Complementary events: a week of ready-to-teach maths

Five lessons for Year 8 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a probability line, twelve event cards, a spinner to make, a two-dice grid, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: the probability line and event cards for Day 1, the spinner and dice grid for Days 2 and 3.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A probability is a number from 0 (impossible) to 1 (certain). The complement of an event A, written “not A”, is everything in the sample space that is not A. Since an outcome is either in A or not in A, the two probabilities add to one: P(A) + P(not A) = 1, so P(not A) = 1 − P(A). This is more than a rule to remember: it is a shortcut. The chance of at least one six in two rolls is awkward to count directly, but its complement, no sixes at all, is easy: 25 of 36, so at least one six is 11 of 36.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

Fractions, decimals and percentages

Probabilities appear in all three forms, as they do in real life: 1/4, 0.25 and 25% are the same. Answers keep the form of the question and are simplified where it helps; a forecast’s 70% has a complement of 30%.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 places probabilities and their complements on one line; Day 2 counts them in sample spaces; Day 3 uses the complement as a shortcut; Day 4 takes it into real contexts; and Day 5 solves longer problems. It builds on sample spaces and probabilities as fractions, decimals and percentages in Year 7, and prepares two-way tables and Venn diagrams later in Year 8 and compound events in Year 9.

Materials for the week (one trip)

• From the classroom: pencils, coloured pencils, scissors, this pack printed.
• Cut-out sheet 1, one per pair: the probability line and 12 event cards (Day 1).
• Cut-out sheet 2, one per pair: the spinner (Day 2) and the two-dice grid (Day 3).
• For Day 2: a paper clip and a pencil per pair for the spinner; a pack of cards to show the class.

Words for the word wall

probability, outcome, sample space, event, equally likely, impossible, unlikely, even chance, likely, certain, complement, not A, at least one, none, fraction, decimal, percentage, relative frequency, forecast

Dear families

This week in maths, Year 8 is learning about complementary events: an event and everything that is not that event. Their probabilities always add to one, so if there is a 30% chance of rain, there is a 70% chance of no rain. Students also learn to use this as a shortcut for problems like “at least one six”.

Try this at home

• Check tomorrow’s forecast. If it gives a chance of rain, what is the chance of no rain?
• Ask: what is the chance of not rolling a 6 on a die? (5/6.)
• Ask: toss two coins. What is the chance of at least one head? (3/4: only tails-tails has none.)
• Talk about a game you play: what is the chance of a good result, and what is the chance of anything else?

My chances this week

Fill one row a day. Write an event, its probability, and the probability of its complement.

Printed from the free seegongsik Complementary events teaching pack · seegongsik.com/au/y8/probability/AC9M8P01/pack

The probability scale

Every probability is a number from 0 to 1, and every event has a complement at the mirror point. Students place events on a probability line, write probabilities three ways, and pair each event with its complement.

We are learning to

• place an event on a scale from 0 (impossible) to 1 (certain),
• write a probability as a fraction, a decimal and a percentage,
• name the complement of an event and see that the two probabilities add to 1.

Success criteria

• I can explain why a probability can never be more than 1.
• I can say the complement of “a card is a diamond” and give both probabilities.

You need

The worksheet each; per pair, the probability line and the twelve event cards from cut-out sheet 1, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the cards; start the second with Part B.

Watch for these ideas

• Giving a probability bigger than 1, or a negative one.
• Taking an opposite-sounding event as the complement when other outcomes are left out.
• Thinking 70% and 0.7 are different probabilities.

Answers

• Part A: complement pairs are cards 3 and 8; 6 and 11; 1 and 10; 4 and 7; 2 and 9; 5 and 12. Each pair adds to 1.
• Part B: 3/4 = 0.75 = 75%; 2/5 = 0.4 = 40%; 7/20 = 0.35 = 35%; 1/8 = 0.125 = 12.5%.
• Part C: 1. It does not rain tomorrow. 2. A die shows an odd number. 3. The bus is on time or early.
• Part D: no: 1.2 is more than certain and −0.1 is less than impossible.

The probability scale

A. Event cards on the line

Lay the cards along the line. Then pair each card with its complement and write the pairs and their probabilities.

B. Three ways

C. Name the complement

D. Stretch: impossible numbers

Sam says the chance of the school team winning is 1.2 and the chance of it losing is −0.1. Explain what is wrong with both numbers.

Complements in sample spaces

List the outcomes, mark event A, and everything else is not A. Students find complements for a die, a spinner and a deck of cards with P(not A) = 1 − P(A), then spin a spinner to see the rule in results.

We are learning to

• list a sample space and mark an event and its complement,
• use P(not A) = 1 − P(A), and check by counting,
• compare predicted results with the results of an experiment.

