Grade 8 / Probability 22 exercises (100% corrected)

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ROUSSETTES AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ ChingQuizz : 7 exercises available for Quizz assessment : 1. Probability calculation, probability comparison E.9143 Definition: A évènemnet certain is an event that is certain to occur. Its probability is 1 . An impossible event is an event that never comes true. Its probability is 0 . Consider the random experiment of simultaneously throwing two balanced six-sided dice numbered from 1 to 6 and sum-ming their two sides. 1 What is the probability of the event ˇ the number ob-tained is 13 ı? What is this event called? 2 What is the probability of the event ˇ the number ob-tained is less than or equal to 12 ı? What is this event called? E.3768 The red bat is a species of bat, endemic to the territory of New Caledonia. It will be the official mascot of the XIV ième Pacific Games in 2011. In an urn, there are ten balls indistinguishable to the touch bearing the letters of the word ROUSSETTES. A ball is drawn at random from this urn and the letter written on the ball is looked at. 1 What are the six possible outcomes from a draw? 2 Determine the following probabilities : a the letter drawn is a R . b the letter drawn is a S . c the letter drawn is not a S . 3 Julie says she has a better chance of getting a vowel than a consonant from a draw. Is she correct? Justify your answer. E.4331 The four suits of a deck of cards are: Heart ; Tile ; Clover ; Pick Player A draws from a deck of 32 cards (each suit has the cards: 7 , 8 , 9 , 10 , Jack, Queen, King and Ace) . The player B draws from a deck of 52 cards (each suit has the cards: 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , Jack, Queen, King and Ace) . Each player draws a card at random. 1 Calculate the probability each player has of drawing the 7 of Diamonds. 2 Does each player have the same probability of drawing a Heart? Justify. 3 Who has the highest probability of drawing a Queen? Justify. E.5918 The hotel ˇ la ora na ı welcomes 125 tourists : 55 New Caledonians of whom 12 also speak English. 45 Americans speaking only English. The rest being Polynesians of whom 8 also speak English. New Caledonians and Polynesians all speak French 1 If I choose a random tourist from the hotel, what is the probability of the following events : a Event A : ˇ The tourist is a américain ı b Event B : ˇ The tourist is a Polynesian not speaking anglais ı c Event C : ˇ Tourist speaks anglais ı 2 If I approach a tourist in this hotel, am I more likely to be understood by speaking in English or French? Justify your answer. (Any trace of research, even if incomplete, will be consid-ered in the evaluation) . E.5048 In a jar with a red lid, we put 6 strawberry candy and 10 mint candy. In a jar with a blue lid, we put 8 strawberry candy and 14 mint candy. The candies are wrapped in such a way that you can’t tell them apart. Antoine prefers strawberry candies. From which jar is he more likely to choose a strawberry candy? 2. Probability and frequency https://chingmath.fr chapExoCorrec/9143 sacados/9143 chapExoCorrec/3768 sacados/3768 ROUSSETTES chapExoCorrec/4331 sacados/4331 Asie Juin 2011 AsAsAsAsRRRRDDDDVVVV10101010999988887777♥♦♠♣ chapExoCorrec/5918 sacados/5918 chapExoCorrec/5048 sacados/5048
12ABCDEFGHNombredemontée-descenteEntre0et999Entre1000et1999Entre2000et2999Entre3000et3999Entre4000et4999Plusde5000TotalNombredevoletsroulantstombésenpanne20541371868419 E.7623 Definition: The frequency f of a character value is equal to the quotient : f = nombre of appearances of this valeur effectif total The percentage frequency F of a character value is equal to the quotient : f = nombre of appearances of this valeur effectif total × 100 A manufacturer of electric roller shutters is carrying out a statistical study to find out how reliable they are. He there-fore runs a sample of 500 shutters without stopping, until an eventual breakdown. He enters the results in the spreadsheet below : 1 What formula should be entered in the cell H2 of the spreadsheet to obtain the total number of shutters tested? 2 An employee randomly selects a shutter from this sam-ple. What is the probability that this shutter will operate more than 3 000 up down? 3 The manufacturer considers its shutters reliable if more than 95 % of the shutters operate more than 1 000 up and down. Is this batch of roller shutters reliable? Explain your reasoning. E.9142 An opaque bottle contains 20 marbles that can be different colors. Each marble has only one color. Turning the bottle upside down causes only one marble to appear at the neck at a time. The marble cannot come out of the bottle. Ninth graders are trying to determine the colors of the mar-bles in the bottle and their numbers. They turn the bottle upside down 40 times and obtain the following table : Color apparue rouge bleue verte Nombre of appearances de the color 18 8 14 Do these results support the claim that the bottle contains exactly 9 red beads, 4 blue beads, and 7 green beads? E.7636 An opaque bag contains 120 balls all indistinguishable by touch, of which 30 are blue. The remaining balls are red or green. Consider the following random experiment : We draw a ball at random, look at its color, put the ball back in the bag, and mix. 1 What is the probability of drawing a blue ball? Write the result as an irreducible fraction. 