Grade 11 / Probability and random variables 49 exercises (100% corrected)

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AB AAB AAB BAB ABAB ABAB ABAB ABAB ABAB ABAB ABAB ABAB AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ ChingQuizz : 3 exercises available for Quizz assessment : 1. Reminders E.3115 A chess tournament pits two teams, each containing a man and a woman, against each other. A game is played between one person from each team. One person from each team is chosen at random to compete in a game. Consider the three events that ˇ ompose ı the universe of possibilities : A : ˇ Two men clash in this partie ı B : ˇ Two women clash in this partie ı C : ˇ A man and a woman clash in this partie ı 1 Conjecture the probability of each of these events. 2 The following notation is used to denote the composition of each group : G 1 = H 1 ; F 1 ; G 2 = H 2 ; F 2 a Describe all the games that can be organized during this tournament. b Give the probability of the events A , B and C . E.5866 1 Below are represented the universe Ω of a random exper-iment and two events A and B from Ω . For each of the representations below, hatch the requested set. 2 Give, without justification, a simplified expression for the sets : a A ∩ B b A ∪ B E.5179 Consider a deck of 32 cards and the follow-ing three events : A : ˇ The card drawn is a heart " B : ˇ The card drawn is a figure " C : ˇ The card drawn is a number whose value is strictly between 7 and 10 ı 1 A card is drawn at random from the deck of cards. Determine the probabil-ity of the following events : a A b B c C d A ∩ B e A ∪ B f A ∪ C 2 The card ˇ King of hearts ı has been removed from the deck, then a card is drawn at random. Determine the probability of the following events : a A b B c C E.7578 Consider a random experiment com-prising n elementary events. The law of equiprobability ap-plies to this random experiment. Two events A and B are respectively composed of 432 and 72 elementary events and verify the following properties : P A ∩ B = 0.25 ; P A ∪ B = 0.5 What is the number of elementary events making up the uni-verse Ω : a 496 b 540 c 643 d 672 E.2929 After studying a rigged die whose faces are numbered from 1 to 6, we obtain the following prob-ability law : x i 1 2 3 4 5 6 p i 0.2 0.15 0.12 0.17 0.08 0.28 Determine the probabilities of each of the following : 1 A: ˇ Result is greater than or equal to 4ı . 2 B : ˇ The result is an odd number ı. 3 C : ˇ The result is an even number ı. https://chingmath.fr chapExoCorrec/3115 sacados/3115 chapExoCorrec/5866 sacados/5866 AB AAB AAB BAB ABAB ABAB ABAB ABAB ABAB ABAB ABAB ABAB chapExoCorrec/5179 sacados/5179 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ chapExoCorrec/7578 sacados/7578 chapExoCorrec/2929 sacados/2929
5points0point3points0point ABC ABC ABCCB ABCACB ABC ABC ABCBA ABCCBA 133421243221124331 E.7802 A game involves throwing darts at a target. The target is divided into four sectors, as shown in the figure below : It is assumed that the throws are in-dependent and that the player hits the target every time. The player throws a dart. Note: p 0 the probability of obtaining 0 point ; p 3 the probability of obtaining 3 points ; p 5 the probability of obtaining 5 points. Knowing that p 5 = 1 2 · p 3 and that p 5 = 1 3 · p 0 , determine the val-ues of p 0 , p 3 and p 5 . E.8183 1 Express each of the hatched parts below using the sets A , B and C : a b 2 Hatch the set shown on each of the figures below : a b E.8184 1 Express each of the hatched parts below using the sets A , B and C : a b 2 Hatch the set shown on each of the figures below : a b E.6630 An urn contains 18 wooden pieces of different shapes, colors and numbers. An element is drawn at random from this urn. The draw is assumed to be equiprobable. Consider the following events : A : ˇ the room is a triangle ı B : ˇ the coin is colored blanche ı C : ˇ the part has the number 2 ı D : ˇ the part is not a cercle ı E : ˇ the part has a number pair ı Without justification, give the probability of the following events : a A b A ∩ C c C ∩ B ∪ A d A ∩ C e A ∩ D f A ∩ E ∪ C ∩ D g C ∩ E h C ∪ D i A ∪ C 2. Probability law of a random variable E.5170 An urn contains four blue balls num-bered 1 to 4 , three red balls numbered 1 to 3 and two green balls numbered 1 to 2 . 1 Note X the random variable that associates with each ball the number written on it. Determine the probability distribution of the random variable X . 2 The following rules apply to the drawing of a ball from this urn : If the ball drawn is blue and bears an even integer, the player wins 2 e . If the ball drawn is not blue and carries an even integer, https://chingmath.fr chapExoCorrec/7802 sacados/7802 5points0point3points0point chapExoCorrec/8183 sacados/8183 ABC ABC ABCCB ABCACB chapExoCorrec/8184 sacados/8184 ABC ABC ABCBA ABCCBA chapExoCorrec/6630 sacados/6630 133421243221124331 chapExoCorrec/5170 sacados/5170
