Combined 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 outcome grids, tree diagrams, twelve student cards and a large Venn mat to sort them on, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet. The grids and trees are on the worksheets.
- Print the two cut-out sheets, one copy per pair: the student cards and the Venn mat are both for Day 4.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
When two things happen, such as tossing a coin and spinning a spinner, every result of the first can pair with every result of the second, so a coin and a 1-2-3 spinner give 2 × 3 = 6 combined outcomes. Three diagrams make sure none is missed: a two-way table puts one event down the side and the other along the top; a tree diagram branches once for each step; and a Venn diagram sorts people or things by two properties at once. When the outcomes are equally likely, a probability is the number of outcomes we want over the number there are. Venn diagrams and survey tables give probabilities from counts in the same way.
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.
And and or
In this pack “A and B” means both happen (the overlap of a Venn diagram) and “A or B” means at least one happens, including both. Students say which they mean before they count.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Listing combinations | List every outcome of two events in order, count combinations by multiplying, and find probabilities from the list | The combined sample space |
| 2 | Two-way tables | Fill an outcome table for two dice, read probabilities of sums, and read a survey table of counts | The two-way table |
| 3 | Tree diagrams | Draw trees for two-step events, count the leaves, draw from a bag twice, and choose two captains | The tree diagram |
| 4 | Venn diagrams | Sort students into a Venn diagram, fill one from survey numbers, and tell and from or | The Venn diagram; student cards |
| 5 | Choosing a diagram | Pick the best diagram for each problem, fix two counting mistakes, and play the quiz | Reading a probability |
How the week builds
Day 1 lists combinations systematically; Days 2, 3 and 4 give three diagrams that keep the list complete; and Day 5 chooses between them. It builds on single-event sample spaces in Year 7 and complementary events earlier in Year 8, and prepares chance experiments with compound events later in Year 8 and compound events with and without replacement in Year 9.
Materials for the week (one trip)
- From the classroom: pencils, coloured pencils, rulers, scissors, this pack printed.
- Cut-out sheet 1, one per pair: 12 student cards (Day 4).
- Cut-out sheet 2, one per pair: the Venn mat to sort the cards on (Day 4).
- Optional: a coin, a die and a 1-2-3 spinner to show on Day 1.
Words for the word wall
outcome, combined event, sample space, list, two-way table, row, column, tree diagram, branch, path, leaf, Venn diagram, overlap, region, and, or, neither, equally likely, with replacement
Dear families
This week in maths, Year 8 is finding all the outcomes when two things happen together, such as two dice, or a meal and a drink, using tables, tree diagrams and Venn diagrams. They then use the outcomes to work out probabilities.
Try this at home
- Count the outfits you could make from 3 tops and 2 pairs of pants. (6: each top with each pair.)
- Ask: two dice are rolled. Which total is most likely? (7: it can be made 6 ways out of 36.)
- Ask: in a group, who plays a sport, who plays an instrument, who does both, and who does neither?
- Make a tree of the choices for a takeaway meal: a main, then a drink.
My combinations this week
Fill one row a day. Name two events, how many combined outcomes they have, and one probability.
| Day | The two events | Outcomes | A probability |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Combined events teaching pack · seegongsik.com/au/y8/probability/AC9M8P02/pack
Listing combinations
Every result of the first event can pair with every result of the second. Students list combined outcomes in an order that misses none, count combinations by multiplying, and find probabilities from the list.
We are learning to
- list all the outcomes of two events in a system, holding one fixed while the other changes,
- count combinations by multiplying the number of choices,
- find the probability of a combined outcome from the list.
Success criteria
- I can list the outcomes of tossing a coin and spinning a 1-2-3-4 spinner, and say how many there are.
- I can explain why 2 choices and 6 choices make 12 combinations, not 8.
You need
The worksheet each; a coin and a die to show.
Lesson flow (about 60 minutes)
| 10 min | Pair them up Show the board (see the box). Ask: “Tracing H gives three outcomes. Why does tracing T give three more, and not one?” |
| 25 min | A coin and a die Students list the outcomes in Part A, H with each number first, then T, and find the probabilities. Ask: “How can you be sure your list has every outcome exactly once?” |
| 25 min | Multiply the choices Parts B and C; early finishers do Part D. Ask: “Adding a pair of shoes doubled the outfits. Why does a third choice multiply?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Adding the choices instead of multiplying: 2 + 6 instead of 2 × 6.
