Outside the high school program / Probability 20 exercises (including 8 corrected)

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AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ 1. Enumeration and equiprobability E.140 A game consists of ticking 8 boxes on a A ( n ◦ 1 to 20) grid and 1 box on a B ( n ◦ 1 to 4) grid Grille A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Grille B 1 2 3 4 ˇ Results of probability calculations will be given to 0.0001 près ı 1 Determine the number of possible ways to tick 8 boxes in the grid A . A draw determines 8 ˇ good numéros ı in grid A and 1 ˇ good numéro ı in grid B . 2 Case 1 : the player recovers his stake when he has ticked 4 ˇ vouchers numéros ı in the grid A and 1 ˇ vouchers numéros ı in the grid B . a How many ticked grids A have 4 ˇ good numéros ı and 4 ˇ (wrong numbers) ı? Deduce the probability of ticking 4 ˇ right numbers ı in grid A . b Determine the probability of ticking the ˇ good numéros ı in the B grid. c Deduce that the probability of the player recovering his bet is 0.068 8. 3 Case 2 : The player wins if he has ticked 5, 6, 7 and 8 ˇ good numéros ı in the grid (with or without the winning grid number B ) . a How many ticked grids A have 5 ˇ good numéros ı and 3 ˇ (bad numéros ı? b Deduce the probability of ticking 5 ˇ good numéros ı in the A grid. 4 The probability of winning is assumed to be 2 11 . A player decides to play the same numbers on 4 consecutive draws. Determine the probability that this player will be a win-ner 2 times out of the 4 draws. E.4788 A random experiment consists of randomly drawing a card from a deck of 32 cards. 1 Determine the probabilities of the fol-lowing events : A : ˇ The card drawn is a spade ı ; B : ˇ The card drawn is a face card ı ; C : ˇ The card drawn is black ı ; D : ˇ The card drawn is the jack ı ; 2 Determine the probabilities of the following events : a A ∩ B b A ∩ C c A ∪ B d B ∪ C e C ∩ D f C ∪ D g C ∩ D h C ∪ D E.4838 On consider a deck of 32 cards and a person is asked to draw a card at random from this deck. Consider the following events : R : ˇ the card drawn is a roi ı ; C : ˇ the card drawn is a coeur ı ; N : ˇ the card drawn is noire ı color. In this exercise, results will be given as irre-ducible fractions 1 Determine the probability of events R , C and N . 2 Determine the probability of the following events : a C ∩ R b R ∩ C c R ∪ N E.7500 Consider a deck of 32 cards shown opposite. We draw a card at random from this deck and consider the events below : A : ˇ the card drawn is a carreau ı B : ˇ The card drawn is a figure ı C : ˇ The card drawn has the number 8 or 9 ı Determine the probabilities below : a P C b P A ∩ C c P A ∪ B d P B ∩ C https://chingmath.fr chapExoCorrec/140 sacados/140 chapExoCorrec/4788 sacados/4788 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ chapExoCorrec/4838 sacados/4838 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ chapExoCorrec/7500 sacados/7500 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣
Diviseursde240Multiplesde10Diviseursde2Diviseursde5240120806040302010×××××××× E.135 Part A 1 Determine the 20 positive divisors of 240. 2 In the table below, among these 20 in-tegers arranged in ascending order, we have ticked the multiples of 10 Reproduce and complete the table, ticking the multiples of 2 and 5 Part B We study the random event of randomly drawing a number from the 20 divisors of 240. 1 What is the probability of drawing the number 2? the number 7? 2 Consider the following events : A : ˇWe draw a multiple of 10ı B : ˇWe draw a multiple of 2ı C : ˇWe draw a multiple of 5ı Determine the probabilities P ( A ) , P ( B ) and P ( C ) of the events A , B , C . 3 This random event is repeated four times in succession under the same conditions. a What is the probability of drawing a multiple of 10 four times in a row? b What is the probability of never drawing a multiple of 10? c What is the probability of drawing a multiple of 10 at least once? d For any natural n between 1 and 4, note A n the event : ˇ Get a multiple of 10 for the first time on the n -th draw ı Calculate the probabilities P ( A 2 ) , P ( A 3 ) and P ( A 4 ) of the events A 2 , A x , A 4 . E.4836 The table below provides an overview of daily newspaper readership in France, based on a sample of 800 people surveyed in 2005 . Every day or almost Once or twice a se-maine Seule ment during certain