Likelihood and Independent Events: a week of ready-to-teach maths

Five days of lessons for Year 4 Probability. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus cut-out event cards, a spinner sheet, a mini-check and every answer.

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. Cut out the event cards once for Day 1, and print the spinner sheet for each pair for Days 2 and 3.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. Year 4 moves from chance words to counting: how many possible outcomes there are, and how many of them give the result we want.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Chance as a count

This pack writes a chance as “3 in 8”: 3 of the 8 equal parts of a spinner, or 3 of the 8 marbles in a bag. Fractions of chance come in later years. Two events are independent when the first result does not change the chance of the second, such as two coin flips; they are dependent when it does, such as taking a marble out and keeping it.

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 describes events in words. Days 2 and 3 describe chance experiments: first every possible outcome, then how likely each one is, so outcomes can be put in order. Days 4 and 5 look at two events in a row and ask the question that sorts them: does the first result change the chance of the next? It builds on chance words and chance experiments in Year 3 (AC9M3P01, AC9M3P02).

Materials for the week (one trip)

• From the classroom: a coin and a die for each pair, paper clips, pencils, scissors, and this pack printed.
• For Day 5: a paper bag or a sock for each pair, with 3 cubes or counters of one colour and 2 of another.
• Cut out once: the event cards and the five chance words. Print once for each pair: the spinner sheet.

Words for the word wall

chance, impossible, unlikely, even chance, likely, certain, event, outcome, possible, likelihood, equally likely, experiment, independent, dependent

Dear families

This week in maths, Year 4 describes how likely things are, lists all the possible outcomes of games of chance, and asks whether one result changes the chance of the next.

Try this at home

• Use chance words about your day: is it impossible, unlikely, an even chance, likely or certain?
• Play a board game with a die. Which numbers could come up? Is any number more likely?
• Flip a coin ten times. After a few heads in a row, is tails more likely next? (No: a coin has no memory.)
• Take two socks from a drawer without looking. Does taking the first one change your chance of a matching second sock?

Chance words I used this week

Fill one row a day. Write an event and the chance word that fits it.

Printed from the free seegongsik Likelihood and Independent Events teaching pack · seegongsik.com/au/y4/probability/AC9M4P01/pack

The chance scale

Chance words describe how likely an event is: impossible (it cannot happen), unlikely, an even chance, likely, and certain (it must happen). Children place event cards on a chance line across the board, then describe everyday events on the worksheet. Everyday events can depend on where and when, so accept a different word when the reason is sound.

We are learning to

• describe everyday events with chance words,
• place events on a chance line from impossible to certain,
• give a reason for the chance word we choose.

Success criteria

• I can choose a chance word for an event and say why.
• I can tell likely from certain, and unlikely from impossible.

You need

The event cards and the five chance words (cut-out sheet 1), one set for the class. Sticky tape or putty for the board. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the chance line; start Session B with Part A.

Watch for these ideas

• Mixing up likely and certain, or unlikely and impossible.
• Thinking a result they want is more likely than one they do not.
• Thinking an even chance means it happens every second time.

Answers

• Chance line: impossible C and F; unlikely D and J; even chance B and G; likely E and I; certain A and H.
• Part A (other words are fine with a good reason): 1 likely, 2 impossible, 3 even chance, 4 unlikely, 5 certain, 6 likely, 7 unlikely, 8 certain. Part B: answers vary.
• Part C: no; likely means it will probably happen, but it still might not. Only certain means it must happen.

The chance scale

A. Write a chance word for each event.

B. Write an everyday event of your own for each chance word.

C. Ben says: “Rain is likely tomorrow, so it is sure to rain.” Is he right? Explain.

Possible outcomes

An outcome is one result a chance experiment can give, and describing the experiment means listing every possible outcome: a coin has 2, a die has 6. The outcomes of a spinner are its colours, not its parts: spinner 1 has 8 parts but only 3 colours. Two coins have four outcomes, because heads on the first and tails on the second is not the same as tails on the first and heads on the second.

We are learning to

• list every possible outcome of a chance experiment,
• tell the outcomes of a spinner from its parts,
• list the outcomes of two things at once in an organised way.

