Outside the high school program / Conditional probability 23 exercises (100% corrected)

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ABE∑ectifs:81ABE∑ectifs:54ABE∑ectifs:27ABE∑ectifs:18 0;70RRF0;35RRCRRS 0;60;2B0;120;8B0;48A0;40;7B0;280;3B0;12A 1. Introduction to conditional probabilities E.7127 In a random experiment, consider two events A and B that realize the partition of the universe shown below : 1 Determine the following probabilities : a P A b P B c P A ∩ B 2 We modify the random experiment by considering only the universe consisting of the event A . We denote P the new probability on A . Determine the following probabil-ity: P B 3 Similarly, consider only events arising from B and note P  the probability over this universe. Determine the fol-lowing probability: P  A 2. Conditional probability calculation E.7134 A waiter working in a pizzeria notes that, on average, 40 % of customers are families, 25 % of customers are singles, and 35 % of customers are couples. He also notes that : 70 % of families leave a tip ; 90 % of singles leave a tip ; 40 % of couples leave a tip. On a given evening, this waiter randomly picks an occupied table in the pizzeria. We are interested in the following events : F : ˇ the table is occupied by a famille ı S : ˇ the table is occupied by a person seule ı C : ˇ the table is occupied by a couple ı R : ˇ the server receives a pourboire ı We note A the opposite event of A and p B ( A ) the probability of A , knowing B . 1 From the data in the statement, specify the probabilities p ( F ) and p S ( R ) . 2 Copy and complete the following weighted tree : 3 Calculate p F ∩ R E.7157 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 tourist finds himself at the top of the cliff on two con-secutive days. On the first day, he chooses one of the two directions at random. 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 t =1 or t =2 , we note E t the event : ˇ The tourist heads east on the t -th jour ı and O t the event : ˇ The tourist heads west on the t -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. E.7124 In a random experiment, consider the two events A and B . The study of this random experiment has produced the probability tree below : Give, if possible, by reading the tree the following probabili-ties : a P A b P A ∩ B c P A ∩ B d P B A e P A B f P A B 3. Conditional probability and abre construction https://chingmath.fr chapExoCorrec/7127 sacados/7127 ABE∑ectifs:81ABE∑ectifs:54ABE∑ectifs:27ABE∑ectifs:18 chapExoCorrec/7134 sacados/7134 Extrait Liban Mai 2014 0;70RRF0;35RRCRRS chapExoCorrec/7157 sacados/7157 Extrait France Septembre 2006 chapExoCorrec/7124 sacados/7124 0;60;2B0;120;8B0;48A0;40;7B0;280;3B0;12A
E.7125 A car rental agency has three types of vehicle: sedan, utility or luxury, and offers, at the time of rental, an insurance option with no deductible. A statistical study found that : 30 % of customers rented a sedan and 10 % rented a lux-ury vehicle. 40 % of customers who rented a sedan chose the zero de-ductible insurance option. 9 % of customers who rented a luxury vehicle chose the no-deductible insurance option. 21 % of customers rented a commercial vehicle and chose the no-deductible insurance option. A customer’s file is taken at random and the following events are considered : B : the customer has rented a sedan. L : the customer has rented a luxury vehicle. U : the customer has rented a commercial vehicle. A : the customer has chosen the no-excess insurance op- tion. Produce a probability tree representing the above situation and incorporating the data in the statement. E.7126 A company manufactures circular medals in large quantities. The entire production is carried out by two machines M A and M B . The M A machine supplies 40 % of total production and M B the rest. The M A machine produces 2 % defective medals and the M B machine produces 3 % defective medals. A medal produced by the company is taken at random and the following events are considered : A : ˇ medal comes from machine M A ı ; B : ˇ the medal comes from the machine M B ı ; D : ˇ the medal is défectueuse ı ; D is the opposite event of the event D . Translate this situation into a weighted tree. 4. Total probability formula E.7135 An investor wants to buy an apart-ment with the aim of renting it out. To this end, he is inter-ested in the rental profitability of this apartment. The three parts can be treated independently. Results will be rounded, if necessary, to 10 − 4 . Two types of apartment are considered : One- and two-room apartments, noted T 1 and T 2 respec-tively; Apartments with more than two rooms. A study of the records of apartments rented in one sector showed that : 35 % of rented apartments are of type T 1 and T 2 ; 45 % of rented apartments of type T 1 or T 2 are prof-itable; 30 % of rented apartments, which are neither type T 1 nor type T 2 , are profitable. A file is chosen at random and the following events are con-sidered : T : ˇ the apartment is of type T 1 or T 2 ı ; R : ˇ the rented apartment is rentable ı ; T is the opposite event of T and R is the opposite event of R . 