Sample space and probability: a week of ready-to-teach maths

Five lessons for Year 7 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 chance cards, three spinners to make, 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 at 100%: chance cards (Day 3), one set per pair, and spinners (Days 4 and 5), one sheet per pair.
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 chance experiment is one action whose result cannot be known in advance: a roll, a spin, a draw. Its sample space lists every outcome that can happen, each one once. A probability is a number from 0 (impossible) to 1 (certain). When the outcomes are equally likely, each gets 1 divided by the number of outcomes; when they are not, each gets its share of the counts in a bag or of the 360° of a spinner. An event is a set of outcomes, and the probabilities of all the outcomes add to 1, so the chance that an event does not happen is 1 minus the chance that it does. A probability also predicts: over n trials, an outcome with probability p should happen about p × n times, though the real count varies.

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.

How we write probability

Write a sample space in braces, such as {red, blue, white}, and a probability as P(red) = 1/4. Give fractions in lowest terms, and give decimals to three places and percentages to one place when they do not end sooner. Protractors are needed on Day 4, and calculators on Days 2 and 5.

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 lists the sample space; Day 2 gives equally likely outcomes their equal shares; Day 3 builds events from outcomes; Day 4 weights outcomes that are not equally likely; and Day 5 turns probabilities into predictions. It builds on the probability scale (AC9M6P01) and leads into chance experiments and simulations (AC9M7P02) and complementary events in Year 8 (AC9M8P01).

Materials for the week (one trip)

• From the classroom: pencils, rulers, protractors, scissors, glue, calculators, this pack printed at 100%.
• For the board on Day 1: a coin, a die and a bag of coloured counters.
• Cut-out sheet 1, one set per pair: 12 chance cards (Day 3).
• Cut-out sheet 2, one per pair: three spinners (Days 4 and 5), with a paper clip and a pencil, and light card to glue them to if you have it.

Words for the word wall

chance experiment, trial, outcome, sample space, equally likely, probability, event, favourable outcome, not (the complement), impossible, unlikely, even chance, likely, certain, sector, relative frequency, expected number, prediction

Dear families

This week in maths, Year 7 is learning to list every outcome of a chance experiment, give each outcome a probability, and use those probabilities to predict how often things should happen.

Try this at home

• List every outcome of a board game’s spinner or die, and decide together whether the outcomes are equally likely.
• Shuffle a pack of playing cards and work out the probability of a heart, a king and a red card. Then draw, note and replace 20 times.
• Ask: if the forecast gives a 30% chance of rain, what is the chance of no rain? On about how many of 10 such days would it rain?
• Before rolling a die 30 times, predict how many sixes. How close did you get, and would 300 rolls come closer?

Chance this week

Fill one row a day with a chance experiment you met, its sample space, and one probability.

Printed from the free seegongsik Sample space and probability teaching pack · seegongsik.com/au/y7/probability/AC9M7P01/pack

Sample spaces

The sample space of a chance experiment lists every outcome that can happen, once each, in braces. Students list sample spaces, fix lists that are not quite right, and see that what we record decides the sample space.

We are learning to

• describe a single-stage chance experiment and its outcomes,
• list a sample space in braces, with every outcome once,
• choose the sample space that fits what is being recorded.

Success criteria

• I can list the sample space of a single-stage experiment and count its outcomes.
• I can say what is wrong with a list that leaves out, repeats or adds an outcome.

You need

A coin, a die and a bag of coloured counters for the board, and the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Listing an outcome twice, such as the second A in KANGAROO.
• Leaving one out, such as 0 as the last digit of a phone number.
• Adding an outcome that cannot happen.

Answers

• Part A: (a) {Monday, Tuesday, ..., Sunday}, 7; (b) {K, A, N, G, R, O}, 6; (c) {0, 1, ..., 9}, 10; (d) {NSW, Vic, Qld, WA, SA, Tas, ACT, NT}, 8; (e) {hearts, diamonds, clubs, spades}, 4.
• Part B: (a) o is missing: {a, e, i, o, u}; (b) A and N are listed more than once: {B, A, N}; (c) green cannot happen: {red, white, blue}.
• Part C: (a) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; (b) {odd, even}; (c) {one digit, two digits}; (d) {0, 1, 2}.
• Part D: Mia, as a sample space lists each outcome once. Kai’s list counts sectors, which shows that blue and green (3 each) are more likely than yellow (2): Day 4.

Sample spaces

Write each sample space in braces, with every outcome once.

A. List and count

B. Find the mistake

Each list should be a sample space but has one mistake. Say what is wrong, then write the sample space correctly.

1. Choosing a vowel at random: {a, e, i, u}
   What is wrong? ____________________   Correct: ______________________
2. Choosing a letter at random from the word BANANA: {B, A, N, A, N, A}
   What is wrong? ____________________   Correct: ______________________
3. Spinning a spinner with sectors coloured red, white and blue: {red, white, blue, green}
   What is wrong? ____________________   Correct: ______________________

C. What do we record?

A card is drawn at random from ten cards numbered 1 to 10. Write the sample space when we record:

1. the number on it.   ______________________________
2. whether the number is odd or even.   ______________________________
3. how many digits the number has.   ______________________________
4. the remainder when the number is divided by 3.   ______________________________

D. Stretch

A spinner has 8 equal sectors: 3 blue, 3 green and 2 yellow. For the colour it lands on, Kai writes the sample space {blue, blue, blue, green, green, green, yellow, yellow}. Mia writes {blue, green, yellow}. Who is right? What does Kai’s list tell us that Mia’s does not?

Equally likely outcomes

With n equally likely outcomes, each has probability 1/n, and the n shares add to 1. Students write shares as fractions, decimals and percentages, check totals, and decide when outcomes are equally likely.