the player wins 3 e . Otherwise the player wins nothing. We denote Y the random variable that associates the draw of a ball with the win obtained. Determine the probability distribution of the random variable Y . E.5188 A game consists of throwing a per-fectly balanced dodecahedron whose faces are numbered from 1 to 12 . The game consists of rolling the die once. Consider the random variable X which associates with each value of a face the number of divisors of that value. Determine the probability distribution of the random variable X . E.7336 Consider an urn containing 11 balls. Some are round, others square. Some are white, others are striped. They are shown below : It is assumed that by pressing a button, the balls come out of the urn at random. We have just set up a random experiment according to the law of equiprobability. A payoff is associated with each ball as follows : A ball pays 1 e while a square pays 2 e . In addition, if the element is scratched, the payout is increased by 1 e . This association of a value with each elementary event consti-tutes a random variable. Note X . 1 Determine the probability distribution of the variable X when the contents of the urn are shown below : 2 Determine the probability distribution of the variable X when the contents of the urn are shown below : 3. Random variables and distribution function E.4801 Definition: The set X 2 is defined as the set of el-ementary events taking a value less than or equal to 2 : X 2 = ! ∈ Ω ⏐ ⏐ X ( ! ) 2 In the case Let X be a random variable taking values in N , we have : X 2 = X =0 ∪ X =1 ∪ X =2 Let X be a random variable taking integer values from 1 to 6 and whose probability distribution is given in the table below : x i 1 2 3 4 5 6 P X = x i 0 ; 05 0 ; 12 0 ; 15 0 ; 23 0 ; 17 1 Complete the table of the probability distribution of X . 2 Determine the following probabilities : a P X 3 b P X > 3 E.8461 In a game based on a random ex-periment, the random variable X measures the participant’s winnings. The following table shows the probability distribu-tion of the variable X : x 0 1 2 3 6 P ( X = x ) 0.34 0.3 0.19 0.15 0.02 Determine the following probabilities : a P ( X < 3) b P ( X 3) c P (2 X < 5) E.8462 We have a balanced die with 6 faces and we associate to each face a gain as follows : the face ˇ 6 ı reports 5 e . face ˇ 1 ı reports 2 e . other even-numbered faces report 1 e . the other faces yield nothing. We denote X the random variable which, for each throw of the die, associates the gain realized. 1 Determine the value of the probability P X =1 . 1 6 2 6 3 6 4 6 2 Determine the probability value P X > 1 . 1 6 2 6 3 6 4 6 https://chingmath.fr chapExoCorrec/5188 sacados/5188 chapExoCorrec/7336 sacados/7336 chapExoCorrec/4801 sacados/4801 chapExoCorrec/8461 sacados/8461 chapExoCorrec/8462 sacados/8462
E.8463 We have a balanced die with 6 faces and we associate to each face a gain as follows : face ˇ 6 ı reports 5 e . faces ˇ 1 ı and ˇ 3 ı report 2 e . other even-numbered faces report 1 e . the ˇ 5 ı face pays nothing. We denote X the random variable which, for each throw of the die, associates the gain made. 1 Determine the value of the probability P X =1 . 1 6 2 6 3 6 4 6 2 Determine the probability value P X > 1 . 1 6 2 6 3 6 4 6 E.8211 Consider the random experiment consisting of a throw of a 6 -sided rigged die. Consider the random variable X which, on each throw, returns the number of the face obtained. The cumulative distribution law of the random variable X is given below : k 1 2 3 4 5 6 P X k 0.275 0.34 0.51 0.6 0.84 1 1 Determine the following probabilities : a P X > 3 b P X 4 2 a Justify that : P X =2 =0.065 b Determine the following probabilities : P X =3 ; P X =6 3 Determine the following probabilities : a P 2 X 5 b P 3 < X 5 E.4799 Let n be a non-zero natural integer ( n ∈ N ∗ ) . An urn contains : 8 red balls numbered from 1 to 8 . 