Chance and Probability

Transcription

1 Student Teacher Chance and Probability My name Series G

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3 Series G Chance and Probability Contents Topic Section Chance Answers and (pp. probability 0) (pp. 0) probability chance and scale probability / / using samples to predict probability / / tree diagrams Section Assessment with answers (pp. 6) chance experiments probability scale_ using tables using samples to predict probability location, location apply using tables to look at sample space extension lucky throw solve Date completed / / / / / / / / / / Section Outcomes (p. 7) Series Authors: Rachel Flenley Nicola Herringer Copyright

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5 Chance and probability probability scale Probability measures how likely something is to happen. impossible even certain unlikely likely Probability measures how likely something is to happen. Events that are certain to happen are given a probability of. Events that will never happen are given a probability of 0. Events that could happen are rated between 0 and. Event When you flip a coin, it will land on heads. You will grow wings and fly today. A spinner with 0 even segments with the numbers to 0 will land on. people are lined up and every second person in the line has gloves on. What is the chance that one person is not wearing gloves? You have 0 cards. have hearts, have stripes and the rest are blank. What is the chance you will choose a blank card? Probability as a fraction 0 0 or or or or Probability as a decimal What is the probability of spinning a striped segment on each of these wheels? Write your answer as a rating between 0 and using decimals. a b c d Reuben is going to put ten blocks in a bag and ask a friend to choose one without looking. Circle the blocks he could put in the bag to make the probability of choosing a cube 0. Sample answer: Answers will vary. cubes and a total of 8 cones and cylinders. Chance and Probability G SERIES TOPIC

6 Chance and probability probability scale 00 guests each buy a ticket for a raffle at a fundraising dinner. The winning ticket will be selected at random. This table on the right shows the colours of all of the tickets in the raffle. Red 0 Purple 0 Orange 0 Total 00 What is the probability of the winning ticket being red, purple or orange? Draw arrows on this probability scale to show the probability of each colour and write the colour beneath the arrow red purple orange Inside a box there are rectangles, triangles and squares. Without looking, Ellie chooses one shape from the box. a Draw each shape on this probability scale to show the probability of Ellie choosing each type of shape b more rectangles, more triangles and more squares are added to the same box. Draw each shape on this probability scale to show the probability of Ellie choosing each shape from the box c What do you notice? Probability does change. 6 Sam did an experiment with 0 cubes that were either red, white or blue. She took a cube from a jar without looking, tallied which colour it was then put it back in the same jar. She repeated the process 0 times. After tallying her results, she created this pie chart to show the results of the experiment. a How many times did Sam take each colour out of the jar? Remember she performed the experiment 0 times. Red White Blue White Blue Red b Draw the combination of cubes there could have been inside the jar. Remember there are only 0 cubes. red and a total of white and blue. W R W B R R R R W B Sample answer G SERIES TOPIC Chance and Probability

7 Chance and probability using samples to predict probability Surveys are used to collect data about certain topics or questions. Once the data is collected, it is presented in a table so it is easy to understand. Surveys can be conducted to ask all kinds of questions. We can use probability to see an even bigger picture than the survey tells us. This table shows the data collected when 0 people were surveyed to find their favourite milkshake flavour. Chocolate Strawberry Vanilla Banana We can use probability to predict the number of people who will choose each flavour in a larger survey. When 00 people are surveyed, it is likely that chocolate will be the favourite milkshake flavour of 8 people. When 000 people are surveyed, it is likely that chocolate will be the favourite milkshake flavour of 80 people. Faisal has had enough of selling clothes. If one more woman asks him, Do I look fat in this?, he will scream. He holds a crazy closing down sale and sells the following items in hour: Shirts Jackets Skirts Dresses 8 7 Predict how many: a jackets would sell in hours c shirts would sell in hours 8 b skirts would sell in hours d dresses would sell in hours e shirts and jackets would sell in hours 8 f items of clothing would sell in 8 hours 6 Here is a table showing the results from a survey of 0 boys and 0 girls who were asked, Which fruit do you like best? Rate the probability that a person selected randomly will be: a a boy b a girl who likes apples or Girls Boys Apple 7 Banana 8 Orange 6 c someone who likes pears 00 Pear 9 d Is the probability of someone choosing a banana greater than or less than? Less Chance and Probability G SERIES TOPIC

