50 bulbs of lot 1 follow? b What is the mathematical expectation of this law? c Give an expression for the probability of the gardener obtaining n yellow tulips. d Calculate the probability that the gardener obtains 15 yellow tulips. The result will be rounded to the thou-sandth. 2 Conditional probabilities : a Show that : P B J n = 50 n · 2 − 50 b Deduce the probability that the gardener will get n yellow tulips. c Let p n be the conditional probability of the event A knowing that J n is realized. Establish that : p n = 3 50 − n 3 50 − n + 2 50 d For what values of n has p n 0.9 ? How can this result be interpreted? 3. Independent events E.3736 A square with side 20 cm is di-vided into the following 10 zones : a disk D of radius 1 cm ; 8 sectors S 1 , S 2 , . . . , S 8 of the same area bounded by the borders of the disk D and the disk D of the same center and radius 9 cm ; an area R between the disk D and the edge of the square. A point is placed randomly in the square. The probability of placing the point in any area of the square is proportional to the area of that area. 1 a Determine the probability p ( D ) for the point to be placed in the disk D . b Determine the probability p ( S 1 ) for the point to be placed in the sector S 1 . 2 For this question 2 , the following approximate values will be used : p ( D ) = 0.008 ; p ( S k ) = 0.078 5 for k ∈ 1;2;3;4;5;6;7;8 . The following game is associated with this random situ-ation : a point placed in the disk D wins 10 euros ; a point placed in sector S k earns k euros for any k belonging to 1; 2; 3; 4; 5; 6; 7; 8 ; a point placed in the zone R loses 4 euros. The random variable equal to the algebraic gain obtained is X : a Calculate the probability p ( R ) for the point to be placed in the area R . Calculate the expectation of X . b We play twice in a row. Two points have been placed independently in the square. Calculate the probability of obtaining a positive or zero total win. c Let n be a natural number greater than or equal to two. We play n times in a row. We have therefore placed n points independently in the square. Calculate the probability p n of obtaining at least one point placed in the D disk. Determine the smallest value of n such that p n 0.9 . https://chingmath.fr chapExoCorrec/3143 sacados/3143 chapExoCorrec/3736 sacados/3736 France Septembre 2002 RS0S1S2S3S4S5S6S7S8
‡ E.3732 An urn contains 5 black balls and 5 white balls. We take n of them successively and with re-placement, n being a natural number greater than or equal to 2 . We consider the following two events : A : ˇ We obtain balls of both colors ı ; B : ˇ We obtain at most one white ball ı. 1 a Calculate the probability of the event : ˇ All the balls drawn are of the same color ı. b Calculate the probability of the event : ˇ We get exactly one white ball ı. c Deduce that the probabilities p ( A ∩ B ) , p ( A ) , p ( B ) are: p ( A ∩ B ) = n 2 n ; p ( A ) = 1 − 1 2 n − 1 ; p ( B ) = n + 1 2 n 2 Show that p ( A ∩ B )= p ( A ) × p ( B ) if, and only if : 2 n − 1 = n + 1 3 Let ( u n ) be the sequence defined by: u n = 2 n − 1 − ( n + 1) for all n ∈ N such that n 2 . Calculate u 2 , u 3 , u 4 . Prove that the sequence ( u n ) is strictly increasing. 4 Deduce the value of the integer n such that events A and B are independent. 4. Conditional probability and binomial distribution E.3145 A disease has appeared in a coun-try’s cattle herd. It affects 0 ; 5 of the herd (or 5 per thou-sand) . 1 An animal is chosen at random from the herd. What is the probability that it is sick? 2 a We randomly select 10 animals in succession. We call X the random variable equal to the number of sick animals among them. Show that X follows a binomial distribution, the pa-rameters of which we will give. Calculate its mathematical expectation. b Let A denote the event : ˇ none of the 10 animals is sick ı. We denote by B the event : ˇ at least one animal is sick among the 10 ı. Calculate the probabilities of A and B . 3 We know that the probability of an animal testing posi-tive for this disease, given that it is sick, is 0 ; 8 . When an animal is not sick, the probability of testing negative is 0 ; 9 . Let T be the event ˇ testing positive for this disease ı and M be the event ˇ êhaving this disease ı. a Represent the data in the statement using a weighted tree. b Calculate the probability of event T . c What is the probability that an animal is sick given that the test is positive? E.3834 In a fair a game organizer has 2 wheels with 20 squares each : The A wheel has 18 black squares and 2 red squares. Wheel B has 16 black squares and 4 red squares. When throwing a wheel all squares have the same probability of being obtained. The rule of the game is as follows : The player bets 1 e and rolls the wheel A . If he gets a red square, then he rolls the wheel B , notes the color of the square obtained and the game ends. If he gets a black square, then he re-rolls the wheel A , notes the color of the square obtained and the game ends. 1 Translate the statement using a weighted tree. 2 Let E and F be the events : E : ˇ at the end of the game, the 2 boxes obtained are rouges ı ; F : ˇ at the end of the game, only one of the two squares is rouge ı. Show that : p ( E )=0.02 ; p ( F ) = 0.17 . 