1 green ball numbered 1 . n blue balls numbered from 1 to n . Assume that the balls are indistinguishable to the touch, and consider the random experiment of randomly drawing a ball from this urn. Note X which associates each ball drawn with a number of points according to the following rules : A red ball brings 1 point, the green ball brings 2 points, a blue ball brings 3 point. An odd-numbered ball brings 1 extra points. In addition, we know that : P X 3 = 3 4 Determine the value(s) of n achieving all these conditions. Hint: we will perform a case disjunction on the parity of the integer n . E.4800 Let X be a random variable whose probability distribution is given below : x i 0 1 2 3 P X = x i 0.15 0.24 0.35 0.26 1 Justify that the table below represents a probability dis-tribution. 2 Determine the following probabilities : a P X 2 b P X < 2 c P {X =1 } ∪{X =3 } 4. Expectations E.4805 Consider the random variable X whose probability distribution is given in the table below : k 0 1 2 5 10 P X = k 0.4 0.38 0.15 0.05 0.02 Determine the expectation of the random variable X . E.4804 Consider the random variable X whose probability distribution is given in the table below : k 0 1 2 3 P X = k 0.51 0.08 0.17 0.24 Determine the expectation of the random variable X . E.3735 To keep the heating system in good working order, a property company has the boilers in its housing stock inspected during the summer. We know that 20 % boilers are under warranty. The following events are considered : A : ˇ The boiler is garantie ı ; B : ˇ The boiler is défectueuse ı. Here are the probabilities of some items : E A A ∩ B A ∩ B P ( E ) 0.2 0.08 0.72 The inspection is free if the boiler is under warranty. It costs 80 euros if the boiler is no longer under warranty and is not faulty. It costs 280 euros if the boiler is no longer under war-ranty and is defective. The random variable representing the cost of checking a boiler is X . https://chingmath.fr chapExoCorrec/8463 sacados/8463 chapExoCorrec/8211 sacados/8211 chapExoCorrec/4799 sacados/4799 chapExoCorrec/4800 sacados/4800 chapExoCorrec/4805 sacados/4805 chapExoCorrec/4804 sacados/4804 chapExoCorrec/3735 sacados/3735 Inspiree d'Antilles-Guyane Juin 2002
AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ ABBCCCDDDDEEEEE Determine the probability distribution of X and its mathe-matical expectation. E.5351 A game consists of drawing a ball at random from an urn. The win of the game is associated with the color of the ball drawn : A red ball yields 10 e . A blue ball yields 1 e . A green ball yields no win. 1 The A urn has 1 red ball, 10 blue balls and 5 green balls. We denote X the random variable associated with the win of a ball drawn from the A urn. a Give the probability law of the random variable X . b Determine the expectation of the random variable X . 2 The urn B has 3 red balls, 3 blue balls and 20 green balls. We denote Y the random variable associated with the win of a ball drawn from the B urn. a Give the probability law of the random variable Y . b Determine the expectation of the random variable Y . (we’ll round the value to the nearest hundredth) . 3 Paul would like to take part in the game. Which urn is best for him? E.5911 We have a balanced die with 6 faces and we associate to each face a gain as follows : the face ˇ 6 ı reports 5 e . face ˇ 1 ı reports 2 e . other even-numbered faces report 1 e . the other faces yield nothing. We denote X the random variable which, for each throw of the die, associates the gain made. Determine the expectation of the random variable X . 9 6 10 6 11 6 12 6 E.5912 We have a balanced die with 6 faces and we associate to each face a gain as follows : face ˇ 6 ı reports 5 e . faces ˇ 1 ı and ˇ 3 ı report 2 e . other even-numbered faces report 1 e . the ˇ 5 ı face pays nothing. We denote X the random variable which, for each throw of the die, associates the gain made. Determine the expectation of the random variable X . 