- Listing at random and missing or repeating outcomes.
- Treating H6 and T6 as the same outcome.
Answers
- Part A: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6: 12 outcomes. P(head and even) = 3/12 = 1/4; P(tail and more than 4) = 2/12 = 1/6; P(a 6) = 2/12 = 1/6.
- Part B: 3 × 4 = 12 outfits; with 2 pairs of shoes, 12 × 2 = 24.
- Part C: pasta and water; pasta and juice; curry and water; curry and juice; burger and water; burger and juice: 6 meals; P(curry with juice) = 1/6.
- Part D: 10 × 10 × 10 = 1000 codes from 000 to 999; with 4 digits, 10 000.
Listing combinations
A. A coin and a die
List every outcome in order: H with 1 to 6, then T with 1 to 6. Then answer the questions.
| How many outcomes are there? | |
| P(a head and an even number) | |
| P(a tail and a number more than 4) | |
| P(a 6) |
B. Outfits
Mia has 3 shirts and 4 pairs of pants. How many different outfits can Mia make? Mia also has 2 pairs of shoes. How many outfits now?
C. A meal deal
A meal deal is one main (pasta, curry, burger) and one drink (water or juice). List every meal deal. If one is chosen at random, what is the chance it is curry with juice?
D. Stretch: lock codes
A lock has three wheels, each showing a digit from 0 to 9. How many codes are possible? How many with four wheels?
Two-way tables
A two-way table puts one event down the side and the other along the top, so every cell is one combined outcome. Students fill a table for two four-sided dice, read probabilities of sums, and read a survey table of counts.
We are learning to
- fill a two-way table so that every combined outcome has one cell,
- count cells to find the probability of an event,
- read a table of counts and find probabilities from its totals.
Success criteria
- I can find P(sum of 6) for two four-sided dice from a table.
- I can find a probability from a survey table, using the right total.
You need
The worksheet and two coloured pencils each.
Lesson flow (about 60 minutes)
| 10 min | Row meets column Show the board (see the box). Ask: “The table has 6 cells. Why is that the same as 2 × 3?” |
| 25 min | Sums of two dice Students fill the sum table in Part A and colour the cells for each event. Ask: “Why is a sum of 5 the most likely? Why is 2 so rare?” |
| 25 min | A survey table Parts B and C; early finishers do Part D. Ask: ““A bus student has a phone” and “a phone owner catches the bus”: which total do you divide by for each?” |
Two shorter sessions instead? Split after Part A; start the second with the survey.
Watch for these ideas
- Thinking every sum is equally likely, because each total appears once on a list of sums.
- Dividing by the whole survey when the question is about one row or column.
- Counting 2 + 3 and 3 + 2 as one outcome.
Answers
- Part A: 16 cells. P(sum 5) = 4/16 = 1/4; P(sum 2) = 1/16; P(sum 6 or more) = 6/16 = 3/8; most likely sum 5.
- Part B: totals Bus 24, Walk or ride 16; has a phone 28, no phone 12; all 40.
- Part C: 1. 18/40 = 9/20; 2. 12/40 = 3/10; 3. of the bus students, 18/24 = 3/4 have a phone.
- Part D: Ash. A total of 5 is 4 of 16 outcomes (4/16 = 1/4), the likeliest single total, but any other total is 12/16 = 3/4.
Two-way tables
A. Two four-sided dice
- Fill each cell with the sum of the two dice.
- How many outcomes are there?
- Find P(sum is 5), P(sum is 2) and P(sum is 6 or more).
- Which sum is most likely?
B. A survey table
A Year 8 class was asked how they travel to school and whether they have a phone. Fill in the totals.
| Has a phone | No phone | Total | |
|---|---|---|---|
| Bus | 18 | 6 | |
| Walk or ride | 10 | 6 | |
| Total |
C. Read the table
- A student is chosen at random. What is the chance they catch the bus and have a phone?
- What is the chance they have no phone?
- A bus student is chosen at random. What is the chance they have a phone?
D. Stretch: a fair bet?
Jo bets that two four-sided dice will total 5. Ash bets on any other total. Who is more likely to win? Explain with the table.