3 If the 2 squares obtained are red the player receives 10 e ; if only one of the squares is red the player receives 2 e ; otherwise he receives nothing. X denotes the random variable equal to the player’s al-gebraic gain in euros (reminder the player bets 1 e ) . a Determine the probability law of X . b Calculate the mathematical expectation of X and give an interpretation. 4 The player decides to play n consecutive, independent games ( n denotes a natural number greater than or equal to 2 ) a Show that the probability p n that he throws the wheel B at least once is such that : p n =1 − (0.9) n . b Justify that the sequence of general term p n is conver-gent and specify its limit. c What is the smallest value of the integer n for which p n > 0.9 ? https://chingmath.fr chapExoCorrec/3732 sacados/3732 Polynesie Juin 1999 chapExoCorrec/3145 sacados/3145 ‡ chapExoCorrec/3834 sacados/3834
AB(0ptC(10ptD(0ptE(10ptF(0ptG(10pt891989198919 E.4142 During an epidemic in cattle, we realized that if the disease is diagnosed early enough in an animal, it can be cured : otherwise, the disease is fatal. A test was developed and tested on a sample of animals, 1 % of which were carriers of the disease. The results are as follows : if an animal is a carrier of the disease, the test is positive in 85 % of cases ; if an animal is healthy, the test is negative in 95 % of cases. We choose to take these observed frequencies as probabili-ties for the entire population and use the test for preventive screening of the disease. We note : M the event : ˇ the animal is a carrier of maladie ı ; T the event : ˇ the test is positif ı. 1 Construct a weighted tree modeling the proposed situa-tion. 2 An animal is chosen at random. a What is the probability that he is a carrier of the dis-ease and that his test is positive? b Show that the probability of his test being positive is 0.058 . 3 An animal is chosen at random from those with a posi-tive test. What is the probability that it is a carrier of the disease? 4 Five animals are chosen at random. The size of this herd means that the tests can be considered independent and the draws can be treated as draws with a discount. Note X the random variable which, for the five animals chosen, associates the number of animals with a positive test. a What is the probability distribution followed by X ? b What is the probability that at least one of the five animals has a positive test? 5 The cost of caring for an animal that has reacted posi-tively to the test is 100 euros, and the cost of slaughtering an animal not detected by the test that has developed the disease is 1 000 euros. The test is assumed to be free of charge. Based on the above data, the probability distribution of the cost to be incurred per animal undergoing the test is given by the following table : Coût 0 100 1 000 Probabilité 0.9405 0.0580 0.0015 a Calculate the mathematical expectation of the random variable associating with an animal the cost to be in-curred. b A breeder has a herd of 200 animals. If the whole herd is to be tested, how much should he plan to spend? E.4167 A player throws a ball that starts from A and then necessarily takes one of the branches shown on the tree below to arrive at one of the points D , E , F and G . Each branch of the tree has been marked with the probability that the ball will take it after passing through a node. The numbers in brackets indicate the points won by the player when the marble passes. Note X the random variable corre-sponding to the total number of points won at the end of a game; i.e. once the ball has arrived in D , E , F or G . 1 In this question, results are expected in fractional form. a Determine the probability law of X . b Calculate the expectation of X . c Calculate the probability that the ball followed the branch AC knowing that the player got exactly 10 points. 2 The player plays 8 games and it is assumed that these eight games are independent. A game is considered won if the player obtains 20 points in that game. a Calculate the probability of him winning exactly 2 games. We’ll give the result rounded to the thou-sandth. b Calculate the probability that he wins at least one game. The result will be rounded to the thousandth. https://chingmath.fr chapExoCorrec/4142 sacados/4142 Antilles-Guyane Septembre 2010 4 points chapExoCorrec/4167 sacados/4167 AB(0ptC(10ptD(0ptE(10ptF(0ptG(10pt891989198919