péri-odes Rare-ment Jamais Total Agriculteurs exploitants 1 10 2 8 79 100 Artisans, commerçants , chefs d’entreprise 11 11 5 7 66 100 Cadres 17 16 10 18 39 100 Professions inter-médiaires 8 15 7 15 55 100 Employees 6 7 4 9 74 100 Ouvriers (y compris agricoles) 4 5 3 5 83 100 Retraités 6 7 2 6 79 100 Autres inactifs 5 9 4 9 73 100 Total en effectif 58 80 37 77 548 800 Pourcentages du total 7, 25 % 10 % 4.625 % In this exercise, results will be given in decimal form and rounded to 0.001 nearest. Part A. 1 The last row of the table below shows the proportion of each category in relation to the total sample. Calculate the missing values in this last row. 2 Give the probability that a person chosen at random from among the executives never reads. Part B. A person is chosen at random from this sample of 800 people. In this part, we note the following events : J the event : ˇ the person chosen never reads ı ; O the event : ˇ the person chosen is a worker ı. 1 Calculate the probabilities of events J and O . 2 Calculate the probability of event J ∩ O . 3 Calculate the probability of event J ∪ O https://chingmath.fr sacados/135 Diviseursde240Multiplesde10Diviseursde2Diviseursde5240120806040302010×××××××× chapExoCorrec/4836 sacados/4836
EvènementsélémentairesNNNNNNNBBNNNNNNNNBBNBNNBNNBBBB1ertirage2etirage E.4787 An urn contains two black balls and one white ball; the game is played with the ball being returned : i.e., once the ball has been drawn, it is returned to the urn before the next draw. Here’s a decision tree based on the drawing of two balls : 1 Taking into account the order in which the balls are drawn, what is the possible number of different draws? 2 Determine the probability of the following events : a A : ˇ The first ball drawn is blanche ı. b B : ˇ The two balls drawn are différents ı colors. c C : ˇ The second ball is a noire ı ball. 3 Give the probabilities of the following events : a A ∩ B b B c C 2. Adequation of a probability distribution E.4146 A study looks at purchases made by internet users via the internet. Out of 1 000 internet users who purchased this equipment, the following statistics were compiled : European websites Canadian website Indian website Number of buyers 335 310 355 1 We note respectively f 1 , f 2 and f 3 the frequencies asso-ciated with the previous numbers. We set : d 2 = k =3 k =1 f k − 1 3 2 . Calculate d 2 then 1000 · d 2 . 2 We simulate 3 000 times the experiment of randomly se-lecting a number from 1 ; 2 ; 3 with equal probabil-ity. For each of these simulations, we obtain a value of 1000 · d 2 . Here are the results : Min. First decile First quartile Median Third quartile Ninth decile Max. 0.000 5 0.076 3 0.211 1 0.488 45 0.940 1 1, 5104 5.925 6 At the risk of 10 % , can we consider that the choice of a European, North American, or Asian site is made on an equiprobable basis? E.4183 We have a regular tetrahedral die ; we want to know if the die used can be considered perfectly balanced. To do this, we number the four sides of the die from 1 to 4 , then we roll the die 160 times, noting the number n i of times each side is hidden ; we obtain the following results : Face i 1 2 3 4 Number n i 34 48 46 32 We note f i the relative frequency for face n i and d 2 obs the ac-tual : 4 i =1 f i − 1 4 2 We then simulate 1 000 times the experiment consisting of ran-domly selecting a number 160 times from the set 1 ; 2 ; 3 ; 4 then, for each simulation, we calculate: d 2 = 4 i =1 F i − 1 4 2 where F i is the frequency of occurrence of the number i . The 9 e decile of the statistical series of 1 000 values of d 2 is equal to 0.009 8 . Based on the experiment conducted and the risk of 10 % , can the die be considered perfectly balanced? https://chingmath.fr chapExoCorrec/4787 sacados/4787 EvènementsélémentairesNNNNNNNBBNNNNNNNNBBNBNNBNNBBBB1ertirage2etirage sacados/4146 sacados/4183 Extrait de France Septembre 2005