9 6 10 6 11 6 12 6 E.4806 Consider a deck of 32 cards. A game consists of drawing a card at random from among these cards. Consider the following three events : A : the card drawn is a heart ; B : the card drawn is a figure ; C : the card drawn is 7 , 8 or 9 . A payoff is associated with the card drawn as follows : cards from A ∩ C reports 1 point ; cards from A ∩ B yields 2 points ; cards from A ∩ B yields 4 points. other cards do not score points. Note X the random variable that associates the number of points earned with a card. Determine the mathematical expectation of the random vari-able X . E.7800 One game uses a bag filled with to-kens, each with a letter on one side. Here are the contents of the bag : The player draws a token before returning it to the bag, and each token is associated with a number of points as follows : Each vowel earns the same number of points. Each consonant earns double the points of a vowel. We denote X the random variable that associates with each of the tokens the associated number of points. Knowing that the expectation of the random variable X is 8 5 , determine the number of points assigned to each vowel. https://chingmath.fr chapExoCorrec/5351 sacados/5351 chapExoCorrec/5911 sacados/5911 chapExoCorrec/5912 sacados/5912 chapExoCorrec/4806 sacados/4806 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ chapExoCorrec/7800 sacados/7800 ABBCCCDDDDEEEEE
AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ E.5198 Definition: Let X be a random variable taking n +1 values denoted x 0 , x 1 , . . . , x n . We call the expectation of the random variable X , the number denoted E X defined by: E X = x 0 ×P X = x 0 + x 1 ×P X = x 1 + ··· + x n ×P X = x n This sum is also noted : n k =0 x k ·P X = x k In a game based on a random experiment, the random vari- able X measures the gain realized by the participant. The following table shows the probability distribution of the vari-able X : k 0 1 2 3 6 P ( X = k ) 0.34 0.3 0.19 0.15 0.02 Determine the expectation of this random variable. Note: the random expectation corresponds to the average value taken by the random variable X when the random experiment is repeated a large number of times. 5. Variances E.3117 At the end of the year, a high school students’ association organizes a tombola: 100 tickets are sold at 10 euros each. Here are the various winning tickets : 2 tickets win 50 e ; 10 tickets win 20 e ; 20 tickets win 10 e . 1 What is the sum of the winnings from this raffle? Consider the random experiment of choosing a ticket at ran-dom and the random variable X which associates each ticket with its value. 2 Determine the probability distribution of the random variable X . 3 a Determine the expectation E ( X ) of the random vari-able X . b Determine the variance V ( X ) and standard deviation ff ( X ) of the random variable X . (values will be rounded to the nearest tenth) . Note: the table below can be completed to determine the variance of the random variable X . k 0 10 20 50 k − E ( X ) k − E ( X ) 2 P X = k E.3118 At the end of the year, a high school students’ association organizes a tombola, selling 100 tickets at 10 euros each. Here are the various winning tickets : 2 tickets win 100 e ; 15 tickets win 10 e ; 1 a What is the sum of the winnings in this raffle? b If all the tickets are sold, what will be the profit made by the organizers? Consider the random experiment of randomly selecting a ticket and the random variable X which, to each ticket, asso-ciates its value. 2 Determine the probability distribution of the random variable X . 3 a Determine the expectation E ( X ) of the random vari-able X . b Determine the variance V ( X ) and ff ( X ) of the random variable X . (values will be rounded to the nearest hun-dredth) . E.5189 A game consists of drawing a card at ran-dom from a deck of 32 cards. Each card is associated with a payoff : an ace earns x points a figure earns 4 points A 10 earns 3 points Other cards earn no points. where x is a strictly positive number.y We model the number of points won by the random variable X . Determine the value of x so that the standard deviation of the random variable X is equal to 2 . https://chingmath.fr chapExoCorrec/5198 sacados/5198 chapExoCorrec/3117 sacados/3117 chapExoCorrec/3118 sacados/3118 chapExoCorrec/5189 sacados/5189 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣
