Math June Review: Probability and Voting Procedures

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1 Math - June Review: Probability and Voting Procedures A big box contains 7 chocolate doughnuts and honey doughnuts. A small box contains doughnuts: some are chocolate doughnuts, and the others are honey doughnuts. If one doughnut from each box is chosen at random, the probability of choosing honey doughnuts is 7. One doughnut from each box is chosen at random. What is the probability of choosing two chocolate doughnuts? A survey involved asking 000 people to name their favourite season. The following table shows the data collected. Women Men Fall Winter 0 0 Spring 0 0 Summer One of these 000 people is chosen at random. iven that we know this person is a man, what is the probability that his favourite season is spring? A) % C) % ) 0% D) %

2 A circular wheel is divided into three different-coloured sectors. In the orange sector, the central angle measures 0. Orange 0 Yellow reen A pointer is attached to the centre of the wheel. When someone spins the pointer, it can stop anywhere on the wheel. The probability that the pointer will stop on the green sector is. What is the probability that the pointer will stop on the yellow sector? Mary is driving from one city to another. The cities are 0 km apart. Her car breaks down somewhere between these two cities. What is the probability that, when her car breaks down, she is located over 00 km from the city she was leaving? A game of chance involves successively drawing marbles from a bag. The first marble drawn must not be put back into the bag. The bag contains marbles. The marbles are blue, red, or green. The probability of drawing a blue marble followed by a red marble is. Two marbles are randomly drawn from the bag. What is the probability that the two marbles are green?

3 A newspaper interviewed 7 girls and 70 boys in Secondary V order to find out whether or not they had rented formal wear for their upcoming graduation dance. The data collected led to the following conclusions: of Secondary V students have rented formal wear for their upcoming graduation dance. 0% of boys in Secondary V have rented formal wear for their upcoming graduation dance. One of the girls interviewed was selected at random. What is the probability that the girl selected has rented formal wear for her upcoming graduation dance? 7 A vase contains four marbles: a red one (R), a white one (W), a black one () and a green one (). The tree diagram below represents all possible combinations that can be obtained when drawing three of these marbles without replacement. R W R W W W R R WR WR W W W RR W R R In this draw, what is the probability of getting a red marble, a green marble, and a white marble? A) C) ) D)

4 Two dice are rolled simultaneously. The dice are the same size and their sides are numbered to. What is the probability that the sum of the numbers on the top side of the dice is greater than eight? A mathematics teacher places glasses on a table. The glasses are identified by the letters A,, C, D and E. A C E D Julie is asked to fill these glasses with water in any order she chooses. Louis is then asked to write down three different combinations of letters corresponding to the order in which the glasses could be filled. He is not aware of the order Julie chose. What is the probability that Louis will choose the same order as Julie? 0 In an archery contest, each participant must shoot the first time while standing 0 m from the target and the second time while standing 0 m from the target. The probability of Maria succeeding from a distance of 0 m is the probability of her succeeding from a distance of 0 m is. What is the probability of her succeeding on both her shots? A) 0 C) 0 ) 7 0 D) 7 0

5 A box contains eight balls, five of which are red and three black. simultaneously from the box. Three balls are picked What is the probability that two of these balls will be red and one black, regardless of order? A) C) ) D) There are girls and 0 boys on a school bus. The driver lets the students off in small groups. After stops, a total of boys and girls have got off the bus. At the sixth stop, students get off, one after the other. What is the probability that the second student at the sixth stop is a boy? A drawing is going to be held among the 0 students who organized the high school graduation dance. Among the student organizers, are girls and are boys. The first three names drawn without replacement will win a trip to La Ronde. What is the probability the group of winners will comprise at least one girl and one boy? A group is composed of boys and girls. Teams of are to be randomly formed. Rounded to the nearest thousandth, what is the probability that the first team thus formed will be made up of girls? A) 0.0 C) 0.0 ) D) 0.