Success criteria

• I can list all the outcomes of a coin, a die, a spinner or a bag.
• I can find all four outcomes of two coins, and say how I know there are no more.

You need

A coin and a die for each pair. The spinner sheet (cut-out sheet 2) for each pair, with a paper clip and a pencil. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the two coins in Part B.

Watch for these ideas

• Counting heads then tails, and tails then heads, as one outcome.
• Counting the parts of a spinner instead of its colours.
• Leaving out an outcome because it seems unlikely, such as the green part of spinner 1.

Answers

• Part A: 1 heads, tails (2); 2 1, 2, 3, 4, 5, 6 (6); 3 blue, yellow, green (3); 4 orange, blue, yellow (3); 5 blue, orange (2); 6 Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday (7).
• Part B: HH, HT, TH, TT: 4 outcomes.
• Part C: H orange, H blue, H yellow, T orange, T blue, T yellow: 6 outcomes.
• Part D: no; spinner 1 has 3 possible outcomes (blue, yellow, green). The parts change how likely each colour is, not how many colours there are.

Possible outcomes

A. List every possible outcome, then count them.

B. Flip two coins: a 5c coin and a 10c coin. Fill in every outcome.

C. Flip a coin and spin spinner 2. List every outcome, such as “H orange”.

D. Sam says spinner 1 has 8 possible outcomes because it has 8 parts. Is Sam right? Explain.

Order by likelihood

To order outcomes by likelihood, count how many of the equal parts, or how many of the marbles, give each one. On spinner 1, blue 4 in 8, yellow 3 in 8, green 1 in 8, so from least to most likely the order is green, yellow, blue. Two outcomes with the same count are equally likely. Children order the outcomes of both spinners and a bag, then order five die events from impossible to certain.

We are learning to

• order outcomes from least to most likely by counting,
• write a chance as a count, such as 3 in 8,
• recognise outcomes that are equally likely.

Success criteria

• I can order the colours of a spinner from least to most likely, and say why.
• I can order events from impossible to certain.

You need

The spinners from Day 2, with paper clips and pencils. The class chance line from Day 1. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the bag in Part C.

Watch for these ideas

• Thinking a favourite colour or a lucky number is more likely.
• Thinking a 6 is harder to roll than a 1 because 6 is a bigger number.
• Forgetting that two outcomes can be equally likely.

Answers

• Part A: blue 4 in 8, yellow 3 in 8, green 1 in 8; least to most likely green, yellow, blue.
• Part B: orange 3 in 6, blue 2 in 6, yellow 1 in 6; least to most likely yellow, blue, orange.
• Part C: least to most likely green, blue, yellow, red (blue and yellow equally likely); red 5 in 10.
• Part D: 1 4, 2 1, 3 3, 4 5, 5 2; impossible: roll a 7; certain: roll a number from 1 to 6.

Order by likelihood

A and B. Count the parts of each colour, then order the colours.

C. A bag holds 5 red, 2 blue, 1 green and 2 yellow marbles. Pick one without looking.

D. Number these die events from 1 (least likely) to 5 (most likely).

Independent events

Two events are independent when the result of the first does not change the chance of the second. A coin has no memory: after four heads in a row, the next flip is still an even chance, heads or tails. The same is true of rolling a die again or spinning a spinner again. Children flip a coin twenty times, look at the runs of heads and tails, and meet the common belief that tails is “due” after a run of heads.

We are learning to

• identify independent events,
• explain why one flip of a coin does not change the next,
• recognise that runs of the same result happen by chance.

Success criteria

• I can say why two flips of a coin are independent.
• I can explain why tails is not more likely after a run of heads.

You need

A coin for each pair, and a die for Part C. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after the twenty flips; start Session B with Part B.

Watch for these ideas

• Thinking tails is “due” after a run of heads.
• Thinking a run of four or five the same means the coin is unfair.
• Thinking twenty flips must give exactly ten heads.

Answers

• Part A: answers vary; most pairs see a run of three or more of one side.
• Part B: 10 heads and 10 tails; the longest run is 4 heads; no, a run like this happens often with a fair coin.
• Part C: no; the die has no memory, so a 6 is still 1 in 6. Mia is not right: tails is still an even chance, the same as heads.
• Part D: answers vary, for example spinning a spinner twice, or rolling a die and then flipping a coin.