E.10453 In a probabilized space (Ω ; P ) , consider the random variable X taking its values in the set 0 ; 3 ; 4 . We have the following information : P X =0 = x ; P X =3 = 2 x ; V X = 2 where x is a number belonging to the interval 0 ; 1 . Determine the possible values of x fulfilling these conditions. E.10626 Consider a random experiment associ-ated with a random variable X whose probability distribution is given below : k 0 3 6 P X = k 0.2 0.5 0.3 1 Show that : E X =3.3 . 2 a Complete the table below : k 0 3 6 k − E ( X ) k − E ( X ) 2 P X = k b Deduce the value of the variance and standard devia-tion of the random variable X . E.7801 Definition: We consider Ω; P a random experiment and X a random variable on Ω taking the values k 1 , k 2 , . . . , k n . We call the variance of X the number, de-noted by V ( X ) defined by: V ( X ) = n i =1 k i − E ( X ) 2 ×P X = k i We call the standard deviation of X the number, denoted ff X defined by: ff X = V ( X ) Consider a random experiment associated with a random vari-able X whose probability distribution is given below : k − 2 1 3 5 8 P X = k 0.3 0.1 0.2 0.1 0.3 1 Show that : E X =3 . 2 a Complete the table below : k − 2 1 3 5 8 k − E ( X ) k − E ( X ) 2 P X = k b Deduce the value of the variance. 6. Variances and calculators E.4185 A player once rolls a well-balanced die with 6 faces. He wins 10 e if the die scores 1 . He wins 1 e if the die marks 2 or 4 . He wins nothing in the other cases. Let X be the random variable equal to the player’s win. 1 Without the use of a calculator, give the exact value of the variance of the random variable X . 2 Using the calculator, give the value of the standard de-viation of the random variable X rounded to the nearest hundredth. E.5156 Let X be a random variable whose probability distribution is given above : x i 0 1 2 3 P X = x i 0.15 0.24 0.35 0.26 1 Justify that the table below represents a probability dis-tribution. 2 Determine the following probabilities : a P X 2 b P X < 2 c P {X =1 } ∪{X =3 } 3 Give, using the calculator and rounded to the thou-sandth, the expectation and standard deviation of the variable X . 7. Independent successions of random experiments E.4796 A game consists of tossing a bal-anced coin four times in succession. At each toss, the face obtained is noted. 1 Construct a choice tree representing this random experi-ment. The outcomes of this experiment are assumed to be equiprob-able. Each outcome is associated with a payoff as follows : the gain is 0 e si the face side does not appear; the gain is 1 e si the face side appears 1 times ; the gain is 2 e si the face side appears 2 times ; the gain is 4 e si the face side appears 3 times ; the gain is 10 e si the face side appears 4 times ; 2 Establish that : P X =4 = 1 4 3 Complete the table below giving the probability distribu-tion of the random variable X : https://chingmath.fr chapExoCorrec/10453 sacados/10453 chapExoCorrec/10626 sacados/10626 chapExoCorrec/7801 sacados/7801 chapExoCorrec/4185 sacados/4185 chapExoCorrec/5156 sacados/5156 chapExoCorrec/4796 sacados/4796
RRSpRRSnRRSr FGMCFGMC k 0 1 2 4 10 P X = k 4 Complete the table below giving the cumulative distribu-tion law of the random variable X : k 0 1 2 4 10 P X k 1 E.5197 A sports store rents out down-hill skis, snowboards and touring skis. Its rental equipment consists of 60 % downhill skis, with the remainder divided equally between snowboards and touring skis. After the rental day, the equipment is checked and, if neces-sary, repaired. Regardless of the type of equipment rented, 30 % requires repair. Each pair of skis and each snowboard is listed on a card that details its follow-up. A card is drawn at random. Consider the following events : S p : ˇ The plug is for a pair of skis from piste ı ; S n : ˇ The plug is for a snowboard ı ; S r : ˇ The plug is for a pair of skis from randonnée ı ; R : ˇ The equipment requires a réparation ı ; R is its op-posite event. All results of the first four questions will be rounded to 10 − 3 . 1 Copy and complete the weighted tree opposite : 2 a Calculate the probability that the card drawn concerns a pair of piste skis not in need of repair. b Calculate P S p ∪ R : the probabil-ity that the card drawn concerns a pair of piste skis or equipment not requiring repair. 