2 Cecile performed 20 times this random experiment and got 8 times a green ball. Choose the number of green balls in the bag (no justification is required) from the following answers a 48 b 70 c On can’t know d 25 3 The probability of drawing a red ball is equal to 0.4 . a What is the number of red balls in the bag? b What is the probability of drawing a green ball? 3. Two-entry table E.6302 In a middle school class, after the physical, the following chart was developed : Porte of lunettes Ne doorstep de bezel Fille 3 15 Garçon 7 5 Individual fact sheets fall to the floor and scatter. 1 If the nurse randomly picks up some, what is the proba-bility that this card is : a that of a girl wearing glasses? b The one of a boy? 2 The students who wear glasses in this class represent 12.5 % of those who wear glasses in the entire middle school. How many students in the middle school wear glasses? E.5051 On board a cruise ship pass-ing through Tahiti, there were 4 000 people, none of them children. Each person aboard the boat is : either a tourist, or a crew member. Here is the table that gives the composition of the people on this boat. Men Women Total Touristes 1400 1700 Membres de l’équipage 440 Total 4000 1 Copy and then complete the table above. 2 One person is chosen on the boat, at random. a Can we say that there is more than a 50/50 chance https://chingmath.fr chapExoCorrec/7623 sacados/7623 12ABCDEFGHNombredemontée-descenteEntre0et999Entre1000et1999Entre2000et2999Entre3000et3999Entre4000et4999Plusde5000TotalNombredevoletsroulantstombésenpanne20541371868419 chapExoCorrec/9142 sacados/9142 chapExoCorrec/7636 sacados/7636 chapExoCorrec/6302 sacados/6302 chapExoCorrec/5051 sacados/5051
77527674 BAKLAVA that it is a man? Justify. b What is the probability that this person is one of the tourists? c What is the probability that this person is not a male crew member? E.9134 In the window of a store A are pre-sented a total of 45 models of shoes. Some are con˘ ues for the city, others for sports and come in three different colors: black, white or brown. 1 Complete the table below : Model For the city For the sport Total Noir 5 20 Blanc 7 Marron 3 Total 27 45 2 A random shoe model is chosen from this display case. a What is the probability of choosing a black model? b What is the probability of choosing a model for sports? c What is the probability of choosing a model for the city color brown? 3 In a store window B , there are 54 models of shoes, of which 30 are black. A shoe model is randomly selected from the store’s A window and then from this one in the B store. In which of the two windows is there a greater chance of getting a black model? Justify. 4. Modification of the exits E.5046 For each of the following two questions, there are several proposed answers. Only one of the proposals is correct. No justification is expected. 1 Alice is participating in a game show. She has three closed doors in front of her. Behind one of the doors is a car; behind the others, there is nothing. Alice must choose one of these doors. If she chooses the door behind which there is the car, she wins that car. Alice randomly chooses a door. What is the probability that she wins the car? a 1 2 b 1 3 c 2 3 d We cannot savoir 2 If there are four doors instead of three and still only one car to be won, how does the probability that Alice wins the car change? a augmente b diminue c reste same d We cannot savoir E.7638 In an urn, there are eight balls indistinguishable by touch, each bearing a number: 1 If we randomly draw a ball from this urn, what is the probability that it has the number 7 ? 2 Wacim is about to draw a ball. He says that there is a better chance of drawing an even number than an odd number. Is he right? 3 Finally, Wacim draws the ball with the number 5 and keeps it: he does not put it back in the urn. Baptiste is about to draw a ball from the urn. What is the probability that this ball has the number 7 ? E.7631 Baklava is a traditional pas-try in several countries like Bulgaria or Morocco. It is a time-consuming dessert made of puff pastry, honey, nuts or pistachios or hazelnuts, depending on the region. In a non-transparent bag, there are seven indistinguishable baklavas to the touch bearing the letters of the word BAKLAVA . We randomly draw a cake from this bag and look at the letter written on the cake. 1 What are the outcomes of this experiment? 