--- title: "Sample spaces and probability" subtitle: "Data and chance · Chance · Lesson 6" --- # Sample spaces and probability **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A spinner is half red. Red, blue or green: three colours, so is each one a third? Spin it a thousand times and see. :::teacher-note **In one sentence:** Students list sample spaces for coins, dice, bags and spinners, assign probabilities as favourable ÷ total when outcomes are equally likely and from sizes or counts when they are not, check that probabilities add to 1, and test a prediction with 1000 spins (AC9M7P01, nodes 1 and 2). **Students should be able to say:** I can list the sample space for a single event. I can work out a probability as a fraction, decimal or percentage. I can tell when outcomes are not equally likely, and use sizes or counts instead. **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 Three colours, but are they the same size? :::keep-together **A spinner is half red, a quarter blue and a quarter green. What is the probability of blue?** ☐ $\tfrac{1}{2}$ ☐ $\tfrac{1}{3}$ ☐ $\tfrac{1}{4}$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 1/4. There are three colours, but they are not equally likely: blue is a quarter of the spinner, so P(blue) = 1/4. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## List, then count :::keep-together ### List, then count Here is how to find a probability. 1. $\{1, 2, 3, 4, 5, 6\}$ — List the sample space: every possible outcome of rolling a dice. 2. $\text{equally likely?} \;\checkmark$ — A fair dice: each face has the same chance. 3. $\text{even} = \{2, 4, 6\}$ — The favourable outcomes for the event. 4. $P(\text{even}) = \tfrac{3}{6} = \tfrac{1}{2} = 0.5 = 50\%$ — Favourable ÷ total, in any form. 5. $P(\text{red}) = \tfrac{1}{2},\ P(\text{blue}) = P(\text{green}) = \tfrac{1}{4}$ — Not equally likely: use the sizes. Check: they add to 1. Now work out P(even) on a dice. **Roll a fair dice. What is the probability of an even number? Give a fraction in simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/2 Sample space {1, 2, 3, 4, 5, 6}; even: 2, 4, 6. P = 3/6 = 1/2. ::: :::keep-together ### Find the mistake **Kai works out probabilities for the half-red spinner. Click the line where his reasoning goes wrong.** ``` 1 The spinner has three colours: red, blue and green. 2 So the sample space is {red, blue, green}. 3 There are 3 outcomes, so each has probability 1/3. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: favourable ÷ total only works when the outcomes are equally likely. Red is half the spinner: P(red) = 1/2, P(blue) = P(green) = 1/4. ::: :::keep-together ### Counters in a bag **A bag has 3 red, 5 blue and 2 green counters. You take one without looking. What is $P(\text{blue})$? Simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/2 10 counters, 5 blue: 5/10 = 1/2. ::: :::typical ## One change at a time **A different event.** Each task changes one thing. **Repeated labels.** **The missing probability.** **As a percentage.** **Impossible and certain.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Probabilities :::keep-together ### A bag **A bag has 4 yellow and 6 purple counters. $P(\text{yellow})$? Simplest form.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2/5 4 of 10 = 2/5. ::: :::keep-together ### Letters **Pick a letter at random from the word MATHS. What is the probability it is a vowel? (A fraction.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1/5 5 letters, 1 vowel (A): 1/5. ::: :::keep-together ### Sample space **Which list is the full sample space for spinning this spinner: four equal sectors labelled 1, 2, 2, 3?** ☐ {1, 2, 2, 3} ☐ {1, 2, 3} ☐ {2} ::: :::teacher-note **Answer:** {1, 2, 3} The possible outcomes are 1, 2 and 3; 2 has probability 1/2, the others 1/4 each. ::: :::keep-together ### Review: comparing **Review: Group A's median is 12 (range 4); Group B's is 12 (range 15). Which is right?** ☐ B is better ☐ Same centre; B is much more spread ☐ They are the same data ::: :::teacher-note **Answer:** Same centre; B is much more spread Equal medians, but B varies far more. ::: ## Make: design a bag Then: design a bag with $P(\text{red}) = \tfrac{1}{3}$. What is the smallest bag that works? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A bag has 2 red, 3 green and 5 yellow counters. List the sample space and give each probability.** {{lines-3}} ::: :::teacher-note **Answer:** {red, green, yellow}: P(red) = 2/10 = 1/5, P(green) = 3/10, P(yellow) = 5/10 = 1/2. They add to 1. :::