--- title: "Grids and tree diagrams" subtitle: "Data and chance · Chance · Lesson 12" --- # Grids and tree diagrams **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A game asks you to toss a coin and roll a dice: you win with heads and an even number. Is that a good chance or a poor one? First you need every way the two can land. :::teacher-note **In one sentence:** Students list the outcomes of two events in a grid and a tree diagram, count the outcomes in an event and find its probability (AC9M8P02, node 1). **Students should be able to say:** I can list every outcome of two events with a grid or a tree diagram. I can count the outcomes in an event and give its probability. I can choose between a grid, a tree and a Venn diagram. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **You toss a coin and roll a dice. How many different outcomes are there?** ☐ 6 ☐ 8 ☐ 12 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 12. Each of the 2 coin results goes with each of the 6 dice results: 2 × 6 = 12 pairs. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## List every pair :::keep-together ### Count the pairs Here is the routine. 1. $\text{Spinner with equal red and blue parts};\ \text{then a coin}$ — Two events, one after the other. 2. $\text{red} \to \text{H, T};\ \text{blue} \to \text{H, T}$ — Split each branch into every outcome of the second. 3. $2 \times 2 = 4 \text{ outcomes}$ — Count the leaves of the tree. 4. $P(\text{blue, H}) = \tfrac{1}{4}$ — One leaf out of four. **A spinner has 3 colours and a dice has 6 faces. How many outcomes when you spin, then roll?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 18 3 × 6 = 18 pairs. ::: :::keep-together ### Find the mistake **Max finds the chance of one head and one tail with two coins. Click the line where he goes wrong.** ``` 1 Max tosses two coins and lists what can happen. 2 The equally likely outcomes are: HH, TT, and one of each. 3 That is 3 outcomes, and one of them is "one of each". 4 So P(one head and one tail) = 1/3. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: one of each can happen two ways, HT and TH. The four equally likely outcomes are HH, HT, TH, TT, so P(one of each) = 2/4 = 1/2. ::: :::keep-together ### Three coins **You toss three coins. Which is the best way to list the outcomes?** ☐ A tree diagram ☐ A list of the totals ☐ A Venn diagram ☐ A grid ::: :::teacher-note **Answer:** A tree diagram A tree can keep branching: a column of branches for each event, 2 × 2 × 2 = 8 outcomes. ::: :::typical ## Other pairs **Spinner and coin.** **Sum of 8.** **The same letter.** **Grid or tree.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Two events :::keep-together ### Sum of 7 **Two dice are rolled. What is P(the sum is 7)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/6 6 of the 36 pairs make 7: 6/36 = 1/6. ::: :::keep-together ### Two heads **Two coins are tossed. What is P(two heads)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/4 HH is 1 of the 4 outcomes HH, HT, TH, TT. ::: :::keep-together ### Meals **A café offers 4 sandwiches and 3 drinks. How many different sandwich-and-drink meals are there?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 Each sandwich goes with each drink: 4 × 3 = 12. ::: :::keep-together ### Review: neither **Review: 30 students, 18 play netball, 10 play hockey, 4 play both. How many play neither?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 Netball only 14, both 4, hockey only 6: 24. So 30 − 24 = 6. ::: ## Make: a fair game Write the rule of your game in words, then check a friend's game with the grid. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A spinner shows 1, 2 or 3 (equal parts) and a coin is tossed. List the outcomes. What is P(an odd number and heads)?** {{lines-3}} ::: :::teacher-note **Answer:** 1H, 1T, 2H, 2T, 3H, 3T: 6 outcomes. 1H and 3H, so 2/6 = 1/3. :::