Grade 11 / Probability and independent events 23 exercises (100% corrected)

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AB CD 0112101102011112 BBABBA 0;20;41B0;59BA0;80;41B0;59BA CBA 1. Definition of independence E.10582 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.5834 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.3733 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.8321 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. E.10583 Consider the random experiment of universe Ω represented below and provided with the law of equiprobability: Also represented are the events A , B and C . Of the two pairs of events ( A ; B ) and ( A ; C ) , which (s) is an independent pair of events? 2. Properties https://chingmath.fr chapExoCorrec/10582 sacados/10582 AB CD chapExoCorrec/5834 sacados/5834 0112101102011112 BBABBA chapExoCorrec/3733 sacados/3733 Extrait France Juin 2000 chapExoCorrec/8321 sacados/8321 0;20;41B0;59BA0;80;41B0;59BA chapExoCorrec/10583 sacados/10583 CBA
NombredebulbesdetulipejauneNombredebulbesdetuliperougeNombredebulbesdetulipenoireTotalNombredebulbesdetulipequiπeurirontNombredebulbesdetulipequineπeurirontpasTotal1000 E.10584 Definition: let A and B be two eventmets. The events A and B are independent if, and only if,: P A ∩ B = P A ×P B . In a probabilized space Ω ; P , consider the two events A and B such that : P A = 0.2 ; P B = 0.25 ; P A ∪ B = 0.4 1 Determine the probability P A ∩ B . 2 Justify that events A and B are independent. E.136 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.3734 We have a cubic die whose faces are numbered from 1 to 6 . We denote by p k the proba-bility of obtaining, on a throw, the face numbered k ( k is an integer and 1 k 6 ) . This die has been piped such that : the six faces are not equiprobable ; the numbers p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , in this order, are six consecutive terms of an arithmetic sequence of reason r ; the numbers p 1 , p 2 , p 4 in this order, are three consecutive terms of a geometric sequence. 1 Demonstrate that : p k = k 21 for any integer k such that 1 k 6 . 2 We roll this die once and consider the following events : A : ˇ the number obtained is pair ı ; B : ˇ the number obtained is greater than or equal to 3 ı ; C : ˇ the number obtained is 3 or 4 ı. a Calculate the probability of each of these events. b Calculate the probability that the number obtained is greater than or equal to 3 knowing that it is even. c Are the events A and B independent? Are the events A and C independent? E.10578 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. 3. Independence: and probability trees https://chingmath.fr chapExoCorrec/10584 sacados/10584 chapExoCorrec/136 sacados/136 NombredebulbesdetulipejauneNombredebulbesdetuliperougeNombredebulbesdetulipenoireTotalNombredebulbesdetulipequiπeurirontNombredebulbesdetulipequineπeurirontpasTotal1000 chapExoCorrec/3734 sacados/3734 Extrait de Polynesie Septembre 2000 chapExoCorrec/10578 sacados/10578
BBABBA 12Circuit en parallèleA12Circuit en sérieB D2D2D1D2D2D1 E.10585 Consider a probabilized space (Ω ; P ) and two independent events A and B . We have the following information : P A = 0.6 ; P B = 0.7 1 Complete the probability tree below : 2 Determine the following probabilities : P A ∩ B ; P A ∪ B E.6764 An electronic circuit consists of two identical components numbered 1 and 2 . We note D 1 the event ˇ the component 1 fails before a an ı and we note D 2 the event ˇ the component 2 fails before a an ı. It is assumed that the two events D 1 and D 2 are independent and that : P D 1 = P D 2 =0.39 Two possible setups are considered, shown below : 1 Complete the probability tree below : 2 a When the two components are mounted ˇ in paral-lèle ı, the circuit A fails only if both components fail at the same time. Calculate the probability that circuit A will fail before one year. b When the two components are mounted ˇ in série