6 On Casino Night, Ray rolled two identical dice whose faces were numbered to. What is the probability that the sum of the numbers that appear on the top face of the dice is greater than or equal to but less than 0? A) C) ) D) The Super 7 drawing is held every Friday night. The winning combination is determined by drawing seven balls and one complementary ball from a barrel. Each ball is numbered from to 7. Once a ball has been drawn, it is not replaced in the barrel. What is the probability that the first two balls drawn will be a followed by a? A) 0 C) 0 ) D) 7 There are balls in a container, red and black. Three balls are drawn simultaneously. Regardless of the order, what is the probability that, among the balls drawn, will be red and will black? A) C) ) D)

7 You are holding a deck of cards. What is the probability that you will draw, without replacement, spades in a row from this deck? (Choose the closest answer.) A) C) 0.0 ) 0.00 D) 0.0 A bag contains 0 marbles, of which are green and the rest are yellow. Two marbles are drawn out of the bag one at a time without putting them back after each draw. Represent this situation by a tree diagram. On each branch, indicate the probability. 0 A bag contains all the letters of the word "CONFIDENT". Walter is asked to pick out letters at random, one at a time. If Walter does not put any of the letters back into the bag, what is the probability that he will pick out the letters of the word "DON" in order? There are students in Jade's music class : girls and boys. The teacher has two concert tickets that he wants to give to two students. The winners' names will be drawn from the bag. No student is permitted to win both tickets. What is the probability that the two names drawn belong to students of the opposite sex? Club Matawani organized a canoe race for the June th provincial holiday. The 0 participating canoes were each given a number from to 0. What is the probability that the first, second and third place winners of the race will be the canoes numbered, 7 and respectively? A) C) ) 0.00 D) 0.00

8 Neil picked out one card at random from a deck of playing cards and did not return the card to the deck. He did the same thing again, twice. What is the probability that the Queen of Hearts will still be in the deck after his third draw? Simon was playing with a regular cube-shaped die. However, four of the faces bore the number, while the other faces bore the number. Simon threw the die three times and recorded his outcome on each throw. What is the probability that the sum of his outcomes was 0? Samantha rolled a fair die whose faces are numbered to. iven that Samantha rolled an even number, what is the probability that she rolled a number greater than? You are about to roll two fair dice. The six faces of each die are numbered to. Consider the sum of the numbers that come up when you roll this pair of dice. What is the probability that a will come up on one of the dice, given that your roll totals 7? 7 In a group of 0 tourists, have a camera. The probability that a person chosen at random from this group will be a woman is. The probability that a person chosen at random from this group will be a man without a camera is. One person from the group is chosen at random. iven that the person chosen is a woman, what is the probability that she will have a camera?

9 A woodchuck makes its way through a square-shaped piece of land. This piece of land is subdivided into isometric squares. Key: Area contaminated with PCs Farmland If the woodchuck is in an area contaminated with PCs, what is the probability that it would be a farmland? The following table provides information of the distribution of the 00 people working in a company. Men Women Total Full-Time Workers Part-Time Workers 0 0 Total 0 00 One of these 00 people is chosen at random to take part in a survey on working conditions. What is the probability that the person chosen will be a man working full time? A) 0 C) 7 ) D) 7

10 0 % % % st lue Yellow Red Choice nd Yellow Red Yellow rd Red lue lue For the table above determine the winner, if any, based on each of the following methods: a) Majority b) Plurality c) orda Count d) Condorcet e) Elimination 7 votes votes votes votes st A C D nd C D D rd C A th D A A C For the table above determine the winner, if any, based on each of the following methods: a) Majority b) Plurality c) orda Count d) Condorcet e) Elimination

11 Answers Number of honey doughnuts in the small box Let y represent the number of honey doughnuts in the small box y 7 y 7 7 y The small box contains honey doughnut. Number of chocolate doughnuts in the small box The small box contains chocolate doughnuts. Probability of choosing two chocolate doughnuts 7 7 Answer: The probability of choosing two chocolate doughnuts is. 7 The probability that the pointer will stop on the yellow sector is 0.. The probability that, when her car breaks down, she is located over 00 km from the city she was leaving is.