8 Chance and probability tree diagrams Tree diagrams are used to display all possible outcomes in a simple chance experiment. Here is an example: Matilda s father is making her lunch and has given sandwich her the following choice: white or brown bread, lettuce or sprouts, tuna or egg. We can then follow each branch along to see the different options. white bread brown bread lettuce sprouts lettuce sprouts tuna egg tuna egg tuna egg tuna egg By using a tree diagram, we can see that Matilda has 8 different options for her sandwich. When customers buy a new car from Joe s Motors they can pay an additional cost for each of these optional extras: Alloy wheels instead of standard wheels Metallic paint instead of standard paint Automatic transmission instead of Leather seats instead of standard seats manual transmission a Complete this tree diagram to work out all the possible combinations that customers can choose: Automatic transmission alloy wheels standard wheels metallic paint standard paint metallic paint standard paint leather seats standard seats leather seats standard seats leather seats standard seats leather seats standard seats Manual transmission alloy wheels standard wheels metallic paint standard paint metallic paint standard paint leather seats standard seats leather seats standard seats leather seats standard seats leather seats standard seats b How many possible combinations are there? 6 G SERIES TOPIC Chance and Probability

9 Chance and probability tree diagrams You have an after school job at the local ice-cream shop. Your boss has asked you to run a special on the strawberry and banana ice-cream flavours as she mistakenly ordered far too much of each. You decide to offer a double scoop special customers can choose scoops and a topping for the price of a single scoop cone. As all ice-cream connoisseurs know, it matters which flavour goes on top so customers may choose a strawberry-banana combo, a banana-strawberry combo or scoops of the same flavour. Work out the different combinations customers could order if they could choose from cone types, the flavours and different toppings. Decide which cones and toppings you will offer. strawberry strawberry banana plain chocolate plain chocolate nuts sprinkles nuts sprinkles nuts sprinkles nuts sprinkles banana strawberry banana plain chocolate plain chocolate nuts sprinkles nuts sprinkles nuts sprinkles nuts sprinkles Think about this: a How many different combinations are there in total? 6 b If a customer hates banana ice-cream flavour, how many options do they have? c What would be your pick? Answers will vary. Chance and Probability G SERIES TOPIC

10 Chance and probability chance experiments Complete the tree diagram to show all the possible outcomes when you spin Spinner and then Spinner. The first one is done for you. blue blue red yellow Spinner yellow Spinner red What is the probability of landing on: a a yellow or b blue and There were possible outcomes in Question. 60 is, so I would expect each number to be times greater. c a or d yellow and If you did this 60 times, how many times would you expect to get: a blue and b a red 0 c a d a 6 G SERIES TOPIC Chance and Probability

11 Chance and probability using tables When we work out all the possible outcomes of an event that could happen, we are finding out the theoretical probability. When we do the experiment and look at the probability of what actually happened, we call it experimental probability. Theoretical probability is: number of favourable outcomes total number of possible outcomes Experimental probability is: number of times the event occurred total number of trials When we roll dice together, we can get a number of totals. Fill in this table to show the possible outcomes when regular dice are rolled and added together: Die Die a How many different ways can the dice be rolled? 6 b Which total occurred the most often? Shade this in the grid. c Which totals occurred the least often? Circle these on the grid. Graph the outcomes from the table above in the grid below. Express the theoretical probability of the following as a fraction: Number of outcomes a 7 = c = = 6 b 9 = d 0 = 6 6 = 9 = Possible totals Now try this experiment. You will work with a partner and roll dice 6 times. First make your predictions as to how often you will roll each answer. Write this in the first row. This is the theoretical probability. Now actually roll two die 6 times. In the bottom row, tally the number of times each total appears. This is the experimental probability. Total Number of times you expect to see each total Number of times you actually get each total 6 Look at the difference between the two rows. Is this what you expected? Answers will vary. Chance and Probability G SERIES TOPIC 7

12 Chance and probability using tables Now we are going to investigate the sample space of when the dice are different to regular dice. For this you will need regular dice and some white stickers to stick over the sides of the dice. Cover dice with white stickers so that the sides are covered on each die. Colour of the faces yellow and colour faces red: + Y Y Die Y Y R R a Complete the table to show the sample space. b What are the chances of rolling yellows? Colour the table to show this. Y YY YY YY YY YR YR 6 Y YY YY YY YY YR YR 6 = 9 Y YY YY YY YY YR YR c What are the chances of rolling yellow and red? 6 Y YY YY YY YY YR YR 6 = 9 R RY RY RY RY RR RR d What are the chances of rolling reds? R RY RY RY RY RR RR 6 = 9 Die 6 Look at the next table for the sample space of a set of dice. Die Die + Y Y G G Y YY YY YG YG Y Y Y YY YY YG YG Y Y G GY GY GG GG G Y G GY GY GG GG Y Y Y Y Y Y G G G G a Complete the rest of the table to show the sample space. b Show what one die looks like on this net of a cube. Y Y G G c What is the chance of rolling: yellows? 6 = 9 dots? 6 = 9 7 Make up your own crazy set of dice. Show the sample in the space on the left and show what they look like on the two nets of cubes on the right. + Die Die Answers will vary. Die Die 8 G SERIES TOPIC Chance and Probability