1 Translate this situation into a weighted tree. 2 Show that the probability of a rented apartment being profitable is equal to 0.3525 . E.7110 At a freeway exit, the toll plaza has three lanes. A statistical study has shown that : 28 % of motorists use the left lane, reserved for sub-scribers ; a motorist using this lane always crosses the toll in less than 10 seconds ; 52 % of motorists use the center lane, reserved for pay-ment by credit card ; of these, 75 % pass through the tollgate in less than 10 seconds ; the remaining motorists use the right-hand lane using another means of payment (coins or billets) . A motorist is chosen at random and the following events are considered : G : ˇ the motorist uses the gauche ı ; C lane : ˇ the motorist uses the centre ı ; D lane : ˇ the motorist uses the droite ı ; T lane : ˇ the motorist passes through the tollbooth in less than 10 secondes ı. Let T be the opposite event of event T . 1 Construct a weighted tree translating this situation. This tree will be completed as the exercise progresses. 2 Calculate the probability p ( C ∩ T ) 3 L. The study also showed that 70 % of motorists pass the toll in less than 10 seconds. a Justify that p ( D ∩ T )=0.03 . b Calculate the probability that a motorist using the right lane will pass the toll in less than 10 seconds. https://chingmath.fr chapExoCorrec/7125 sacados/7125 Extrait Liban Mai 2015 chapExoCorrec/7126 sacados/7126 Extrait d'Antilles-Guyane Juin 2016 chapExoCorrec/7135 sacados/7135 chapExoCorrec/7110 sacados/7110
......AAB......AAL...AAC 0;74OOM0;26OOK E.7115 A car rental agency has three types of vehicle: sedan, utility or luxury, and offers a zero-deductible insurance option at the time of rental. A statistical study found that : 30 % of customers rented a sedan and 10 % rented a lux-ury vehicle. 40 % of customers who rented a sedan chose the zero-deductible insurance option. 9 % of customers rented a luxury vehicle and chose the no-deductible insurance option. 21 % of customers rented a commercial vehicle and chose the no-deductible insurance option. A customer’s file is taken at random and the following events are considered : B : the customer has rented a sedan. L : the customer has rented a luxury vehicle. U : the customer has rented a utility vehicle. A : the customer has chosen the no-excess insurance op-tion. 1 Recopy and complete the probability tree opposite with the data in the statement. 2 What is the probability that the customer rented a sedan and chose the no-excess insurance option? 3 Calculate the probability that a customer chose the no-excess insurance option. 4 Calculate P L ( A ) , the probability that the customer has taken out excess-free insurance knowing that he has rented a luxury car. 5. Condition reversal E.7138 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. The situation is represented using the following weighted tree : 1 Reproduce the probability tree and then complete it. 2 Show that the probability P O is equal to 0.0268 . 3 A patient has just had an osteopathic session with a prac-titioner from one of the two categories. Determine the probability that the practitioner is a phys-iotherapist. Give the result rounded to the hundredth. E.7111 One manufacturer produces tires in two categories, the ˇ snow tire ı category and the ˇ tire clas-sique ı category. On each of them, quality tests are carried out to improve safety. The following information is available on the production stock: the stock contains 40 % snow tires ; of the snow tires, 92 % have passed quality tests. among classic tires, 96 % passed quality tests. A customer chooses a tire at random from the production stock. We note : N the event : ˇ The tire chosen is a tire neige ı C the event : ˇ The tire chosen is a tire classique ı Q the event : ˇ The chosen tire has passed the qualité ı tests. Throughout this exercise, results will be rounded to the thou-sandth. 1 Illustrate the situation with a weighted tree. 2 Calculate the probability of the event N ∩ Q and interpret this result with a sentence. 3 Show that : p ( Q )=0.944 4 Knowing that the chosen tire has passed the quality tests, what is the probability that this tire is a snow tire? https://chingmath.fr chapExoCorrec/7115 sacados/7115 Extrait d'Antilles-Guyane Juin 2016 ......AAB......AAL...AAC chapExoCorrec/7138 sacados/7138 Extrait Antilles-Guyanes Juin 2014 0;74OOM0;26OOK chapExoCorrec/7111 sacados/7111
