Conditional Probability Grades 7/8 Conditional Probability Find conditional probabilities; use tree diagrams If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

Lesson Plan Lesson Overview Progression of Learning Objective(s) Understand conditional probability; use two way tables, Venn Diagrams and tree diagrams Grade 7/8 Prior Knowledge Basic probability, AND and OR rules Duration 80 minutes (variable). Resources Slides 24 onwards are printable versions of some of the earlier slides. Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Introduction; collecting bivariate data, using two way tables and Venn Diagrams This activity may be useful to students needing to be reminded of two way tables and Venn Diagrams, or by those for whom this is a confidence-building entrée into the more challenging probability questions to follow. Students need to understand the differences between the proportion of items of data out of the whole population and out of a part of the population PRINT SLIDE 24 Three practice questions follow PRINT SLIDE 25 20 Reasoning Reasoning activity, links to common multiples; one set of conditional probabilities match the “non-conditional” set of probabilities, dependent on the link between the total number of elements and the multiples in use PRINT SLIDE 26 Examination style question to conclude the two way tables/Venn Diagram part of the lesson PRINT SLIDE 27 15 Tree diagrams Teacher led input identifies differences between probabilities in “with” and “without” replacement situations. Slides 10 to 12 are animated examples; slide 10 is, effectively, “with replacement”, slide 11 uses bags of individual items to clarify the “without replacement” situation; students have differentiated practice, and examination style practice. PRINT SLIDES 28, 29 AND 30 25 Understanding conditional probability in exam questions (from specimen papers) PRINT SLIDE 31 and 31. This includes 8 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated using slides 16 to 23. Next Steps Assessment PLC/Reformed Specification/Target7/Probability/Conditional Probability

Key Vocabulary Outcomes Two way table Venn Diagram Tree diagram

Conditional Probability You are a market researcher. You want to find out whether people will buy your new product. You think that the results will be different depending on gender, so you want to collect the results for men and women separately; however, you need to have a single display for all the results. Design a data collection sheet for this.

Conditional Probability Two way table Venn Diagram Men Women Total Will buy 18 27 45 Won’t buy 32 23 55 50 100 Men Will buy 32 18 27 23 Here is the same information presented in two different ways. What are the advantages and disadvantages of each?

Conditional Probability Now choose either the two way table, or the Venn Diagram, from which to find the following probabilities: …that someone chosen at random is a woman. …that someone chosen at random will buy the product. …that someone chosen at random is a woman who will buy the product. …that a woman chosen at random will buy the product. …that someone chosen at random, who says they will buy the product, is a woman. Questions 3, 4 and 5 are all about women who will buy the product. In what way are your answers to 3, 4 and 5 alike? In what way are they different? Have you used the two-way table or the Venn Diagram to find these probabilities? Whichever one you chose; was it the better choice? Could you have used either?

Click on a coloured box to show the number. Conditional Probability This table summarises the favourite crisp flavours of some girls and boys The Venn Diagram summarises whether members of a group of men and women voted at the last general election. Plain Salt & vinegar Cheese & onion Total Girls 12 42 Boys 23 48 45 120 Men Voted 22 8 18 % 24% 48% 36 19 78 10% 27 Find the following probabilities: …that a person chosen at random is a boy. …that a person chosen at random prefers cheese & onion crisps. …that a person chosen at random is a girl who prefers plain crisps. …that a boy chosen at random prefers salt and vinegar crisps. …that someone who prefers salt and vinegar crisps chosen at random is a boy. Find the following probabilities: …that a person chosen at random is a man. …that a person chosen at random voted. …that a person chosen at random is a man who voted. …that a woman chosen at random voted. …that someone who voted chosen at random is a woman. There are 100 adults in a hotel. 40 are men. 22 of the women are foreign. 53 of the adults at the hotel are British. What is the probability that a man, chosen at random, is foreign? Click on a coloured box to show the number.

Conditional Probability Reasoning Ɛ = Integers from 1 to 17 inclusive 18 Complete the Venn Diagram by placing all the members of the universal set Ɛ into the correct places. (The first four have been done for you). For the integers in Ɛ, find the probability that … … a randomly chosen member of Ɛ is a member of A. … a randomly chosen member of B is also a member of A. … a randomly chosen member of Ɛ is a member of B. … a randomly chosen member of A is also a member of B. A = Multiples of 2 B = Multiples of 3 10 2 18 1 14 6 3 4 9 12 5 15 8 17 9 18 8 16 7 2 5 3 6 11 13 17 5 17 6 18 2 8 3 9 How do the conditional probabilities differ between those for 1 to 17 and 1 to 18? Why is this?

Conditional Probability Examination style question 5 13 8 4 8 21

Click on a box to see the probability Conditional Probability Pete wins a quarter of the games of chess that he plays. He plays two games of chess. Draw a tree diagram to show the possible outcomes for Pete’s two games of chess. Work out the probability that he wins both games; Work out the probability that he wins at least one game. W 1 4 L 3 4 W 1 4 L 3 4 W 1 4 L 3 4 Click on a box to see the probability

Conditional Probability There are four red and three blue balls in a bag. Two are taken out; the first is not put back into the bag before the second one is taken. Show this on a tree diagram. R 3 6 R B R R B 4 7 R B B 3 6 R R B R B R R B 4 6 3 7 B R R B B R B 2 6 R Click on a bag to see what is in it Click on a box to see the probability

Click on a box to see the probability Conditional Probability A packet of dog biscuits contains 4 yellow and 6 pink biscuits. My dog takes two biscuits at random from the packet; he eats the first before taking the second. Find the probability that he takes two the same colour. Y 3 9 P Y 4 10 6 9 12 90 30 + 42 90 = P Y 6 10 4 9 P 5 9 Click on a box to see the probability

