Grade 11 - STMG / conditional probability 18 exercises (including 16 corrected)

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AB BBABBA AB 1. Probability tree construction E.11536 Consider a set Ω and two of its parts A and B shown below, whose elements are represented by crosses : Equally likely, we choose an element at random. 1 Determine the probability of event A and event A . 2 Assuming that the chosen element does not belong to A : a what is the probability that this event belongs to B ? b what is the probability that this event does not belong to B ? 3 Assuming that the chosen element belongs to A : a what is the probability that this event belongs to B ? b what is the probability that this event does not belong to B ? 4 Complete the probability tree below : E.11537 Consider the universe Ω and its two events A and B are rep-resented : An element of the universe Ω is chosen at random. 1 Determine the probability of event A and event A . 2 Assuming that the chosen element does not belong to A : a what is the probability that this event belongs to B ? b what is the probability that this event does not belong to B ? 3 Assuming that the chosen element belongs to A : a what is the probability that this event belongs to B ? b what is the probability that this event does not belong to B ? 4 Complete the probability tree below : https://chingmath.fr chapExoCorrec/11536 sacados/11536 AB BBABBA chapExoCorrec/11537 sacados/11537 AB BBABBA
CTRRCTRR MMEMME E.11538 A game consists of shaking and turning a bottle upside down to remove one of its contents. Here is the content of this bottle: We assume that the law of equiprobability applies to this ran-dom experiment. We consider the 4 events below : R : ˇ The item that comes out is a stripe ı ; C : ˇ The item that comes out is a square ı ; T : ˇ The item drawn is a triangle ı ; R : ˇ The item drawn is a circle ı ; Construct the probability tree : E.11539 A toy company specializes in manufacturing dolls that talk and walk. Each doll can have two defects and only two: one mechanical defect and one electrical defect. A statistical study shows that : 8% of the dolls have the mechanical defect ; 5% of the dolls have the electrical defect ; 2% of the dolls have both defects. Daily production is 1,000 dolls. 1 Copy and complete the table below, which describes daily production : dolls with mechanical defect Dolls without mechanical defect total Dolls with electrical defect Dolls without electrical defects Total 80 1 000 In the rest of the exercise, each numerical result will be given in decimal form. 2 A doll is randomly selected from the day’s production. Consider the two events below : E : " the doll selected has an electrical defect " M : " The sampled doll has a mechanical defect " 3 a Determine the probability of event E and event E b Assuming that the doll selected has an electrical defect, what is the probability that this doll has a mechanical problem? c Assuming that the doll selected does not have an elec-trical defect, what is the probability that this doll has a mechanical problem? 4 Complete the probability tree below : https://chingmath.fr chapExoCorrec/11538 sacados/11538 CTRRCTRR chapExoCorrec/11539 sacados/11539 MMEMME
Demi-pensionnaireGarçon AB E.11540 In a high school with 2000 students, the distribution of stu-dents by gender and choice of first foreign language is given as follows 1 A student claims " that there are as many girls as boys in this high school" . Is he right? Justify your answer. A student is chosen at random, with equal probability, from this high school. Consider the following events : Girls Boys English 712 728 Other LV1 288 272 F : "thestudent is a girl"; A : "thestudent has chosen English as their first foreign language" In the following questions, the results will be given as frac-tions that do not need to be simplified. 2 Determine the probability of event A F . 3 Determine the probability of event A given that event F has occurred. 4 Are events A and F independent? Justify your answer. 5 We know that the chosen student is a boy. Consider the following statement : " The probability that he chose English as his first foreign language is more than three times greater than the prob-ability that he did not choose English as his first foreign language. ". Is this statement true? Justify your answer. E.11541 We have a biased coin for which the probability of getting heads when tossed is equal to 1 4 . 1 Determine the probability of getting tails. 2 We flip this coin three times in a row, with each flip be-ing independent, and note the result (heads or tails) for each flip. a Represent the situation using a probability tree. b What is the probability of getting heads exactly once in these three tosses? c What is the probability of never getting heads? 