13 Location, location apply Getting ready Play this game with a friend. You will need one copy of this game board, a counter each and two dice. What to do Place your counter in the start hexagon. Take turns rolling both dice and adding the numbers. If your answer is a, or move one space towards the striped hexagons. If your answer is a, 6, 7, 8 or 9 move one space towards the dotted hexagons. If your answer is a 0, or move one space towards the checked hexagons. When your counter gets to a hexagon on the edge, record your and start again. Play games. Who is the grand winner? START What to do next Why are the allocated as they are? Why does it matter what your dice roll is? Explain your reasoning to a friend. You score more with numbers that are less likely to come up. Look back at page 7 to see the table of dice. Chance and Probability G SERIES TOPIC 9

14 Lucky throw solve Getting ready This is a version of a very old game, played by children all over the world. You will need 0 counters, playing pieces (you could use erasers or chess pieces) pop sticks and a partner. What to do Decorate side only of each of the pop sticks. Arrange the counters in a circle like this: START START Place your playing pieces on opposite sides of the circle and mark your starting point. The aim of the game is to be the first person to move around the circle and get back to your starting point. Take turns throwing the pop sticks up and looking at the result. The number of counters you can move depends on your combination of decorated and undecorated pop sticks: decorated sides = move 0 counters plain sides = move counters decorated sides and plain side = move counters decorated side and plain sides = move counter If the other player lands on you, you must return to your starting point. The first person back to the Start wins. What to do next After you finish the game, make a tree diagram of all the possible throw outcomes. Use the diagram to answer the following questions: What is the likelihood of throwing decorated sides? 8 What is the likelihood of throwing plain sides? 8 What is the likelihood of throwing decorated and plain sides? What is the likelihood of throwing decorated and plain sides? Based on this, do you think the scoring system is fair? How would you change the scoring system to make it fairer? Play the game again with your new scoring system. Does this improve the game? Or do you prefer the original game? Why? G Chance and Probability SERIES TOPIC

15 Probability scale Name Draw an arrow to connect each numerical expression of chance to the correct place on the probability line impossible even chance (0%) certain Match each statement to the probability of it occurring: I will pick out a pink pencil from a pencil case filled with 0 different coloured pencils people are in a queue, people have socks on, the rest do not. What is the chance that one person pulled out at random will not have socks on? 8 A spinner with 0 segments labelled to 0 lands on an even number. 0. coins are tossed and land on HHH. 0 Dan has the following cubes: 6 red cubes 6 blue cubes 6 yellow cubes He is going to put 0 cubes in a bag and then ask a friend to choose one cube without looking. Colour the cubes to show which cubes Dan will put in this bag to make it the probability of choosing a yellow 0.. Skills Not yet Kind of Got it Orders fractions or decimals along a probability line between 0 and Identifies possible outcomes for familiar events Series G Topic Assessment

16 Probability scale Name Draw an arrow to connect each numerical expression of chance to the correct place on the probability line impossible even chance (0%) certain Match each statement to the probability of it occurring: I will pick out a pink pencil from a pencil case filled with 0 different coloured pencils people are in a queue, people have socks on, the rest do not. What is the chance that one person pulled out at random will not have socks on? 8 A spinner with 0 segments labelled to 0 lands on an even number. 0. coins are tossed and land on HHH. 0 Dan has the following cubes: 6 red cubes 6 blue cubes 6 yellow cubes He is going to put 0 cubes in a bag and then ask a friend to choose one cube without looking. Colour the cubes to show which cubes Dan will put in this bag to make it the probability of choosing a yellow 0.. Answers will vary there must be yellow cubes. R B Y R B Y R B Y Y Skills Not yet Kind of Got it Orders fractions or decimals along a probability line between 0 and Identifies possible outcomes for familiar events Series G Topic Assessment