E.4189 We decide to test the tetrahedral die to see if it is well balanced or if it is loaded. To do this, we roll the die 200 times and obtain the following table : Face k 1 2 3 4 Number of outputs of the face k 58 49 52 41 1 Calculate the output frequencies f k observed for each of the faces. 2 We set d 2 = 4 k =1 f k − 1 4 . Calculate d 2 . 3 We now perform 1 000 simulations of 200 rolls of a well-balanced tetrahedral die and calculate the number d 2 for each simulation. For the statistical series of 1 000 values of d 2 , we obtain the following results : Min. D 1 Q 1 Médiane Q 3 D 9 Max. 0.00124 0.00192 0.00235 0. 00281 0.00345 0.00452 0.01015 At the risk of 10 % , can we consider that this die is loaded 7 . E.4196 1 The 1000 first decimal places of ı are given here by a computer : 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3233066470 9384460959 0582235725 3594085234 8111745028 4102701930 5211055596 4462294895 4930301964 4288109756 6593344612 8475648233 7867831652 7120190914 5648566923 4603486534 5432664825 3393607260 2491412737 2450700660 6315580574 8815209209 6282925409 1715364367 8925903600 1133053054 8820466525 3841469519 4151160943 3057270365 7595919530 9218611738 1932611793 1051185480 7446297996 2749567355 8857527240 9122793318 3011949129 8336733624 4065664308 6025394946 3952247371 9070217986 0943702770 5392171762 9317675238 4674818467 6691051320 0056812714 5263560827 7857753427 9778900917 3637178721 4684409012 2495343054 6549585371 0507922796 8925892354 2019956112 1290219608 6403441815 9813629774 7713099605 1870721134 9999998372 9780499510 5973173281 6096318599 0244594553 4690830264 2522300253 3446850352 6193110017 1010003137 8387528865 8753320830 1420617177 6691473035 9825349042 8755460731 1595620633 8235378759 3751957781 8577805321 7122600661 3001927876 6111959092 1642019894 By grouping these decimals by values between 0 and 9 , we obtain the following table : Valeurs 0 1 2 3 4 Occurrences 93 116 102 102 94 Values 5 6 7 8 9 Occurrences 97 94 95 101 106 Using a table, we simulated 1000 experiments of 1000 random draws of a number between 0 and 9 . For each experiment, we calculated d 2 = k =9 k =0 f k − 0.1 2 where f k represents, for the experiment, the observed frequency of the digit k . We then obtained a statistical series for which we calculated the first and ninth deciles ( d 1 and d 9 ) , the first and third quartiles ( Q 1 and Q 3 ) and the median ( Me )) : d 1 = 0.000 422 ; Q 1 = 0.000 582 ; M e = 0.000 822 Q 3 = 0.00 1136 ; d 9 = 0.00145 By calculating d 2 on the series of 1000 first decimal places of π , we obtain : a 0.000 456 b 0.004 56 c 0.000 314 2 A statistician discovering the table and unaware that these are the decimals of π , hypothesizes that the series is the re-sult of independent random draws following an equipartition law. He takes a risk of 10 % of rejecting this hypothesis when it is true. Does he accept this hypothesis? a Oui b Non c He cannot conclude 3. Other draws: successive with discount E.4164 There are 26 cards containing the 26 letters of the alphabet. Three cards are drawn successively from the deck. Each draw is assumed to be independent. 1 How many three-letter words can be formed in this way? 2 a Determine the probability of the event : A 1 : ˇ The word begins with the letter B ı. b Determine the probability of the event : A 2 : ˇ The second letter of the word is the letter B ı. 3 a Determine the probability of the event : B 1 : ˇ The first letter of the word is the only letter B ı b What is the probability of the event : C : ˇ The word contains the letter B ı only once. 4. Other draws: successive without discount https://chingmath.fr sacados/4189 Extrait de France Juin 2006 sacados/4196 chapExoCorrec/4164 sacados/4164
E.4163 A librarian wants to arrange a four-volume encyclopedia in its proper place. To do this, he pulls the volumes from his cart in succession and arranges them in that order. At the end of this random arrangement, he looks at its composition : 1 What is the probability of the event : A : ˇ Is the volume 1 at its place ı? 2 What is the probability of the event : B : ˇ The volume 2 is at its place ı? 3 What is the probability of the event : C : ˇ The four volumes are perfectly ordonnés ı? 5. Other draws: simultaneous E.3744 Hint: exact values will be given as fractions, as well as values rounded to 10 − 3 near. A child puts 10 red marbles and 3 green marbles into a cube box. In this game, he simultaneously chooses three marbles at ran-dom from the cubic box and looks to see how many red mar-bles he has chosen. We call X the random variable corre-sponding to the number of red marbles chosen. 1 Determine the probability distribution of X . 2 Calculate the mathematical expectation of X . E.4173 A bag contains 10 tokens indistin-guishable by touch : 7 white tokens numbered 1 to 7 and 3 black tokens numbered 1 to 3 . Two tokens are simultaneously drawn from this bag 1 Note A the event ˇ get two tokens blancs ı. Show that the probability of the event A is equal to 7 15 . 