2. Conditional Probability: Introduction E.11511 In a statistical study, we con-sider a class of 24 students, where each student is repre-sented by a box in the graph be-low : G : ˇ the student is a boy ı ; D : ˇ the student is a day boarder ı. The random experiment con-sists of choosing a student at random from the class and see-ing whether or not these two cri-teria are met : 1 a Determine the probability of choosing a " boy ". b Knowing that a " boy "has been chosen, what is the probability of choosing a " day student " c Determine the probability of choosing a " boy "who is a " day student ". d What do we notice? 2 a Determine the probability of choosing a " girl ". b Given that a " girl "has been chosen, what is the prob-ability of choosing a " day student " c Determine the probability of choosing a " girl "who is " a day student ". d What do we notice? E.11513 Consider the universe Ω and its two events A and B are rep-resented : Use the law of equiprobability. 1 a Determine the probability of event A . b Knowing that event A has occurred, determine the probability of event B . c Determine the probability of event A B . d What do we notice? 2 a Determine the probability of event A . b Knowing that event A has occurred, determine the probability of event B . c Determine the probability of event A B . d What do we notice? 3. Conditional probability E.11512 In a statistical study, we consider a class of 24 students, where each student is represented by a box in the graph below : https://chingmath.fr chapExoCorrec/11540 sacados/11540 chapExoCorrec/11541 sacados/11541 chapExoCorrec/11511 sacados/11511 Demi-pensionnaireGarçon sacados/11513 AB chapExoCorrec/11512 sacados/11512
ClasseBDemi-pensionnaireGarçon ::::::G:::GD::::::G:::GD AB CD 0112101102011112 BBABBA 0;20;41B0;59BA0;80;41B0;59BA Two characteristics are studied in the individuals in this study : G : ˇ the student is a boy ı ; D : ˇ the student is a day student ı. The random experiment consists of choosing a student at ran- dom from the class and seeing whether or not these two cri-teria are met. 1 a Determine the probability of choosing a " boy ". b Knowing that a " boy "has been chosen, what is the probability of choosing a " day student " c Determine the probability of choosing a " boy "who is " an external student ". d What do we notice? 2 a Determine the probability of choosing a " girl ". b Given that a " girl "has been chosen, what is the prob-ability of choosing an " external student " c Determine the probability of choosing a " girl "who is " external ". d What do we notice? 4. Independent events E.11591 Consider the random experiment of universe Ω represented below and provided with the law of equiprobability: 1 Consider the two events A and B shown below : Determine the following probabilities : a P A B b P B 2 Consider the two events C and D shown below : Determine the following probabilities : a P C D b P D Definition: let A and B be two events. The events A and B are said to be independent P A B = P B . 3 Are ( A ; B ) and ( C ; D ) pairs of independent events. E.11592 Un The game consists of spinning the wheel opposite once and not-ing the color of the square ob-tained, then spinning the wheel a second time and noting the num-ber obtained. The two throws of the wheel are obviously independent of each other. Consider the two events : A : ˇ the box obtained is grey on the first tirage ı ; B : ˇ the resulting square is numbered 0 on the second tirage ı. 1 a Determine the probabilities : P B ; P A B b Are the events A and B independent? 2 Complete the probability tree : E.11593 In a random experiment, consider two events A and B allowing the probability tree to be con-structed : 1 Determine the probability of the event B . 2 Establish that the events A and B are independent. https://chingmath.fr ClasseBDemi-pensionnaireGarçon ::::::G:::GD::::::G:::GD chapExoCorrec/11591 sacados/11591 AB CD chapExoCorrec/11592 sacados/11592 0112101102011112 BBABBA chapExoCorrec/11593 sacados/11593 0;20;41B0;59BA0;80;41B0;59BA
NombredebulbesdetulipejauneNombredebulbesdetuliperougeNombredebulbesdetulipenoireTotalNombredebulbesdetulipequiπeurirontNombredebulbesdetulipequineπeurirontpasTotal1000 E.11594 In a class of 30 students, there is a drawing club and a theater club. The drawing club has 10 members, and the theater club has 6 members. Two students are members of both clubs. A student from the class is chosen at random and asked a question. We call: D the event : ˇ The student is a member of the drawing club ı ; T the event : ˇ The student is a member of the theater club ı. Show that events D and T are independent. E.11595 A gardener has a bag filled with 1.000 tulip bulbs. These include : 60 % are yellow tulip bulbs ; 25 % are red tulip bulbs ; the rest are black tulip bulbs. In addition : 28 % of all these bulbs will not flower; 80 % of the yellow tulip bulbs will flower; 60 black tulip bulbs will not flower. Part A Copy and complete the table below : Part B The gardener draws a bulb at random from his bag. We note : F the event : ˇ The bulb will bloom ı ; J : ˇ The bulb is that of a tulip jaune ı ; R : ˇ The bulb is that of a tulip rouge ı ; N : ˇ The bulb is that of a tulip noire ı. 1 Determine the probabilities of the following events : J F , J , F , J F . 