Conditional Probability A packet of dog biscuits contains 3 yellow, 2 brown and 5 pink biscuits. My dog takes two biscuits at random from the packet; he eats the first before taking the second. Find the probability that he takes two the same colour. 2 9 Y B 2 9 P 5 9 3 10 Y 6 90 2 20 + 28 90 = 3 9 Y B 2 10 B 1 9 P 5 9 P 5 10 3 9 Y B 2 9 P 4 9

Click on “Silver” or “Gold” to see answers for that section Conditional Probability SILVER GOLD There are five plain and seven milk chocolates in the box. Jen eats two of these chocolates at random. Complete the tree diagram to show the possible outcomes. What is the probability that she takes one milk and one plain chocolate? I take two counters, at random, from a box containing six red and four blue counters. Find the probability that they are the same colour… (a) if I put the first one back before I take the second. (b) if I keep the first one before I take the second. There are eight red and six blue socks in my sock drawer. I take two without looking. What is the probability that I take a pair of “odd” socks? In the draw for the quarter final of a cup competition, there are eight teams, including County and Town, which will be drawn to play four games. What is the probability that County and Town play each other in the quarter final? There are twenty tickets in a bag. Three are “winning” tickets. A competitor takes three tickets from the bag without replacement. To win a prize, a competitor must draw at least two winning tickets. What is the probability of winning a prize? 52 100 42 90 4 11 Plain Plain Milk 5 12 96 182 Milk 7 11 1 7 5 11 Plain 7 12 Milk 6 11 312 6840 70 132 Click on “Silver” or “Gold” to see answers for that section

Conditional Probability Examination style question 14 6 20 5 19 30 380 × = = 0.0789 No 6 20 5 19 6 20 6 20 × ≠ ×

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Conditional Probability You are a market researcher. You want to find out whether people will buy your new product. You think that the results will be different depending on gender, so you want to collect the results for men and women separately; however, you need to have a single display for all the results. Here is the same information presented in two different ways. What are the advantages and disadvantages of each? Now choose either the two way table, or the Venn Diagram, from which to find the following probabilities: …that someone chosen at random is a woman. …that someone chosen at random will buy the product. …that someone chosen at random is a woman who will buy the product. …that a woman chosen at random will buy the product. …that someone chosen at random, who says they will buy the product, is a woman. Questions 3, 4 and 5 are all about women who will buy the product. In what way are your answers to 3, 4 and 5 alike? In what way are they different? Have you used the two-way table or the Venn Diagram to find these probabilities? Whichever one you chose; was it the better choice? Could you have used either? Two way table Men Women Total Will buy 18 27 45 Won’t buy 32 23 55 50 100 Venn Diagram Men Will buy 32 18 27 23

Conditional Probability This table summarises the favourite crisp flavours of some girls and boys The Venn Diagram summarises whether members of a group of men and women voted at the last general election. Plain Salt & vinegar Cheese & onion Total Girls 12 42 Boys 23 48 45 120 Men Voted 24% 48% 10% Find the following probabilities: …that a person chosen at random is a boy. …that a person chosen at random prefers cheese & onion crisps. …that a person chosen at random is a girl who prefers plain crisps. …that a boy chosen at random prefers salt and vinegar crisps. …that someone who prefers salt and vinegar crisps chosen at random is a boy. Find the following probabilities: …that a person chosen at random is a man. …that a person chosen at random voted. …that a person chosen at random is a man who voted. …that a woman chosen at random voted. …that someone who voted chosen at random is a woman. There are 100 adults in a hotel. 40 are men. 22 of the women are foreign. 53 of the adults at the hotel are British. What is the probability that a man, chosen at random, is foreign?

Conditional Probability Reasoning Ɛ = Integers from 1 to 17 inclusive Complete the Venn Diagram by placing all the members of the universal set Ɛ into the correct places. (The first four have been done for you). For the integers in Ɛ, find the probability that … … a randomly chosen member of Ɛ is a member of A. … a randomly chosen member of B is also a member of A. … a randomly chosen member of Ɛ is a member of B. … a randomly chosen member of A is also a member of B. A = Multiples of 2 B = Multiples of 3 2 1 3 4 7 12 Plain Milk How do the conditional probabilities differ between those for 1 to 17 and 1 to 18? Why is this?

Conditional Probability Examination style question

Conditional Probability A packet of dog biscuits contains 3 yellow, 2 brown and 5 pink biscuits. My dog takes two biscuits at random from the packet; he eats the first before taking the second. Find the probability that he takes two the same colour.

Conditional Probability SILVER GOLD There are five plain and seven milk chocolates in the box. Jen eats two of these chocolates at random. Complete the tree diagram to show the possible outcomes. What is the probability that she takes one milk and one plain chocolate? I take two counters, at random, from a box containing six red and four blue counters. Find the probability that they are the same colour… (a) if I put the first one back before I take the second. (b) if I keep the first one before I take the second. There are eight red and six blue socks in my sock drawer. I take two without looking. What is the probability that I take a pair of “odd” socks? In the draw for the quarter final of a cup competition, there are eight teams, including County and Town, which will be drawn to play four games. What is the probability that County and Town play each other in the quarter final? There are twenty tickets in a bag. Three are “winning” tickets. A competitor takes three tickets from the bag without replacement. To win a prize, a competitor must draw at least two winning tickets. What is the probability of winning a prize? Plain Milk 7 12

Conditional Probability Examination style question

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