3 The cost of renting downhill skis or a snowboard is 20 e , that of a pair of touring skis is 15 e . In the event of repair, a surcharge of 15 e is charged. Consider the random variable X which associates the amount of the associated billing with a plug. a Draw up a table representing the probability distribu-tion of the random variable X . b Determine the expectation of the random variable X . E.4810 Five boys and three girls write their names on a piece of paper and insert it into a ballot box. Two pieces of paper are drawn successively from the urn. The two draws are considered to be independent. 1 At each draw, we look to see if the paper drawn desig-nates a boy or a girl. Construct the probability tree for this experiment. 2 Let X be the random variable associating with an out-come of this draw the number of girls selected. a Determine the probability law of X . b Calculate its mathematical expectation of E ( X ) . E.5157 A red fruit producer offers raspber-ries, redcurrants and blueberries for direct sale. Customers can buy either trays of fruit for tasting, or trays of fruit for jam. The producer has noticed that, among his customers, 9 out of 10 buy a tray of fruit for jam. Whatever the type of tray purchased, in 50 % of cases the customer chooses blueberries for the fruit, 30 % raspberries in the other cases, redcurrants are chosen. Note: C the event ˇ the customer buys a punnet of fruit from confiture ı ; F the event ˇ the customer asks for framboises ı ; G the event ˇ the customer requests groseilles ı ; M the event ˇ the customer requests myrtilles ı ; It is assumed that the fruit chosen does not depend on the type of tray purchased and that each customer buys only one tray. 1 Complete the weighted tree be-low : 2 Determine the probability of C ∩ F . 3 The producer prices his trays as follows : The basic price of a tray of jam fruit is 5 euros and that of a tray of tasting fruit is 3 euros ; If the chosen punnet contains raspberries, he adds 1 euro to the price of the punnet ; If the chosen tray contains blueberries, he adds 2 euros to the price of the tray; If the chosen tray contains redcurrants, the base price remains unchanged. The random variable associating the price of the tray purchased with each customer is X . a What are the values taken by the random variable X ? b Draw up a table representing the probability distribu-tion of X . c Determine the expectation of the random variable X . 8. Conditional probability https://chingmath.fr chapExoCorrec/5197 sacados/5197 RRSpRRSnRRSr chapExoCorrec/4810 sacados/4810 chapExoCorrec/5157 sacados/5157 FGMCFGMC
DDADDB E.5572 A factory produces its televisions on two production lines A and B . Consider the events : A : ˇ television is produced on the channel A ı B : ˇ the television is produced on the channel B ı D : ˇ the television produced is défectueuse ı On leaving the production line, a television is chosen at ran-dom. We have the following probabilities : P ( A ) = 0.8 ; P A ( D ) = 0.1 ; P B ( D ) = 0.2 1 Complete the probability tree below : 2 a Justify that : P B ∩ D = 0.16 b Justify that : P D = 0.88 . c Justify that to the nearest thousandth, we have : P D A ) ≈ 0.667 3 We have the following information : A television produced on the A channel has a cost of 250 $ . A television produced on channel B has a cost of 300 $ . The cost of repeating a faulty television is 50 $ . Consider the random variable X which has a television produced in the factory, associates its total production cost. a Determine the probability distribution of the random variable X b Determine, to the nearest hundredth, the expectation of X E.3806 During an epidemic in cattle, we realized that if the disease is diagnosed early enough in an animal, it can be cured ; if not, the disease is fatal. A test was developed and tested on a sample of animals, 1 % of which were carriers of the disease. The following results are obtained : 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 for the disease. The events are noted : M : ˇ the animal is a carrier of maladie ı ; T : 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? Round the probability to the nearest thou-sandth. 4 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 and having 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 0.940 5 0.058 0 0.001 5 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 entire herd is tested, how much money should he plan to spend? https://chingmath.fr chapExoCorrec/5572 sacados/5572 DDADDB chapExoCorrec/3806 sacados/3806