Tree diagrams
A tree diagram branches once for each step, so every path from the start to a leaf is one combined outcome. Students draw and read trees for a lunch order and two draws from a bag, then list the pairs of captains from four players.
We are learning to
- draw a tree diagram for two steps, with one branch for each result,
- read each path as an outcome and count the leaves,
- find probabilities when the paths are equally likely.
Success criteria
- I can draw a tree for tossing a coin twice and list its four outcomes.
- I can find the probability of the same colour twice from a tree.
You need
The worksheet and a ruler each.
Lesson flow (about 60 minutes)
| 10 min | Path by path Show the board (see the box). Ask: “How many leaves will the tree have before we draw it? How do you know?” |
| 25 min | Lunch and a bag Students complete Parts A and B, writing each outcome at the end of its path. Ask: “The counter goes back in the bag. Why does that make each draw the same?” |
| 25 min | Choosing captains Part C; early finishers do Part D. Ask: “Ana and Ben is the same pair as Ben and Ana. How does the list avoid counting it twice?” |
Two shorter sessions instead? Split after Part A; start the second with the bag.
Watch for these ideas
- Drawing the second set of branches only once, instead of from every first branch.
- Counting branches instead of leaves.
- Counting a pair twice when order does not matter.
Answers
- Part A: ham-apple, ham-banana, cheese-apple, cheese-banana, egg-apple, egg-banana: 6 lunches; P(banana) = 3/6 = 1/2; P(not egg) = 4/6 = 2/3.
- Part B: 9 paths; the same colour twice on 3 of them, 3/9 = 1/3; two different colours 6/9 = 2/3.
- Part C: Ana and Ben, Ana and Cho, Ana and Dee, Ben and Cho, Ben and Dee, Cho and Dee: 6 pairs; Ana is a captain in 3, so 3/6 = 1/2.
- Part D: with order (captain, then vice-captain) there are 4 × 3 = 12; each pair appears twice.
Tree diagrams
A. A lunch order
A lunch is one sandwich (ham, cheese, egg) and one piece of fruit (apple or banana), chosen at random. Write each lunch at the end of its path.
How many lunches? P(a banana) = ______ P(not egg) = ______
B. Two draws from a bag
A bag has one red, one blue and one green counter. A counter is drawn, put back, and a second is drawn.
P(the same colour twice) = ______ P(two different colours) = ______
C. Choosing captains
Two captains are chosen at random from Ana, Ben, Cho and Dee. List every pair. What is the chance that Ana is one of the captains?
D. Stretch: captain and vice-captain
Now one player is captain and another is vice-captain, so order matters. How many ways are there? Why is it double?
Venn diagrams
A Venn diagram sorts a whole group by two properties at once: each circle is one property, the overlap is both, and the box outside the circles is neither. Students sort twelve student cards on a Venn mat, fill a Venn diagram from survey numbers, and match and and or to regions.
We are learning to
- sort a group into the four regions of a Venn diagram,
- fill a Venn diagram from two totals and the overlap, starting with the overlap,
- find probabilities from the regions, telling “and” from “or”.
Success criteria
- I can fill all four regions when I know both totals and the overlap.
- I can find P(A and B) and P(A or B) from a Venn diagram.
You need
Cut-out sheets 1 and 2 and scissors, one set per pair; the worksheet each.
Lesson flow (about 60 minutes)
| 10 min | Four regions Show the board (see the box). Ask: “12 play sport and 9 play music, but only 16 are inside a circle. Why not 21?” |
| 25 min | Sorting the cards Pairs cut out the cards, sort them on the Venn mat, then record Part A. Ask: “Gus plays neither. Is Gus still in the diagram, and where?” |
| 25 min | Numbers in regions Parts B and C; early finishers do Part D. Ask: “Why do we write the overlap first?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Adding the two totals and counting the overlap twice: 14 + 11 = 25 is not the number in a circle.
- Leaving out the people in neither, who are still in the box.
- Reading “swims or cycles” as one or the other but not both.
Answers
- Part A: sport only Ari, Cal, Fay, Ivo (4); instrument only Bea, Eli, Hana, Jas, Lia (5); both Dev, Kai (2); neither Gus (1). P(both) = 2/12 = 1/6; P(sport) = 6/12 = 1/2; P(sport or instrument) = 11/12; P(neither) = 1/12.
