--- title: "Complementary events" subtitle: "Data and chance · Chance · Lesson 10" --- # Complementary events **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** The forecast says a 30% chance of rain. What is the chance it stays dry? You know straight away, but why does it work? :::teacher-note **In one sentence:** Students choose events on a dice and two dice, show their complements, and use P(not A) = 1 − P(A), including for 'at least one' (AC9M8P01, nodes 1 and 2). **Students should be able to say:** I can name the complement of an event. I can find P(not A) from P(A), in fractions, decimals and percentages. I can use the complement to find the chance of 'at least one'. **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 **A spinner lands on red with probability 0.35. It has other colours too. What is the probability it does not land on red?** ☐ 0.65 ☐ 0.35 ☐ You need to know the other colours My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 0.65. Not red is every other outcome, so P(not red) = 1 − 0.35, whatever the other colours are. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Using the complement :::keep-together ### Not late Here is the routine. 1. $P(\text{red}) = \tfrac{3}{8}$ — A bag has 3 red counters out of 8. 2. $P(\text{not red}) = 1 - \tfrac{3}{8}$ — Not red is every other counter. 3. $= \tfrac{5}{8}$ — Check: 5 counters are not red. **The chance a bus is late is 0.15. What is the chance it is not late?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.85 P(not late) = 1 − 0.15 = 0.85. ::: :::keep-together ### Find the mistake **Ben works with two events on a dice. Click the line where his reasoning goes wrong.** ``` 1 Ben rolls a dice. Event A is "roll a 1" and event B is "roll a 6". 2 A and B cannot both happen on the same roll. 3 So A and B are complementary events. 4 So P(B) = 1 − P(A) = 1 − 1/6 = 5/6. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: A and B can't happen together, but they don't cover every outcome (2, 3, 4 and 5 are in neither). Complementary events must cover everything. P(B) = 1/6. ::: :::keep-together ### Which pair **On one roll of a dice, which pair of events is complementary?** ☐ Roll a 6; roll a 5 or a 6 ☐ Roll more than 3; roll less than 3 ☐ Roll a 1; roll a 6 ☐ Roll an even number; roll an odd number ::: :::teacher-note **Answer:** Roll an even number; roll an odd number Every number is even or odd, never both: together they cover every outcome once. ::: :::typical ## At least one **No six.** **At least one six.** **Three coins.** **Faulty globes.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Complements :::keep-together ### Win or not **P(a team wins) = 0.45. A draw is not possible. What is P(the team does not win)?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 0.55 1 − 0.45 = 0.55. ::: :::keep-together ### Complement in words **On a dice, what is the complement of 'roll at least 4'?** ☐ Roll a 4 ☐ Roll more than 4 ☐ Roll less than 4 or more than 4 ☐ Roll 3 or less ::: :::teacher-note **Answer:** Roll 3 or less At least 4 is 4, 5, 6, so the complement is 1, 2, 3. ::: :::keep-together ### Cards **A card is drawn from a pack. A is 'a heart' and B is 'a spade'. Are A and B complementary?** ☐ Yes: they are both suits ☐ Yes: they can't both happen ☐ No: they can both happen ☐ No: some cards are in neither ::: :::teacher-note **Answer:** No: some cards are in neither They can't both happen, but clubs and diamonds are in neither. Complementary events cover every outcome: P(not a heart) = 3/4, not P(spade). ::: :::keep-together ### Review: expected count **Review: a dice is rolled 120 times. How many sixes do you expect?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 20 1/6 × 120 = 20. ::: ## Make: a quarter Then describe your event in words, so someone else could check it. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **P(it snows tomorrow) = 0.08. What is P(it does not snow)? Then: two coins are tossed; what is P(at least one tail)?** {{lines-3}} ::: :::teacher-note **Answer:** 0.92. And 1 − P(HH) = 1 − 1/4 = 3/4. :::