Success criteria

• I can find the probability of not rolling a 1 or a 2 on a die in two ways.
• I can say why my spinner results are close to, but not exactly, the predictions.

You need

The worksheet each; per pair, the spinner from cut-out sheet 2, a paper clip and a pencil; a pack of cards to show.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the spinner.

Watch for these ideas

• Thinking each colour on a spinner is equally likely, whatever the size of its sector.
• Subtracting from 100 when the probability is a fraction.
• Expecting results to match predictions exactly after only 30 spins.

Answers

• Part A: a 4 1/6, not a 4 5/6; an odd number 3/6 = 1/2, an even number 3/6 = 1/2; a number greater than 4 2/6 = 1/3, a number 4 or less 4/6 = 2/3; a multiple of 3 2/6 = 1/3, not a multiple of 3 4/6 = 2/3.
• Part B: not a heart: 1 − 13/52 = 1/4 = 39/52 = 3/4; not a picture card (jack, queen or king): 1 − 12/52 = 3/13 = 40/52 = 10/13; not an ace: 1 − 4/52 = 1/13 = 48/52 = 12/13.
• Part C: predicted yellow 1/2, so about 15 of 30; blue 1/3, so about 10 of 30; green 1/6, so about 5 of 30. Not blue is 2/3; real results vary a little.
• Part D: not yellow is 1/2 (blue or green), the same as yellow: an event can be as likely as its complement.

Complements in sample spaces

P(not A) = 1 − P(A). Check by counting the outcomes that are not A.

A. One die

B. A deck of 52 cards

A standard deck has 4 suits of 13 cards: ace, 2 to 10, jack, queen and king.

C. Spin and compare

Predict first from the sector sizes. Then spin 30 times and tally.

D. Stretch: as likely as not

Find an event on the spinner that is exactly as likely as its complement. Is there one on a die? Explain with probabilities of 1/2.

The complement shortcut

“At least one” is awkward to count, but its complement, “none”, is easy. Students shade a two-dice grid, count the complement and subtract, and use the shortcut for coins and guessing.

We are learning to

• recognise when the complement is easier to count than the event,
• use P(at least one) = 1 − P(none),
• check a shortcut answer by counting directly.

Success criteria

• I can find the chance of at least one tail when two coins are tossed, using the complement.
• I can say what the complement of “at least one” is.

You need

The worksheet and two coloured pencils each; the two-dice grid from cut-out sheet 2, or the one on the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the coins.

Watch for these ideas

• Taking “exactly one” as the complement of “at least one”.
• Counting 11 at-least-one-six squares as 12, by counting the double six twice.
• Thinking the shortcut gives a different answer from counting.

Answers

• Part A: no six 25/36, so at least one six = 1 − 25/36 = 11/36; sum 4 or more: the complement is a sum of 2 or 3 (3 squares), so 33/36 = 11/12; not a double 30/36 = 5/6.
• Part B: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT: only TTT has no heads, so at least one head = 1 − 1/8 = 7/8.
• Part C: all 3 wrong has probability 1/8, so at least one right = 1 − 1/8 = 7/8.
• Part D: with four coins, none is 1/16, so at least one head is 15/16; each extra coin halves the chance of none.

The complement shortcut

P(at least one) = 1 − P(none)

A. Two dice

B. Three coins

List all the outcomes of tossing three coins. Use the complement to find P(at least one head).

C. Guessing

D. Stretch: more coins

What is the chance of at least one head with four coins? With ten coins? What happens to the chance of no heads each time you add a coin?

Complements in context

Forecasts, sport, factories and surveys all use complements. Students find complements given as percentages and decimals, work them out from data, and check whether an offered complement really covers everything else.

We are learning to

• find a complement given as a percentage or a decimal,
• estimate a probability from data, then find its complement,
• check that a complement includes every other outcome, not just the opposite-sounding one.

Success criteria

• I can find the chance of no rain from a forecast of a 45% chance of rain.
• I can explain why “lose” is not the complement of “win” in a game that can be drawn.

You need

The worksheet and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the data.

Watch for these ideas

• Subtracting a percentage from 1, or a decimal from 100.
• Treating a probability from data as exact.
• Taking the opposite word as the complement when other outcomes exist.

Answers

• Part A: 82% gives 18%; 2% gives 98%; 15% gives 85%.
• Part B: 120/400 = 3/10 = 0.3, so on time 7/10 = 0.7; 185/250 = 37/50 = 0.74, so come another way 13/50 = 0.26.
• Part C: 1. no: it could also draw: the complement is that it does not win; 2. yes: every other outcome is 1 to 5; 3. no: one 6 and one other is in neither: the complement is that at least one die is not a 6.
• Part D: answers vary; the complement must contain every outcome not in the event.