- Part B: swim only 9, cycle only 6, both 5, neither 10. P(neither) = 10/30 = 1/3; P(swims or cycles) = 20/30 = 2/3; P(swims but does not cycle) = 9/30 = 3/10.
- Part C: 1. the overlap only; 2. both circles, the overlap included; 3. the cycle circle without the overlap; 4. the box outside both circles.
- Part D: swims: 5 and 9, total 14; does not: 6 and 10, total 16; column totals 11 and 19, all 30. The overlap is the top-left cell.
Venn diagrams
A. Sort the class
Sort the twelve student cards on the Venn mat. Write each name in its region here, then find each probability for a student chosen at random.
- P(both) = ______
- P(sport) = ______
- P(sport or instrument) = ______
- P(neither) = ______
B. From survey numbers
In a class of 30, 14 swim, 11 cycle and 5 do both. Write the number in each region, starting with the overlap.
- P(neither) = ______
- P(swims or cycles) = ______
- P(swims but does not cycle) = ______
C. And or or?
Shade the region for each event, for a student from Part B.
D. Stretch: the same numbers in a table
Put the Part B numbers in a two-way table, with swim or not down the side and cycle or not along the top. Which cell is the overlap? Which is neither?
| cycles | does not cycle | Total | |
|---|---|---|---|
| swims | |||
| does not swim | |||
| Total |
Choosing a diagram
Each diagram suits a kind of problem: a list or a table for two events side by side, a tree for one step after another, and a Venn diagram for a group sorted by two properties. Students choose and use a diagram for each problem, fix two counting mistakes, and play the self-check quiz as a class game.
We are learning to
- choose a list, a table, a tree or a Venn diagram to suit a problem,
- use the diagram to find a probability,
- explain a counting mistake and fix it.
Success criteria
- I can say why a Venn diagram suits a survey of two properties.
- I can explain why two coins give four outcomes, not three.
You need
The worksheet each; the self-check quiz at the bottom of the unit page for the game.
Lesson flow (about 60 minutes)
| 10 min | Which diagram? Show the board (see the box), then read the four Part A problems aloud and vote on a diagram for each before anyone solves. Ask: “Could a table do the job of the tree in problem 4? Which would be quicker to draw?” |
| 30 min | Choose and solve Parts A to C; early finishers do Part D. Ask: “HH, HT, TT: which outcome is missing, and how would a tree show it?” |
| 20 min | Self-check quiz as a class game Teams of four hold up one, two or three fingers for their option, then reveal and read the explanation aloud. A team that was wrong says what fooled them. Ask: “Before we reveal it: which answer would someone choose if they added the choices instead of multiplying?” |
Two shorter sessions instead? Split after Part A; start the second with Part B and the quiz.
Watch for these ideas
- Treating HT and TH as one outcome.
- Using a Venn diagram for two dice, where a table shows every pair.
- Dividing by the wrong total after drawing the right diagram.
Answers
- Part A: 1. a list or table, 8 outcomes, 2/8 = 1/4; 2. a table, 6/36 = 1/6; 3. a Venn diagram, 8, 4 and 5 in the circles, so 8/25; 4. a tree, 6 paths, 2/6 = 1/3.
- Part B: 1. 2 × 6 = 12 outcomes, not 2 + 6, so P(H6) = 1/12; 2. HT and TH are different, so there are 4 outcomes and P(one of each) = 2/4 = 1/2.
- Part C: the tree has 8 leaves, H1, H2, H3, H4, T1, T2, T3, T4: the same outcomes as the list and the table.
- Part D: 8 outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT); exactly two heads in 3, so 3/8.
Choosing a diagram
A. Choose and solve
For each problem, write the diagram you chose (a list, a table, a tree or a Venn diagram), draw it and find the probability.
| 1. A coin is tossed and a 1-2-3-4 spinner is spun. What is the chance of a tail and an even number? | Diagram: |
| 2. Two dice are rolled. What is the chance that the total is 7? | Diagram: |
| 3. Of 25 students, 12 like pizza, 9 like sushi and 4 like both. One is chosen at random. What is the chance they like neither? | Diagram: |
| 4. A team picks a shirt (red or white), then socks (red, white or black), at random. What is the chance the shirt and socks match? | Diagram: |
B. Fix two mistakes
- Sam says a coin and a die give 2 + 6 = 8 outcomes, so the chance of a head and a 6 is 1/8.