ı, the circuit B fails as soon as at least one of the two com-ponents fails. Calculate the probability that circuit B will fail before one year. 4. Using independence E.4174 A factory produces bags. Each bag produced can have two defects : defect a and defect b . A bag is said to be defective if it has at least one of the two defects : The probabilities requested will be given with their exact dec-imal values. A bag is taken at random from one day’s production. We note A the event ˇ the bag has the defect a ı and B the event ˇ the bag has the defect b ı. The probabilities of events A and B are respectively: P ( A ) = 0.02 ; P ( B ) = 0.01 ; These two events are assumed to be independent. 1 Calculate the probability of the event : C : ˇ the sampled bag has the defect a and the defect b ı 2 Calculate the probability of the event : D : ˇ the bag is faulty ı 3 Calculate the probability of the event : E : ˇ the bag has no defect ı 4 Knowing that the bag has the defect a , what is the prob-ability that it also has the defect b ? E.3739 A watch factory produces a series of watches. During manufacture, two types of defect may appear, desig-nated a and b . 2 % of the watches manufactured have the a defect and 10 % the b defect. A watch is drawn at random from the production. The fol-lowing events are defined : A : ˇ The watch drawn has the defect a ı ; B : ˇ The pulled watch has the defect b ı ; C : ˇ The pulled watch has neither défauts ı ; D : ˇ The pulled watch has one and only one of the two défauts ı. The events A and B are assumed to be independent. 1 Show that the probability of the event C is equal to 0.882 . 2 Calculate the probability of the event D . 5. Independence: and algebraic manipulation https://chingmath.fr chapExoCorrec/10585 sacados/10585 BBABBA chapExoCorrec/6764 sacados/6764 Extrait d'Antilles-Guyane Juin 2015 12Circuit en parallèleA12Circuit en sérieB D2D2D1D2D2D1 chapExoCorrec/4174 sacados/4174 chapExoCorrec/3739 sacados/3739
V1V2V3 F1F1F2F3 5points0point3points0point E.4322 Let A and B be two indepen-dent events in the same universe Ω such that : P ( A ) = 0.3 ; P ( A ∪ B ) = 0.35 . Determine the probability of the event B . E.4169 We denote by A and B two independent events of a universe provided with a probability law P . We know that : P ( A ∪ B )= 4 5 ; P ( A )= 3 5 Determine the probability of the event B . E.5506 Consider a probabilized space (Ω ; P ) . Let A and B be two independent events. Show that the events A and B are also independent. E.4150 Let A , B and C be three events in the same universe Ω provided with probability P . We know that : A and B are independent ; P ( A ) = 2 5 ; P ( A ∪ B ) = 3 4 P ( C ) = 1 2 ; P ( A ∩ C ) = 1 10 Without justification, indicate whether each of the following propositions is true or false. Proposition 1: P ( B ) = 7 12 Proposition 2: P A ∪ C = 2 5 où A ∪ C denotes the opposite event of A ∪ C . 6. Non-symmetrical tree E.9447 With three identical valves V 1 , V 2 and V 3 , we make the hydraulic circuit shown opposite. The circuit is in working order if V 1 is in working order or if V 2 and V 3 are simultaneously. It is treated as a random experiment whether each valve is or is not in working order after 6 000 hours. Note: F 1 the event : ˇ the V 1 valve is in working order after 6 000 heures ı. F 2 the event : ˇ the valve V 2 is in working order after 6 000 heures ı. F 3 the event : ˇ the valve V 3 is in working order after 6 000 heures ı. E the event : ˇ the circuit is in working order after 6 000 heures ı. It is assumed that the events F 1 , F 2 and F 3 are both indepen-dent and each has a probability equal to 0.3 . 1 The probabilistic tree opposite represents part of the sit-uation. Reproduce this tree and place the probabilities on the branches. 2 Demonstrate that : P ( E )=0.363 . 