2 Determine the following probabilities : a The letter drawn is a L . b The letter drawn is not a A . 3 Enzo buys a bag containing 10 baklava all indistinguish-able by touch. This bag contains 2 pistachio-based baklavas, 4 hazelnut-based baklavas, and the remaining baklavas are nut-based. Enzo randomly picks a cake and eats it; it is a nut-based cake. He wishes he could eat another one. His friend Laura says that if he now wants to have a new cake, he is more likely to pick a nut cake. Is she right? Justify the answer. 5. Sum of probabilities https://chingmath.fr chapExoCorrec/9134 sacados/9134 chapExoCorrec/5046 sacados/5046 chapExoCorrec/7638 sacados/7638 77527674 chapExoCorrec/7631 sacados/7631 BAKLAVA
0123456789101112 E.9144 Proposition: for any random experiment, the sum of the probabilities of realization of each of the outcomes is equal to 1 : An urn contains red, blue and yellow balls. The table be-low gives some probabilities of the outcomes of this random experiment occurring : Event ˇ pull a ball rouge ı ˇ pull a ball bleue ı ˇ pull a ball jaune ı Probabilité 0,3 ? 0,5 Determine the probability of the event ˇ the ball drawn from the urn is blue ı. 6. Contrary event E.5044 Definition: Consider a random experiment and A one of its events. We call the opposite event of A the event that occurs when A does not occur. It is noted A . Note: event A contains all outcomes not belonging to event A . Proposition: If an event has probability p then its oppo-site event has probability 1 − p . In a board game, tokens are square-format holders of the same colors, on which a letter of the alphabet is inscribed. The reverse side is not identifiable. There are 100 tokens. The table below gives the number of tokens in the game for each vowel: Letters from jeu A E I O U Y Effectif 9 15 8 6 6 1 We randomly select a letter from this set. 1 What is the probability of getting the letter I ? 2 What is the probability of getting a vowel? 3 What is the probability of getting a consonant? E.3770 For a random draw, 25 balls of the same size, some white, some black, were placed in an urn. The probability of drawing a white ball is 0.32 . Which balls are more numerous in the urn : white or black? Explain. E.6311 An opaque bottle contains 24 balls that are either blue, red, or green. We know that the probability of a green marble appearing by turning the bottle over is equal to 3 8 and the probability of a blue marble appearing is equal to 1 2 . How many red marbles does the bottle contain? 7. Probability and prime integers E.9137 Consider a game consisting of a rotating board and a ball. Shown opposite, this board has 13 squares numbered from 0 to 12 . The ball is thrown onto the tray. The ball eventually stops at random on a numbered square. The ball has the same probability of stopping on each square. 1 What is the probability that the ball will stop on the numbered square 8 ? 2 What is the probability that the number of the square on which the ball stops is an odd number? 3 What is the probability that the number of the square on which the ball stops is a prime number? E.9133 Consider two randomized ex-periments : experiment n o 1: randomly select an integer between 1 and 11 ( 1 and 11 inclusive) experiment n o 2: throw a six-sided balanced die num-bered from 1 to 6 and announce the number that appears on the top face. Assertion: it is more likely to choose a prime number in the experiment n o 1 than to obtain an even number in the experiment n o 2 . https://chingmath.fr chapExoCorrec/9144 sacados/9144 chapExoCorrec/5044 sacados/5044 Brevet Asie Juin 2012 chapExoCorrec/3770 sacados/3770 chapExoCorrec/6311 sacados/6311 chapExoCorrec/9137 sacados/9137 0123456789101112 chapExoCorrec/9133 sacados/9133
E.9139 A bag contains 20 balls each with the same probability of being drawn. These 20 balls are numbered from 1 to 20 . One ball is drawn at random from the bag. All results will be given as irreducible fractions. 1 What is the probability of drawing the numbered ball 13 ? 2 What is the probability of drawing an even numbered ball? 3 Is it more likely to get a ball with a number that is a multiple of 4 than to get a ball with a number that is a factor of 4 ? 4 What is the probability of drawing a ball with a number that is a prime? E.9140 There are in an urn 12 balls indistinguishable by touch, numbered from 1 to 12 . We want to draw a ball at random. 1 Is it more likely to get an even number or else a multiple of 3 ? 2 What is the probability of getting a number less than 20 ? 3 We remove from the urn all balls whose number is a fac-tor of 6 . We again want to draw a ball at random. Explain why the probability of getting a number that is a prime number is then 0.375 . https://chingmath.fr chapExoCorrec/9139 sacados/9139 chapExoCorrec/9140 sacados/9140