Independent events

A. Flip a coin twenty times. Write H or T in each box.

B. Another class flipped a coin twenty times.

C. Answer and explain.

D. Write one more pair of independent events.

Dependent events

Two events are dependent when the result of the first changes the chance of the second. A bag holds 3 blue and 2 orange marbles, so the chance of blue is 3 in 5. Take out a blue marble and keep it, and now 2 of the 4 marbles left are blue: the chance has changed. Put each marble back and the chance stays 3 in 5 every time, so the draws are independent. Children follow a run of draws, then sort pairs of events.

We are learning to

• identify dependent events,
• work out how a chance changes when something is taken away,
• sort pairs of events into independent and dependent.

Success criteria

• I can say how the chance changes after a marble is taken and kept.
• I can sort events into independent and dependent, and say why.

You need

A paper bag or a sock for each pair, with 3 cubes of one colour and 2 of another. The worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the sort in Part C.

Watch for these ideas

• Thinking the chance of blue stays 3 in 5 after marbles are taken out.
• Thinking any two events in a row must be dependent.
• Forgetting that putting the marble back makes the draws independent.

Answers

• Part A, chance the next is blue: at the start 3 in 5, after draw 1 2 in 4, after draw 2 2 in 3, after draw 3 1 in 2, after draw 4 1 in 1.
• Part B: 3 in 5 every time; the draws are independent, because the bag is the same each time.
• Part C: 1 independent, 2 dependent, 3 dependent, 4 independent, 5 independent, 6 dependent, 7 independent, 8 dependent.
• Part D: they are dependent when the first result changes the chance of the second.

Dependent events

A. A bag holds 3 blue and 2 orange marbles. These marbles are drawn and kept, one at a time: blue, orange, blue, orange, blue. Fill in the table.

B. This time each marble is put back before the next draw. What is the chance of blue each time? Are the draws independent or dependent?

C. Does the first result change the chance of the second? Tick one.

D. How can you tell if two events are dependent?

Chance words and event cards

One set for the class, for Day 1. Stick the five chance words in order along the board, then place each event card where it belongs.

Teacher note: impossible C and F; unlikely D and J; even chance B and G; likely E and I; certain A and H.

Spinner sheet

One per pair, for Days 2 and 3. Cut out each spinner. Put a pencil point through a paper clip at the centre dot, hold the pencil still and flick the clip. The colour under the end of the clip is the outcome.

Try it: ten spins of each

Ten spins can come out in any order. The count of parts, not a few spins, decides which colour is most likely.

What we know: Likelihood and independent events

1. Write a chance word for each event.

   You roll a 7 on a normal die. ______________

   A coin lands on tails. ______________
2. List every possible outcome of this spinner. ______________________

   O orange, B blue, G green
3. A bag holds 6 red, 3 blue and 1 yellow marbles. Order the colours from least to most likely. ______________________
4. Two coins are flipped. List every possible outcome. ______________________
5. Tom rolled a 6 five times in a row. Is a 6 now less likely on his next roll? Explain. ______________________
6. A bag holds 4 green sweets and 1 purple sweet. Jess takes a green one and eats it. What is the chance the next sweet is green? ____ in ____
7. Independent or dependent? Ring one.

   Roll a die, then roll it again.   independent · dependent

   Take two lollies from a bag and eat them.   independent · dependent
8. Which is more likely: to roll an even number or to roll a number less than 3? Why? ______________________

Answers and marking guide

Answers

1. impossible; even chance.
2. orange, blue, green: 3 outcomes (the spinner has 4 parts).
3. yellow, blue, red.
4. HH, HT, TH, TT.
5. No: a die has no memory, so a 6 is still 1 in 6.
6. 3 in 4.
7. independent; dependent.
8. To roll an even number: 3 of the 6 faces, against 2 of the 6.

A quick three-level guide

Eight questions, three ideas. A child at standard describes events and outcomes with chance words, orders outcomes by counting, and tells independent events from dependent ones by asking whether the first result changes the chance of the next.

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: chance words, outcomes, ordering by likelihood, independent events and dependent events.