2 Are the events J and F independent? Justify the answer. E.11596 During the manufacture of a pair of spectacles, the pair of lenses must undergo two treatments noted T 1 and T 2 . One pair of glasses is taken at random from the production. We denote by A the event : ˇ the pair of lenses has a defect for the T 1 treatment. We denote by B the event : ˇ the pair of glasses has a defect for treatment T 2 ı. We note A and B respectively the opposite events of A and B . One study showed that : the probability that a pair of lenses has a defect for T 1 noted P A is equal to 0.1 . the probability that a pair of lenses has a defect for T 2 noted P B is equal to 0.2 . the probability that a pair of lenses has neither defect is 0.75 . 1 Copy and complete the following table with the corre-sponding probabilities. A A Total B B Total 1 2 a Determine, justifying the answer, the probability that a pair of lenses, taken at random from production, will have a defect for at least one of the two treatments T 1 or T 2 . b Give the probability that a pair of lenses, taken at ran-dom from production, has two defects, one for each T 1 treatment and T 2 . c Are the events A and B independent? Justify the an-swer. https://chingmath.fr chapExoCorrec/11594 sacados/11594 chapExoCorrec/11595 sacados/11595 NombredebulbesdetulipejauneNombredebulbesdetuliperougeNombredebulbesdetulipenoireTotalNombredebulbesdetulipequiπeurirontNombredebulbesdetulipequineπeurirontpasTotal1000 chapExoCorrec/11596 sacados/11596
Demi-pensionnaireGarçon E.11485 In a statistical study, we con-sider a class of 24 students, where each student is repre-sented by a box in the graph be-low : G : ˇ the student is a boy ı ; D : ˇ the student is a day boarder ı. The random experiment con-sists of choosing a student at random from the class and see-ing whether or not these two cri-teria are met : 1 a Determine the probability of choosing a " boy ". b Knowing that a " boy "has been chosen, what is the probability of choosing a " day student " c Determine the probability of choosing a " boy "who is a " day student ". d What do we notice? 2 a Determine the probability of choosing a " girl ". b Given that a " girl "has been chosen, what is the prob-ability of choosing a " day student " c Determine the probability of choosing a " girl "who is " a day student ". d What do we notice? E.11620 Une entreprise ferme aux mois de juil-let et août. L’ensemble des 180 employés d’une entreprise prennent leurs congés pendant cette période. On a les informations suivantes : 28 hommes ont pris leurs vacances en juillet. 138 personnes ont pris leurs vacances en août. L’entreprise compte 60 femmes. On crée une expérience aléatoire en choisissant au hasard un employé de cette entreprise. On considère les deux événements ci-dessous : F : "l’employé choisi est une femme "; J : "l’employé choisi prend ses vacances en juillet ". 1 Reproduire et compléter le tableau ci-contre : Juillet Août Total Femmes en vacances Hommes en vacances Total en vacances 2 Donner les probabilités : P ( F ) ; P ( F J ) ; P ( P ( F J ) 3 a Déterminer la probabilité P J ( F ) . b Est-ce que les événements F et J sont indépendants ? Justifier. 5. Shares E.11584 Un village propose aux partici-pants de la fête du sport deux épreuves : une randonnée et un cross. Il n’est pas possible de s’inscrire aux deux épreuves à la fois. On dispose des informations suivantes : 90% des participants ont choisi la randonnée, parmi eux, 5% sont licenciés dans un club. 10% des participants ont choisi le cross, parmi eux, 40% sont licenciés dans un club. Un journaliste interroge un participant au hasard. On considère les événements suivants : R : " Le participant a choisi la randonnée " L : " Le participant est licencié dans un club " 1 Par simple lecture de l’énoncé, indiquer : a La probabilité que le participant interrogé soit licencié dans un club sachant qu’il a choisi la randonnée. b La probabilité que le participant interrogé soit licencié dans un club sachant qu’il a choisi le cross. En prenant connaissance de ces deux probabilités, le journaliste estime que s’il choisit un participant parmi ceux qui sont licenciés dans un club, la probabilité qu’il ait effectué le cross sera largement supérieure à 50%. L’objectif des questions suivantes est de vérifier si cette intuition est correcte. 2 Représenter la situation par un arbre de probabilités. 3 a Déterminer la probabilité que le participant inter-rogé ait choisi le cross et soit licencié dans un club. b Vérifier que la probabilité que le participant interrogé soit licencié dans un club est égale à 850 10 000 ; soit 8 ; 5%. 4 Le journaliste interroge un participant licencié dans un club. Déterminer la probabilité que ce participant ait choisi le cross. L’intuition du journaliste est-elle correcte ? https://chingmath.fr chapExoCorrec/11485 sacados/11485 Demi-pensionnaireGarçon chapExoCorrec/11620 sacados/11620 sacados/11584