E.7112 We look at all home loan applica-tions at three major banks. A study shows that 42 % of loan applications are lodged with Karl Bank, 35 % of loan applications are lodged with Lofa Bank, while this proportion is 23 % for Miro Bank. On the other hand : 76 % of loan applications lodged with Karl bank are ac-cepted ; 65 % of loan applications lodged with Lofa bank are ac-cepted ; 82 % of loan applications lodged with Miro bank are ac-cepted. A home loan application is chosen at random from those sub-mitted to the three banks. Consider the following events : K : ˇ the loan application has been submitted to the bank Karl ı ; L : ˇ the loan application has been filed with the bank Lofa ı ; M : ˇ the loan application has been filed with the bank Miro ı ; A : ˇ the loan application has been accepted. ı Throughout the exercise, values rounded to the thousandth will be given if necessary. 1 Construct a weighted tree illustrating the situation 2 Calculate the probability that the loan application is lodged with Bank Karl and is accepted. 3 Show that : P ( A ) ≈ 0.735 4 La loan application is accepted. Calculate the probabil-ity that it was submitted to Bank Miro. E.7113 A cell phone contains in memory 3 200 songs archived by category: rock, techno, rap, reggae. . . some of which are performed in French. Of all the songs recorded, 960 are classified in the rock cate-gory. One of the phone’s features allows music to be listened to in ˇ random play ı : the songs listened to are chosen randomly and equiprobably from the entire repertoire. During his weekly jog, the owner of the phone listens to a song grâthanks to this playback mode. We note : R the event : ˇ the song listened to is a song from the category rock ı ; F the event : ˇ the song listened to is sung in French ı. 1 Calculate p ( R ) , the probability of the event R . 2 35 % of the songs in the rock category are performed in French ; translate this data using the events R and F . 3 Calculate the probability that the song listened to is a song in the rock category and is performed in French. 4 Among all the songs recorded 38.5 % are sung in French. Show that : p F ∩ R = 0.28 5 En deduce p R F and express in a sentence what this result means. E.7109 A recreation center for young peo-ple aged 11 to 18 has 60 % middle school students and 40 % high school students. The director conducted a statistical study on cell phone own-ership. This study showed that 80 % of the young people own a cell phone and that, among middle school students, 70 % own one. A young person is chosen at random from the recreation cen-ter and the following events are considered : C : ˇ the young person chosen is a middle school student ı ; L : ˇ the young person chosen is a high school student ı ; T : ˇ the young person chosen has a cell phone ı. Reminders: if A and B are two events, p ( A ) denotes the probability that event A will occur and p B ( A ) denotes the probability of A given that event B has occurred. We also denote A as the opposite event of A . 1 Give the probabilities : p ( C ) , p ( L ) , p ( T ) , p C ( T ) . 2 Draw a probability tree representing the situation and begin to fill it in with the data from the statement. 3 Calculate the probability that the young person chosen is a middle school student who owns a cell phone 4 Calculate the probability that the young person chosen is a middle school student, knowing that he or she owns a cell phone. 5 a Calculate p T ∩ L , deduce p L ( T ) . b Complete the tree constructed in question 2 . E.6829 On a tennis court, a ball launcher enables a player to train alone. This device sends out 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 randomly sends the ball to the right or left with the same probability. To increase the difficulty, the player sets the ball launcher to give an effect to the balls thrown. They can be either ˇ lifted ı, or ˇ coupées ı. The probability that the ball launcher will send a ball to the right is always equal to the probability that the ball launcher will send a ball to the left. The settings of the device allow us to state that : Throughout the exercise, results will be rounded to the near-est 10 − 3 . the probability that the ball launcher will send a lifted ball to the right is 0.24 ; the probability of the ball-thrower sending a cut ball to the left is 0.235 . If the ball launcher sends a chopped ball, what is the proba-bility that it will be sent to the right? https://chingmath.fr chapExoCorrec/7112 sacados/7112 chapExoCorrec/7113 sacados/7113 chapExoCorrec/7109 sacados/7109 Extrait Liban Mai 2016 chapExoCorrec/6829 sacados/6829 Extrait Liban Mai 2016