2 Let B be the event ˇ getting two tokens with numbers impairs ı. Calculate the probability of B . 3 Are the events A and B independent? E.4184 An urn contains five black balls and three red balls, indistinguishable to the touch. Three balls are simultaneously extracted from the urn. What is the probability of obtaining two black balls and one red ball? 6. Other prints E.4144 A game involves simultaneously drawing 4 balls indistinguishable by touch from a bag contain-ing a black ball and 9 white balls, then rolling a well-balanced six-sided die numbered from 1 to 6 . If the black ball is drawn, you need to get an even integer with the die to win. If the black ball is not drawn, you must get a six with the die to win. We call: N the event : ˇ the black ball is among the balls tirées ı ; G the event : ˇ the player gagne ı. 1 Determine the probability of the event N . 2 Demonstrate that the probability of the event G is equal to 3 10 . A weighted tree may be used. 3 The player does not win. What is the probability that he drew the black ball? E.4161 For each question, three answers are pro-vided ; indicate the only correct answer without justification. An urn contains 10 balls that are indistinguishable to the touch : 7 are white and 3 are black. 1 3 balls are drawn simultaneously from the urn. The prob-ability of drawing 2 white balls and 1 black balls is equal to : 21 40 ; 7 10 × 6 9 × 1 3 ; 7 10 × 7 10 × 1 3 2 From the same urn, we draw a ball, note its color, and put it back in the urn ; we proceed in this manner with 5 successive draws with replacement. The probability of obtaining 3 black balls and 2 white balls is equal to : 3 3 × 7 2 10 5 ; 5 2 × 3 10 2 × 7 10 3 ; 5 2 × 3 10 3 × 7 10 2 3 From the same urn, a single ball is drawn. If it is white, a cubic die is rolled (whose faces are numbered from 1 to 6 ) . If the ball is black, a tetrahedral die is rolled (with sides numbered from 1 to 4 ) . The dice are assumed to be well balanced. The player wins if they roll the number 1 . Knowing that the player has won, the probability that they rolled a white ball is equal to : 7 60 ; 14 23 ; 7 10 × 1 6 1 2 × 1 6 + 1 2 × 1 4 https://chingmath.fr chapExoCorrec/4163 sacados/4163 sacados/3744 Extrait France Juin 2005 sacados/4173 sacados/4184 sacados/4144 sacados/4161
E.4200 A group of friends has a deck of 32 cards. They play cards three times, changing the rules of the game. Consider the following events : A : ˇ the first two cards drawn are as ı ; B : ˇ a jack, a king, a dame ı are drawn in this order ; C : ˇ the last card drawn is a pique ı. 1 The players decide to successively draw three cards from the deck with a discount. Determine the probabilities of the events A , B and C . 2 Now they play by drawing three cards successively and without surrender. Determine the probabilities of the events A , B , C . Consider the following events : D : ˇ two of the cards drawn are as ı ; E : ˇ a jack, a king, a dame ı were drawn ; F : ˇ at least, one of the cards is a pique ı ; G : ˇ only one of the cards is a pique ı. 3 They change the rules of the game and, now, simultane-ously draw three cards. Determine the probabilities of the events D , E , F and G . E.127 An urn contains five blue balls, num- bered from 1 to 5 , four green balls numbered from 1 to 4 and one red ball numbered 1 . As these balls are indistinguishable by touch, in each of the two games, the different eventualities are equiprobable. Note : The probabilities requested will be presented as frac-tions irréductibles Part 1: simultaneous draws Two balls are drawn simultaneously 1 Calculate the number of possible draws. 2 Calculate the probability of obtaining two green balls 3 Calculate the probability of obtaining two balls of the same color 4 Calculate the probability of getting at least one blue ball. Part 2: successive draws One ball is drawn, its number noted, then without returning this first drawn ball to the urn, another ball is drawn and its number also noted. The two numbers thus obtained are used to form a two-digit natural number. The first number drawn is taken as the tens digit and the second as the units digit. 1 Calculate the number of possible draws. 2 Calculate the probability of obtaining the integer 24 . https://chingmath.fr sacados/4200 sacados/127