E.3130 The scene takes place at the top of a cliff by the sea. To find a beach and go swimming, tourists can only choose between two beaches, one to the east and the other to the west. A- A tourist finds himself at the top of the cliff on two con-secutive days. On the first day, he randomly chooses one of two directions. On the second day, it is assumed that the probability of him choosing a direction opposite to the one taken the day before is 0.8 . For i =1 or i =2 , we note E i the event : ˇ The tourist heads east on the i -th jour ı and O i the event : ˇ The tourist heads west on the i -th jour ı. 1 Draw up a probability tree describing the situation. 2 Determine the following probabilities : p ( E 1 ) ; p E 1 ( O 2 ) ; p ( E 1 ∩ E 2 ) 3 Calculate the probability that this tourist will visit the same beach on two consecutive days. B- It is now assumed that n tourists ( n 3 ) one day find themselves at the top of the cliff. These n tourists all want to swim and each of them chooses at random and independently of the others one of the two directions. Note X the random variable giving the number of these tourists who choose the beach to the east. 1 Determine the probability that k tourists ( 0 k n ) leave in an easterly direction. 2 It is assumed here that the two beaches under consider-ation are deserted at the start. A tourist is said to be happy if he finds himself alone on a beach. a Can there be two happy tourists? b Show that the probability (noted p ) that there is one happy tourist among these n tourists is : p = n 2 n − 1 c Numerical application : When the group comprises 10 people, express the prob-ability, rounded to the hundredth, that there is a happy tourist among the 10 . 5. Bernoulli diagrams and binomial distribution E.4152 Consider a questionnaire with five questions. For each of the five questions asked, three proposed answers are made ( A , B and C ) , only one of which is correct. A candidate answers all the questions posed by writing a five-letter word answer. For example, the word ˇ BBAAC ı means that the candidate answered B to the first and second questions, A to the third and fourth questions and C to the fifth question. 1 a How many possible word-answers are there to this questionnaire? b It is assumed that the candidate answers each of the five questions in this questionnaire at random. Calculate the probability of the following events : E : ˇ the candidate has exactly one correct answer. ı. F : ˇ the candidate has no answer exacte ı. G : ˇ the candidate’s answer word is a palindrome ı. (It should be noted that a palindrome is a word that can be read indifferently from left to right or right to left : for example, ˇ BACAB ı is a palindrome) 2 A teacher decides to submit this questionnaire to his 28 students, asking them to randomly answer each of the five questions on this questionnaire. The number of students whose answer word contains no correct answer is X . a Justify that the random variable X follows the bino-mial distribution with parameters n =28 and p = 32 243 . b Calculate the probability, rounded to 10 − 2 , that at most one student has provided only wrong answers. E.4193 A transportation company wants to optimize its controls in order to limit the impact of fraud and the losses caused by this practice. The company conducts a study based on two trips per day during the twenty working days of a month, for a total of forty trips. It is assumed that the checks are independent of each other and that the probability of any passenger being checked is equal to p . The price of each trip is ten euros, and in the event of fraud, the fine is one hundred euros. Claude systematically commits fraud during the forty trips covered by this study. Let X i be the random variable that takes the value 1 if Claude is checked on the i th trip and the value 0 otherwise. Let X be the random variable defined by: X = X 1 + X 2 + X 3 + · · · + X 40 1 Determine the probability distribution of X . 2 In this section, we assume that p = 1 20 . a Calculate the mathematical expectation of X . b Calculate the probabilities : P ( X =0) ; P ( X =1) ; P ( X =2) c Calculate at 10 − 4 the probability that Claude will be checked at most twice. 3 Let Z be the random variable that takes the value of the algebraic gain made by the fraudster. Justify the equal-ity: Z = 400 − 100 ·X , then calculate the expectation of Z for p = 1 5 . 4 We now want to determine p so that the probability that Claude undergoes at least three checks is greater than 99 % . a Prove that : P ( X 2) = (1 − p ) 38 · 741 · p 2 + 38 · p + 1 https://chingmath.fr chapExoCorrec/3130 sacados/3130 France Septembre 2006 5 points chapExoCorrec/4152 sacados/4152 chapExoCorrec/4193 sacados/4193
b Let f be the function defined on 0 ; 1 by: f ( x ) = (1 − x ) 38 · 741 · x 2 + 38 · x + 1 Show that f is strictly decreasing on [0 ; 1] and that there exists a unique real number x 0 belonging to the interval 0 ; 1 such that : f ( x 0 )=0.01 Determine the natural number n such that : n 100 <x 0 < n + 1 100 c Deduce the minimum value that must be assigned to p so that the probability that Claude undergoes at least three checks is greater than or equal to 99 % . (We will express p in terms of x 0 ) https://chingmath.fr