12 Probability tree diagram x: number of blue marbles in the bag y: number of red marbles in the bag z: number of green marbles in the bag st marble nd marble lue: x lue: x y Red: z reen: Red: y lue: x y Red: z reen: reen: z lue: x y Red: reen: z Number of blue marbles and number of red marbles in the bag Probability of drawing a blue marble followed by a red marble x y xy xy iven that there are only marbles in the bag, the only two possible factors of are and. The bag therefore contains blue marbles and red marbles. Number of green marbles in the bag Probability of drawing two green marbles Answer: The probability that the two marbles are green is.

13 Distribution of the students interviewed Students who have rented formal wear Students who have not rented formal wear Total irls oys 0 70 Total 0 Answer: The probability that the girl selected has rented formal wear for her upcoming graduation dance is. 7 D a) Number of possible cases : st die 7 7 nd die b) The favourable cases ( those having the sum of numbers greater than eight) : 0 number of favourable cases number of possible cases c) P ( E) ( E) 0 0 P The probability is or or 0..

14 The number of possible cases: 0 The number of favourable cases: ( E) P number of favourable cases number of possible cases ( E) P 0 Result : The probability is or or A D Situation after stops girls : boys : 0 Illustration of the situation using a tree diagram Calculation of the probability that the second student at the sixth stop is a boy % Result : The probability that the second student at the sixth stop is a boy is. %.

15 Probability tree Start Sum of probabilities with at least one girl and one boy or 0. Result : The probability that the group of winners will comprise at least one girl and one boy is or 0.. Note : The student could have found the probability by means of the complement : + 0 o r 0. ) ) C ) 7) ) A

16 This situation is represented by the following tree diagram. 0 reen 7 reen Yellow 7 0 Yellow reen Yellow 0 The probability that he will pick the letters of the word "DON" in order is. Tree diagram of possibilities Sum of probabilities Result : The probability that the two students are of the opposite sex is. 0 D

17 Probability of not choosing the Queen of Hearts... on the st draw : on the nd draw : on the rd draw : 0 0 Calculate the probability 0 or 0. 0 Answer : The probability that the Queen of Hearts is still in the deck of cards is. Tree diagram Sum equal to 0 Probability of a favourable outcome Total probability or 7 Sum equal to 0 Sum equal to 0 or or Answer : The probability of the sum of the results being 0 is. iven that Samantha rolled an even number, the probability that she rolled a number greater than is.

18 The probability that a will come up on one of the dice, given that your roll totals 7 is. 7 Men Women Total With a camera Without a camera Total Conditional Probability Number of women with a camera Number of women 0 0. Answer: iven that the person chosen is a woman, the probability that she will have a camera is 0.. If the woodchuck is in an area contaminated with PCs, the probability that it would be on farmland is. A

19 0 Majority: Winner gets more than half of the votes - no majority in this example Plurality: Winner gets the greatest number of votes - lue in this example orda: Weighted system. Voter lists preference in order. For n candidates st place gets (n-) points, nd gets (n-) points, etc. Last place gets 0 points. lue: * + *0 + *0 points Yellow: * + * + * points Red: *0 + * + * points Yellow wins Condorcet: Compare each pair of candidates individually lue over Yellow (%)/ Yellow over lue (%) Yellow beats lue Yellow over Red (%)/ Red over Yellow (%) Yellow beats Red Red vs lue does not matter since yellow beat both of them Yellow is the winner Elimination: take votes of last place and apply them to the nd choice, repeat the process until someone has a majority of votes. lue: % Yellow: % + % (yellow was nd choice for those who gave Red st) % Yellow wins Majority: none (need 0% + - in this case would need votes to win) Plurality: A is the winner orda: A 7* + *0 + *0 + * * + * + * 0 C 7* + * + * + *0 D 7*0 + * + * + * C is the winner Condorcet: A- (7)/ -A ()/ -A ()/ -A () PREFERRED A-C (7)/ C-A ()/ C-A () / A-C () A PREFERRED A-D (7)/ D-A ()/ D-A ()/ D-A () D PREFERRED C- (7)/ -C ()/ C- ()/ -C () C PREFERRED -D (7)/ -D ()/ D- ()/ D- () PREFERRED C-D (7)/ D-C ()/ C-D ()/ D-C () C PREFERRED No clear winner >A A>C D>A C> >D C>D Elimination: Round : Round : votes from D go to votes from C go to... A 7 C no majority A 7 wins

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