17 Using samples to predict probability Name 0 people are asked what their favourite ice-cream flavour is. This table shows the results: strawberry 0 mint caramel 6 Total 0 a Based on the table above, what do you think the results would be if 00 people were also surveyed on their favourite ice-cream flavour? Show what the results are likely to look like by completing this table. strawberry mint caramel Total 00 b For each flavour, rate the probability that a person randomly selected will choose that flavour, by labelling the probability line. The first flavour is done for you. strawberry Here is another table showing a group of 00 girls and boys favourite colours. Rate the probability that a person randomly selected will be: a a girl who likes red b a person who likes yellow c a boy who likes green Girls Boys Blue Red 8 Yellow 9 0 Green 7 9 d Colour the box below that makes this statement correct: The probability of a person who likes blue is 0.? greater than equal to less than Skills Not yet Kind of Got it Uses samples to make predictions about a larger population Relates chance fractions and decimals to an everyday situation Describes the likelihood of events as being more than or less than 0. Series G Topic Assessment

18 Using samples to predict probability Name 0 people are asked what their favourite ice-cream flavour is. This table shows the results: strawberry 0 mint caramel 6 Total 0 a Based on the table above, what do you think the results would be if 00 people were also surveyed on their favourite ice-cream flavour? Show what the results are likely to look like by completing this table. strawberry 0 mint 0 caramel 0 Total 00 b For each flavour, rate the probability that a person randomly selected will choose that flavour, by labelling the probability line. The first flavour is done for you. mint caramel strawberry Here is another table showing a group of 00 girls and boys favourite colours. Rate the probability that a person randomly selected will be: a a girl who likes red b a person who likes yellow c a boy who likes green or 0 Girls Boys Blue Red 8 Yellow 9 0 Green 7 9 d Colour the box below that makes this statement correct: The probability of a person who likes blue is 0.? greater than equal to less than Skills Not yet Kind of Got it Uses samples to make predictions about a larger population Relates chance fractions and decimals to an everyday situation Describes the likelihood of events as being more than or less than 0. Series G Topic Assessment

19 Using tables to look at sample space extension Name Here is a sample space of identical dice. Each side of the die is blue or pink or just a dot. (B stands for blue and P stands for pink). Complete the table that shows the sample space and show what each die looks like on this net of a cube: Die Die + B B P P B BB BB BP BP B B B BB BB BP BP B B P PB PB PP PP P P P PB PB PP PP P P B B P P B B P P The table below shows the sample space or all the possible combinations for turning over of these playing cards. 8 p 7 a Circle the type of pair of cards which is least likely to be turned over: number/shape shape/shape 0 Possible pair combinations shape/number number/number p p b Describe a game that could be played based on the fact that this type of pair of cards are the least likely to be turned over. p p 8 p p p p Skills Not yet Kind of Got it Analyses sample space of a die Explains a simple game based on theoretical probability Series G Topic Assessment

20 Using tables to look at sample space extension Name Here is a sample space of identical dice. Each side of the die is blue or pink or just a dot. (B stands for blue and P stands for pink). Complete the table that shows the sample space and show what the dice look like on this net of a cube: Die Die + B B P P B BB BB BP BP B B B BB BB BP BP B B P PB PB PP PP P P P PB PB PP PP P P B B P P B B P P B B P P The table below shows the sample space or all the possible combinations for turning over of these playing cards. 8 p 7 a Circle the type of pair of cards which is least likely to be turned over: number/shape shape/shape 0 Possible pair combinations shape/number number/number p p b Describe a game that could be played based on the fact that this type of pair of cards are the least likely to be turned over. Answers will vary. p p 8 p p p p Skills Not yet Kind of Got it Analyses sample space of a die Explains a simple game based on theoretical probability 6 Series G Topic Assessment

21 Series G Chance and Probability Curriculum ACMMG Outcomes Describe probabilities using fractions, decimals and percentages National Curriculum NSW AusVELS ACMMG ACMMG6 MA-9SP MA-WM MA-WM ACMMG ACMMG ACMMG6 Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies Compare observed frequencies across experiments with expected frequencies Conducts chance experiments and assigns probabilities as values between 0 and to describe their outcomes Describes and represents mathematical situations in a variety of ways using mathematical terminology and some conventions Gives a valid reason for supporting one possible solution over another Describe probabilities using fractions, decimals and percentages Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies Compare observed frequencies across experiments with expected frequencies Series G Outcomes 7