--- title: "And, or and not" subtitle: "Data and chance · Chance · Lesson 13" --- # And, or and not **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A raffle prize goes to someone who plays sport or music. In a class of 30, 20 play sport and 13 play music. How many are in the draw? :::teacher-note **In one sentence:** Students find P(A and B), P(A or B), P(not A) and P(neither) from Venn diagrams, two-way tables and two-dice grids, counting the overlap once (AC9M8P02, node 2). **Students should be able to say:** I can find 'and', 'or', 'not' and 'neither' on a Venn diagram, a table or a grid. I can work out P(A or B) without counting the overlap twice. I can explain that 'or' in maths includes both. **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 **30 students: 20 play sport, 13 play music, 8 play both. What is P(a student plays sport or music)?** ☐ 8/30 ☐ 25/30 ☐ 33/30 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 25/30. 12 sport only, 8 both and 5 music only: each counted once. 33/30 counts the 8 twice (and a probability can't be more than 1). ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Counting once :::keep-together ### And Here is the routine for "or" from totals. 1. $P(\text{juice}) = \tfrac{35}{60},\ P(\text{milk}) = \tfrac{28}{60},\ P(\text{both}) = \tfrac{15}{60}$ — 60 people. 2. $35 + 28 - 15 = 48$ — Add the totals, then take the overlap away once. 3. $P(\text{juice or milk}) = \tfrac{48}{60}$ — Everyone inside the circles. 4. $P(\text{neither}) = 1 - \tfrac{48}{60} = \tfrac{12}{60}$ — Neither is the complement of "or". **30 students: 13 have a dog, 10 have a cat and 4 have both. What is P(a dog and a cat)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 4/30 Both is the overlap: 4 of 30 = 2/15. ::: :::keep-together ### Find the mistake **Zoe works out P(dog or cat) for the same class. Click the line where she goes wrong.** ``` 1 30 students: 13 have a dog, 10 have a cat and 4 have both. 2 "Dog or cat" means a dog, a cat, or both. 3 P(dog or cat) = 13/30 + 10/30 = 23/30. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: the 4 with both are in the 13 and in the 10, so they are counted twice. 13 + 10 − 4 = 19, so P = 19/30. ::: :::keep-together ### Or on a dice **A dice is rolled. A is 'an even number' and B is 'more than 4'. Which outcomes are in A or B?** ☐ 2, 4, 6, 5, 6 ☐ 6 ☐ 2, 4, 5, 6 ☐ 5 ::: :::teacher-note **Answer:** 2, 4, 5, 6 Even: 2, 4, 6. More than 4: 5, 6. Either or both: 2, 4, 5, 6, so P(A or B) = 4/6. ::: :::typical ## Not, or and neither **Not music.** **Over 9 or a double.** **Neither.** **At least one.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## And, or, not :::keep-together ### Glasses or left-handed **A two-way table of 50 people: 12 wear glasses and are left-handed, 18 wear glasses and are right-handed, 5 are left-handed without glasses, 15 neither. What is P(glasses or left-handed)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 35/50 12 + 18 + 5 = 35, so 35/50 = 7/10. ::: :::keep-together ### Left-handed, no glasses **For the same 50 people, what is P(left-handed and no glasses)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 5/50 One cell: 5 of 50 = 1/10. ::: :::keep-together ### Using the rule **P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.7 0.5 + 0.4 − 0.2 = 0.7. ::: :::keep-together ### Exactly one **In the Sport and Music class (12 sport only, 8 both, 5 music only, 5 neither), what is P(exactly one of sport and music)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 17/30 Sport only and music only: 12 + 5 = 17, so 17/30. This is the everyday 'one or the other, not both'. ::: ## Make: an 'or' event Write your events A and B in words, and how many outcomes are in A, in B and in both. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **40 students: 25 like maths, 18 like art, 10 like both. Find P(maths or art) and P(neither).** {{lines-3}} ::: :::teacher-note **Answer:** 25 + 18 − 10 = 33, so 33/40. Neither: 7/40. :::