- Mo tosses two coins and lists HH, HT and TT, so the chance of one head and one tail is 1/3.
C. One problem, three ways
Problem 1 again, as a tree. Write each outcome at the end of its path. Does the tree match your list or table?
D. Stretch: three coins
Three coins are tossed. Draw a tree with three steps. How many outcomes are there? What is the chance of exactly two heads?
Twelve student cards
For Day 4, one set per pair
Cut out the twelve cards. Each card says whether a student plays a sport and whether they play an instrument. Sort the cards on the Venn mat (sheet 2 of 2), then count each region.
Teacher note: sport only: Ari, Cal, Fay, Ivo; instrument only: Bea, Eli, Hana, Jas, Lia; both: Dev, Kai; neither: Gus. The cards are printed so that no row or column gives a region away.
The Venn mat
For Day 4, one per pair
The box is everyone on the cards. Put each card inside the circle for each thing the student plays: in the overlap for both, and inside the box but outside both circles for neither.
What we know: Combined events
Work on your own. Show your working: a list, a table, a tree or a Venn diagram.
- A coin is tossed and a 1-2-3-4-5 spinner is spun. List the outcomes. How many are there? ______________________
- A canteen has 4 sandwiches and 5 drinks. How many sandwich-and-drink combinations are there? ______________________
- Two dice are rolled. Use a two-way table to find the chance that the total is 9. ______________________
- A survey of 50 students gave this table. What is the chance a student chosen at random is in Year 8 and likes sport? A Year 7 student is chosen at random: what is the chance they like sport?
Likes sport Does not Year 7 16 9 Year 8 14 11 ______________________ - A bag has a yellow and a green counter. One is drawn, put back, and a second is drawn. Draw a tree and find the chance of two greens. ______________________
- In a class of 28, 15 have a dog, 10 have a cat and 4 have both. Draw a Venn diagram. How many have neither? What is the chance a student has a dog or a cat? ______________________
- Two class reps are chosen at random from Ava, Bo, Cy, Di, Ed and Flo. How many pairs are possible? What is the chance Cy is a rep? ______________________
- Tom’s Venn diagram has 8 in A only, 4 in both, 6 in B only and 6 in neither. Tom says P(A) = 8/24. What did Tom miss? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- H1, H2, H3, H4, H5, T1, T2, T3, T4, T5: 2 × 5 = 10 outcomes.
- 4 × 5 = 20.
- 4 of the 36 cells total 9 (3 + 6, 4 + 5, 5 + 4, 6 + 3), so 4/36 = 1/9.
- 14/50 = 7/25; of the 25 Year 7 students, 16 like sport, so 16/25.
- 4 paths (yellow-yellow, yellow-green, green-yellow, green-green); two greens on one, so 1/4.
- Only a dog 11, both 4, only a cat 6: 21 in a circle, so 7 have neither and P(a dog or a cat) = 21/28 = 3/4.
- 15 pairs; Cy is in 5 of them, so 5/15 = 1/3.
- A includes the overlap: P(A) = (8 + 4)/24 = 12/24 = 1/2.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Listing (Q1, Q2) | misses or repeats outcomes, or adds the choices | lists outcomes in a system and multiplies the choices | explains why multiplying counts every pair once |
| Tables (Q3, Q4) | counts 3 + 6 and 6 + 3 as one outcome, or divides by the wrong total | fills a table and reads a probability from its cells or totals | chooses the right total for a question about one row or column |
| Trees (Q5, Q7) | draws the second branches only once, or counts a pair twice | draws a two-step tree and counts its paths | says when order matters and why it doubles the count |
| Venn diagrams (Q6, Q8) | counts the overlap twice or leaves out neither | fills the four regions, starting with the overlap | tells A and B from A or B and checks the regions add to the total |
Eight questions, four ideas. A student at standard answers most questions with a diagram and checks that the outcomes or regions add to the total.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Listing | Tables | Trees | Venn diagrams | Choosing |
|---|---|---|---|---|---|
The five columns are the five days: listing, two-way tables, tree diagrams, Venn diagrams, and choosing a diagram.