BBABBA E.5559 In a random experiment, consider the two events A and B for which the information below is available: P A = 0.7 ; P A B = 0.2 ; P A B = 0.9 1 Using the information in the statement, complete the probability tree below : 2 Determine the probability P A ∩ B . 3 Show that : P B )=0.41 4 Show that : P B A )= 14 41 E.4257 A bicycle repairer bought 30 % of his tire stock from a first supplier, 40 % from a second and the rest from a third. The first supplier produces 80 % flawless tires, the second 95 % and the third 85 % . The repairer takes a tire at random from his stock. 1 Construct a probability tree reflecting the situation, and show that the probability of this tire being defect-free is equal to 0.875 . 2 Knowing that the chosen tire is flawless, what is the prob-ability that it comes from the second supplier? The result will be rounded to 10 − 3 . E.5525 The company Fructidoux manu-factures compotes that it packages in small 50 gram jars. It wishes to label them as ˇ light compotes ı. The law requires that the sugar content, i.e., the proportion of sugar in the compote, be between 0.16 and 0.18 . In this case, the small jar of compote is said to be compliant. The company has two production lines, F 1 and F 2 . Production line F 2 seems more reliable than production line F 1 . However, it is slower. Thus, in total production, 70 % small jars come from line F 1 and 30 % from line F 2 . The F 1 line produces 5 % non-compliant compotes and the F 2 line produces 1 % . A small jar is randomly selected from the total production. Consider the following events : E : ˇ The small pot comes from chain F 2 ı ; C : ˇ The small pot is compliant ı. 1 Construct a weighted tree on which the above data will be indicated. 2 Calculate the probability of the event : ˇ The small pot is compliant and comes from production line F 1 ı. 3 Determine the probability of the event C . 4 Determine, at 10 − 3 close, the probability of event E knowing that event C has occurred. E.3729 A store manager buys electronic compo-nents from two suppliers in the following proportions : 25 % from the first supplier and 75 % from the second. The proportion of defective components is 3 % from the first supplier and 2 % from the second. We note : D : the event ˇ the component is défectueux ı ; F 1 : the event ˇ the component comes from the first four-nisseur ı ; F 2 : the event ˇ the component comes from the second fournisseur ı. 1 Draw up a probability tree corresponding to this situa-tion. 2 Calculate P ( D ∩ F 1 ) , then demonstrate : P ( D )=0.022 5 3 Knowing that a component is defective, what is the prob-ability that it comes from the first supplier? Round off the value to the nearest thousandth. 6. Definition of independence E.3778 Prerequisite: Recall that two events A and B are inde-pendent for probability p if, and only if : P ( A ∩ B ) = P ( A ) ×P ( B ) Let A and B be two events associated with a random experi-ment : 1 Justify that : P B = P B ∩ A + P B ∩ A 2 Demonstrate that, if events A and B are independent for probability p , then so are events A and B . E.6362 Indicate whether the following statement is true or false and justify the answer. In the set E of outcomes of a random experiment, consider two events A and B . ˇ If A and B are independent, then A and B are also indepen-dent. ı 7. Independent repetitions of random experiments https://chingmath.fr chapExoCorrec/5559 sacados/5559 Extrait de Liban Mai 2013 BBABBA chapExoCorrec/4257 sacados/4257 chapExoCorrec/5525 sacados/5525 Extrait Liban Mai 2013 chapExoCorrec/3729 sacados/3729 chapExoCorrec/3778 sacados/3778 chapExoCorrec/6362 sacados/6362
E.3746 An urn contains 50 white balls, 25 black balls and 25 red balls. The elementary experiment consists of drawing a ball. The balls all have the same probability of being drawn. We make 3 independent, discounted draws. 1 Determine the probability of the event : A : ˇ the three balls drawn are blanches ı. 2 Determine the probability of the event : B : ˇ none of the balls drawn are blanche ı. 3 a Determine the probability of the event : C 1 : ˇ The first ball drawn is white ; the other two are not blanches ı. b Deduce the probability of the event : C : ˇ only one of the balls drawn is blanche ı. 4 Deduce the probability of the event : D : ˇ two balls drawn are white and one ball is not blanche ı. 5 Let X be the random variable taking as its value the number of white balls drawn : a Determine the probability distribution of the random variable X . b Determine the expectation of the variable X . E.3747 A game consists of throwing a balanced die 3 times. It is assumed that the various throws are inde-pendent and can be assimilated to discounted draws. 1 a Consider the events : A : ˇ The player makes the number 6 ı three times. B : ˇ The player achieves the number 6 ı twice Determine the probabilities of these events. 2 The stake for this game is set at 3 euros. Each winner takes home : 500 euros when the player makes three times the num-ber 6 . 5 euros when the player makes twice the number 6 . We denote X the random variable associated with the win of a game; the win is the difference between the sum won and the stake. a In a table, give the probability distribution of the vari-able X . b Determine the expectation of the random variable X . https://chingmath.fr chapExoCorrec/3746 sacados/3746 chapExoCorrec/3747 sacados/3747