0;74OOM0;26OOK U1 U2 U3 :::N:::B:::U1:::N:::B:::U2:::N:::B:::U3 E.9714 According to a recent study, there are 216 762 doctors in mainland France among whom 0.6 % practice osteopathy and there are 75 164 physiothera-pists among whom 8.6 % practice osteopathy. One person is chosen at random from among the doctors and physiotherapists. The following events are noted : M : ˇ the person chosen is médecin ı ; K : ˇ the person chosen is kinésithérapeute ı ; O : ˇ the person chosen practices ostéopathie ı. 1 The situation is represented using the following weighted tree : a Reproduce the probability tree and then complete it. b Show that the probability P O is equal to 0.0268 . c A patient has just had an osteopathic session with a practitioner from one of the two categories. Determine the probability that the practitioner is a physiotherapist. Give the result rounded to the hun-dredth. 2 The price of a consultation with a physiotherapist is 30 e and that of a doctor is 35 e . Moreover, if this practitioner practices osteopathy, he increases the price of the consu-lation by 5 e . Consider the random variable X which, for a person cho-sen at random, associates the price of one of his consul-tations. a Draw up a table representing the probability distribu-tion of the random variable X . b Determine the expectation of the random variable X . E.3712 A game consists of a balanced die and the three urns below, each made up of black and white balls : The player draws a ball from one of the urns ; the urn is chosen according to the face of the die obtained during a throw : If the face obtained is 1, he draws the ball from the urn U 1 ; If the face obtained is even, he draws the ball from the urn U 2 ; If the face obtained is 3 or 5 , it will use the U 3 urn. 1 Determine the probability of drawing the ball from the U 1 urn ; from the U 2 urn ; from the U 3 urn. 2 a Drawing the ball from the urn U 1 , what is the prob-ability of drawing a black ball? b Drawing the ball from the urn U 2 , what is the proba-bility of drawing a black ball? c Drawing the ball from the urn U 3 , what is the proba-bility of drawing a black ball? 3 Copy and complete the weighted tree opposite : 4 Determine the probality of the fol-lowing event : A : ˇ The ball drawn is black. ı 5 The player wins 5 e when the last ball drawn is a black ball and loses otherwise ; what is the expectation of this game? https://chingmath.fr chapExoCorrec/9714 sacados/9714 0;74OOM0;26OOK chapExoCorrec/3712 sacados/3712 U1 U2 U3 :::N:::B:::U1:::N:::B:::U2:::N:::B:::U3
vert.........rougeouorange...rougeouorange...vert......... ::::::B2:::N2B1::::::B2:::N2N1 E.7156 Amélie has to cross the main street of a village, which has two traffic lights. To n ∈ 1 ; 2 , we note E n the event ˇ Amélie is stopped by the n e red light or orange ı and E n the opposite event. The orange light is considered a red light. Let p n be the probability of E n and q n that of E n . The probability of the first traffic light being red or orange is 1 8 It is assumed that the following two conditions are met : The probability of the second traffic light being red or amber, if the first traffic light is red, is 1 20 . The probability that the second traffic light is red or or-ange, if the first light is green, is equal to 9 20 . We’re interested, first of all, in the first traffic lights. 1 Copy and complete the weighted tree below. 2 Note X the random variable equal to the number of green lights among these two traffic lights. Determine the prob-ability distribution of X . E.3731 There are two urns U 1 and U 2 containing balls indistinguishable by touch. U 1 contains k white balls ( k natural number greater than or equal to 1 ) and 3 black balls. U 2 contains 2 white balls and one black ball. We draw a ball at random from U 1 and place it in U 2 . A ball is then drawn at random from U 2 . All these operations together constitute a test. We denote B 1 (respectively N 1 ) the event ˇ one has drawn a white ball (resp. black) from the urn U 1 ı. We note B 2 (respectively N 2 ) the event ˇ one has drawn a white ball (resp. black) in the urn U 2 ı. 1 a Copy and complete with the missing probabilities the tree below : b Show that the probability of the event B 2 is equal to : 3 k + 6 4 k + 12 2 In the