Complements in context

A. Real complements

B. From data

Estimate each probability from the data, then find its complement. Give a fraction and a decimal.

C. Is it the complement?

Write yes or no. If no, write the real complement.

D. Stretch: write a trap

Write an event and a wrong complement that sounds right, like the ones in Part C. Then write the real complement.

Problems with complements

Students solve problems that take more than one step, choosing each time whether to count the event or its complement, find a complement mistake, judge a lottery claim, and play the self-check quiz as a class game.

We are learning to

• decide whether to count an event or its complement, and say why,
• use complements in problems with several steps,
• explain a mistake with complements and fix it.

Success criteria

• I can choose the easier of an event and its complement to count.
• I can explain why the complement of “at least one” is “none”.

You need

The worksheet and a calculator each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Forgetting the last step, 1 − P(none), after finding P(none).
• Taking “exactly one” or “all” as the complement of “at least one”.
• Reading a tiny chance as zero.

Answers

• Part A: 1. not red = 5/10 = 1/2; 2. late = 1 − 0.94 = 0.06, about 3 of 50 trips; 3. no repeated digit 720 of 1000, so at least one repeat = 280/1000 = 7/25 = 0.28.
• Part B: there are 8,145,060 ways to choose the numbers, so not winning has probability 1 − 1/8,145,060, about 0.99999988: almost certain, not certain.
• Part C: Ari found P(none) and stopped; at least one six is 1 − 25/36 = 11/36, not 25/36.
• Part D: answers vary; for example the complement of “all three are heads” is “at least one tail”, 7/8.

Problems with complements

Before each problem, decide: count the event, or count the complement and subtract from 1?

A. Problems in steps

B. A lottery claim

In a lottery, 6 numbers are drawn from 1 to 45, and there are 8,145,060 possible tickets. An ad says “someone has to win!” What is the chance one ticket does not win the top prize? What does that tell you?

C. Find the mistake

Ari says the chance of at least one six when rolling two dice is 25/36, “because 25 of the 36 outcomes have no six”. What went wrong?

D. Stretch: make a problem

Write a probability problem that is much easier to solve with the complement. Swap with a partner and solve each other’s.

The probability line and event cards

For Day 1, one set per pair

Cut out the line and the twelve cards. Work out each card’s probability, lay it above the line where it belongs, then pair each card with its complement.

Teacher note: 1: 1/4, 2: 30%, 3: 1/2, 4: 0.8, 5: 4/9, 6: 0, 7: 0.2, 8: 1/2, 9: 70%, 10: 3/4, 11: 1, 12: 5/9. Complement pairs: 3 and 8; 6 and 11; 1 and 10; 4 and 7; 2 and 9; 5 and 12. No card is printed next to or under its complement.

A spinner and a two-dice grid

For Day 2: the spinner, one per pair

Put a paper clip at the centre and hold it there with a pencil point. Flick the clip and read the colour it points to. The sectors are not the same size: yellow 1/2, blue 1/3, green 1/6 of the turn.

For Day 3: the 36 outcomes of two dice

Each square is one outcome: the first die down the side, the second along the top. Shade to show an event.

What we know: Complementary events

Work on your own. Show your working, and give each probability as a fraction, a decimal or a percentage.

1. Give an example of an event with probability 0, and one with probability 1. ______________________
2. P(A) = 0.35. What is P(not A)? ______________________
3. A die is rolled. What is the probability of not getting a number greater than 4? ______________________
4. A card is drawn from a shuffled deck of 52. What is the probability it is not a picture card (jack, queen or king)? ______________________
5. Two coins are tossed. Use the complement to find the probability of at least one tail. ______________________
6. Two dice are rolled. Use the complement to find the probability of at least one 6. ______________________
7. The forecast gives a 15% chance of storms. What is the chance of no storms? ______________________
8. Lily says the complement of “at least one head” when two coins are tossed is “exactly one tail”. Is Lily right? ______________________

Answers and marking guide

Answers with working

1. Any correct pair, for example rolling a 7 on a die (0) and rolling a number less than 7 (1).
2. 1 − 0.35 = 0.65.
3. P(greater than 4) = 2/6 = 1/3, so P(not) = 4/6 = 2/3.
4. 12 picture cards, so 1 − 12/52 = 3/13 = 40/52 = 10/13.
5. No tails is HH only, 1/4, so at least one tail = 1 − 1/4 = 3/4.
6. No six in 25 of 36 outcomes, so 1 − 25/36 = 11/36.
7. 100% − 15% = 85%.
8. No: the complement is “no heads”, two tails. Exactly one tail leaves out TT and is itself part of at least one head.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and checks that the two probabilities add to 1.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the probability scale, sample spaces, the shortcut, complements in context, and problems.