3 Knowing that the circuit is in working order after 6 000 hours, calculate the probability that valve V 1 is in work-ing order at that time. Round to the nearest thousandth. E.4256 A game consists of throwing darts at a target. The target is di-vided into four sectors, as shown in the figure below : It is assumed that the throws are independent 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. We have the following probabilities : p 0 = 1 2 ; p 3 = 1 3 ; p 5 = 1 6 A game of this game consists of throwing a maximum of three darts. The player wins the game if he obtains a total (for the three throws) greater than or equal to 8 points. If after 2 throws, he has a total greater than or equal to 8 points, he does not throw the third dart. Note the events : G 2 : ˇ the player wins the game by 2 lancers ı ; G 3 : ˇ the player wins the game by 3 lancers ı ; P : ˇ player loses the partie ı. We note P ( A ) the probability of an event A . 1 Show, using a weighted tree that : P ( G 2 ) = 5 36 In the following, we will assume that : P ( G 3 ) = 7 36 2 Deduct P ( P ) 7. Independence: with a little algebra https://chingmath.fr chapExoCorrec/4322 sacados/4322 Extrait d'Antilles Juin 2011 chapExoCorrec/4169 sacados/4169 Extrait de Liban Juin 2009 chapExoCorrec/5506 sacados/5506 chapExoCorrec/4150 sacados/4150 chapExoCorrec/9447 sacados/9447 V1V2V3 F1F1F2F3 chapExoCorrec/4256 sacados/4256 5points0point3points0point
0;30;40;2C0;8CB0;60;5C0;5CBA0;70;90;4C0;6CB0;10;2C0;8CBA 354535N325R3N21525N335R3R2N1252515N345R3N23545N315R3R2R1U7U6U5U4U3U2U1 E.5527 We denote by x a real belong-ing to the interval 0 ; 80 . An urn contains 100 small wooden cubes of which 60 are blue and the others red. Of the blue cubes, 40 % have their faces marked with a circle, 20 % have their faces marked with a rhombus and the others have their faces marked with a star. Among the red cubes, 20 % have their faces marked with a circle, x % have their faces marked with a rhombus and the others have their faces marked with a star. A cube is drawn at random from the urn. 1 Demonstrate that the probability of drawing a cube marked with a rhombus is equal to 0.12+0.004 x . 2 Determine x so that the probability of drawing a cube marked with a rhombus is equal to that of drawing a cube marked with a star. 3 Determine x so that the events ˇ draw a cube bleu ı and ˇ draw a cube marked with a losange ı are independent. 4 It is assumed in this question that x =50 . Calculate the probability of drawing a blue cube knowing that it is marked with a diamond. E.10586 An urn contains blue balls and red balls. Some of these balls have a star on them. Here is the table showing the contents of this urn : has a star does not have a star Blue ball 5 6 Red ball 7 8 Consider the random experiment of choosing a ball at random from this urn and noting its color and whether or not it has a star. Consider the two events : A : ˇ the ball drawn is red ı B : ˇ the ball drawn has a star ı 1 Establish that events A and B are not independent. 2 We add 30 balls with a red or blue star so that events A and B are independent. Determine the number of red balls and the number of blue balls that have been added. 8. With several events E.8325 Consider a random experiment and three of its events A , B and C giving the probability tree below : 1 Determine the probability of the event C . 2 Establish that the events A and C are independent? E.3738 Consider three urns, each con-taining black and red balls. An experiment consists of randomly drawing one ball from each urn. For all i ∈ 1 ; 2 ; 3 , consider the following events : N i : ˇ a black ball is drawn from urn U i ı ; R i : ˇ a red ball is drawn from the urn U i ı. We consider the following probability tree : 1 Determine the probability of event N 3 . 2 Are events N 1 and N 3 independent? https://chingmath.fr chapExoCorrec/5527 sacados/5527 chapExoCorrec/10586 sacados/10586 chapExoCorrec/8325 sacados/8325 0;30;40;2C0;8CB0;60;5C0;5CBA0;70;90;4C0;6CB0;10;2C0;8CBA chapExoCorrec/3738 sacados/3738 354535N325R3N21525N335R3R2N1252515N345R3N23545N315R3R2R1U7U6U5U4U3U2U1