following, we consider k =12 . A player bets 8 euros and performs a trial. If, at the end of the trial, the player draws a white ball from the second urn, the player receives 12 euros. Otherwise, he receives nothing and loses his stake. Let X be the random variable equal to the player’s winnings, i.e. the difference between the sum received and the stake. a Show that the possible values of X are 4 and − 8 . b Determine the probability distribution of the variable X . c Calculate the mathematical expectation of X . d Is the game favorable to the player? E.3714 An urn contains 10 white balls and n red balls, n being a natural number greater than or equal to 2. A player is asked to draw balls from the urn. On each draw, all the balls have the same probability of being drawn. For each white ball drawn, he wins 2 euros and for each red ball drawn, he loses 3 euros. The random variable corresponding to the player’s algebraic gain is X . The player draws a ball from the urn twice in succession, with-out replacement. 1 Demonstrate that : P ( X = − 1)= 20 · n ( n +10)( n +9) 2 Calculate, as a function of n the probability correspond-ing to the other two values taken by the variable X . 3 Verify that the mahthematical expectation of the random variable X is : E ( X ) = − 6 n 2 − 14 n + 360 ( n + 10)( n + 9) 4 Determine the values of n for which the mathematical expectation is strictly positive. E.10452 The amusement park ˇSix flagsı near the town of Los Humanos have two parking lots : a parking lot near the main entrance at 15 e during the day and the second parking lot at 10 e . Los Humanos residents benefit from a 5 e discount on the price of parking. This company is studying the revenue generated by these parking lots and by randomly choosing a customer of this theme park, we note the events L and P defined by: L : the customer is a resident of Los Humanos. P : the customer has chosen the parking lot near the front door. This company sends us the following information : 30 % of customers are Los Humanos residents 60 % of Los Humanos residents choose parking near the front door. 60 % of customers not living in Los Humanos choose the off-center parking. Part A 1 Construct the probability tree modeling this study. 2 Determine the probability of the event P . 3 Knowing that a customer has chosen to use the parking lot near the front door, what is the probability that this customer lives in Los Humanos? Part B Consider the random variable X which associates with each customer the price paid for parking. 4 Give the probability law of the random variable X . 5 Using a calculator, give the expectation and standard deviation of the random variable X . https://chingmath.fr chapExoCorrec/7156 sacados/7156 vert.........rougeouorange...rougeouorange...vert......... chapExoCorrec/3731 sacados/3731 Extrait Antilles-Guyannes Juin 2008 ::::::B2:::N2B1::::::B2:::N2N1 chapExoCorrec/3714 sacados/3714 chapExoCorrec/10452 sacados/10452
9. Problems E.3730 An urn A contains four red balls and six black balls. An urn B contains one red ball and nine black balls. The balls are indistinguishable to the touch. Part A A player has a six-sided, perfectly balanced die numbered from 1 to 6 . He throws it once : If he gets 1 , he randomly draws a ball from the urn A ; Otherwise he randomly draws a ball from the urn B . 1 Let R be the event ˇ the player gets a ball rouge ı. Show that P ( R )=0.15 2 If the player gets a red ball, is the probability that it comes from A greater than or equal to the probability that it comes from B ? Part B The player repeats the test described in part A twice, under identical and independent conditions (i.e. at the end of the first test the ballot boxes return to their original composition) . Let x be a non-zero natural number. In each of the two trials, the player wins x euros if he gets a red ball and loses two euros if he gets a black ball. We denote by G the random variable corresponding to the player’s algebraic gain in euros at the end of the two trials. The random variable G therefore takes the values 2 x , x − 2 and − 4 . 1 Determine the probability law of G . 2 Express the expectation E ( G ) of the random variable G as a function of x . 3 For what values of x has E ( G ) 0 https://chingmath.fr chapExoCorrec/3730 sacados/3730 Liban Juin 2008 4 points