We are learning to

• decide whether the outcomes of an experiment are equally likely,
• give each of n equally likely outcomes the probability 1/n, as a fraction, a decimal and a percentage,
• check that the probabilities of all the outcomes add to 1.

Success criteria

• I can give the probability of each equally likely outcome in three forms.
• I can explain why two outcomes do not always mean 1/2 each.

You need

The worksheet and calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Giving any two outcomes 1/2 each, whether or not they are equally likely.
• Writing 1/8 as 0.8 or 1/3 as 0.3. One eighth is 0.125, and one third is 0.333 to three places.
• A probability above 1, such as 8/1 for one face of an 8-sided die.

Answers

• Part A: (a) 10 outcomes, 1/10 = 0.1 = 10%; (b) 5 outcomes, 1/5 = 0.2 = 20%; (c) 8 outcomes, 1/8 = 0.125 = 12.5%; (d) 25 outcomes, 1/25 = 0.04 = 4%; (e) 3 outcomes, 1/3 ≈ 0.333 ≈ 33.3%.
• Part B: (a) yes, a fair die makes every face equally likely; (b) no, two outcomes need not be 1/2 each; (c) no, its shape makes one way more likely; (d) yes, 26 cards of each colour; (e) no, the teams are not equally strong.
• Part C: (a) wrong, they add to 7/6; each should be 1/3; (b) right, 4 × 0.25 = 1; (c) right, 5 × 20% = 100%; (d) wrong, they add to 31/30; each should be 1/6.
• Part D: (a) 20, because 1 ÷ 0.05 = 20; (b) 40, because 1 ÷ 0.025 = 40; (c) no, 1 ÷ 0.3 is not whole: 3 outcomes give 0.9, and 4 give 1.2.

Equally likely outcomes

A. One share each

All the outcomes are equally likely. Give decimals to three places and percentages to one place.

B. Equally likely or not?

1. A fair 20-sided die landing on each of 1 to 20.   Yes or no? ______   Why? ______________________________
2. Tomorrow: rain or no rain.   Yes or no? ______   Why? ______________________________
3. A drawing pin landing point up or point down.   Yes or no? ______   Why? ______________________________
4. A card drawn from a shuffled pack of 52 being red or black.   Yes or no? ______   Why? ______________________________
5. A netball team winning, drawing or losing its next game.   Yes or no? ______   Why? ______________________________

C. Check the total

Each list claims to give the probability of every outcome. If it can be right, say so; if not, say why and fix it.

1. A spinner with 3 equal sectors: 1/3, 1/3, 1/2.   ______________________________
2. A fair 4-sided die: 0.25, 0.25, 0.25, 0.25.   ______________________________
3. Choosing one of 5 names from a hat: 20%, 20%, 20%, 20%, 20%.   ______________________________
4. A spinner with 6 equal sectors: 1/6, 1/6, 1/6, 1/6, 1/6, 1/5.   ______________________________

D. Stretch

1. Each face of a fair die has probability 0.05. How many faces does it have?   ______________
2. Each sector of a spinner with equal sectors has probability 2.5%. How many sectors are there?   ______________
3. Could each outcome of an experiment with equally likely outcomes have probability 0.3? Explain.   ______________

Events

An event is a set of outcomes from the sample space, such as rolling an even number. When the outcomes are equally likely, its probability is the number of outcomes in the event over the number in the sample space. Students sort the chance cards, find the probabilities of events, and find the chance that an event does not happen.

We are learning to

• describe an event as a set of outcomes from the sample space,
• find the probability of an event as favourable outcomes over all outcomes, in lowest terms,
• find the probability that an event does not happen by subtracting from 1.

Success criteria

• I can list the outcomes in an event and find its probability.
• I can order events from impossible to certain and explain the order.

You need

Chance cards from cut-out sheet 1 (one set per pair, cut out before the lesson) and the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Dividing by the outcomes not in the event instead of by all of them: a multiple of 3 is 4 of 12, not 4 of 8.
• Reading “greater than 8” as including 8, or counting 1 as a prime number.
• Thinking an impossible event cannot be written as a probability. It is 0.

Answers

• Part A: A 2/6 = 1/3, B 6/6 = 1, C 3/12 = 1/4, D 5/9, E 5/7, F 0/6 = 0, G 4/20 = 1/5, H 1/4, I 7/8, J 1/2, K 1/10, L 2/7. From least to most likely: F, K, G, C & H, L, A, J, D, E, I, B.
• Part B: (a) {1, 3, 5, 7, 9, 11}, 6/12 = 1/2; (b) {3, 6, 9, 12}, 4/12 = 1/3; (c) {9, 10, 11, 12}, 4/12 = 1/3; (d) {2, 3, 5, 7, 11}, 5/12; (e) no outcomes, 0; (f) {1, 2, 3, 4, 6, 12}, 6/12 = 1/2.
• Part C: (a) 1 − 0.3 = 0.7; (b) 1 − 2/9 = 7/9; (c) 100% − 35% = 65%; (d) 1 − 5/12 = 7/12.
• Part D (one answer each; others work): 0, rolling a 9; 1/8, rolling a 5; 1/4, rolling a number greater than 6; 1/2, rolling an even number; 3/4, rolling a number greater than 2; 1, rolling a number less than 9.

Events

A. Chance cards

Lay the cards in a line from impossible to certain. Then write each card’s probability in lowest terms, and the order of the letters.

B. One card from twelve

A card is drawn at random from twelve cards numbered 1 to 12. List the outcomes in each event and give its probability.

C. Not happening

1. The probability of rain tomorrow is 0.3. What is the probability of no rain tomorrow?   ____________
2. The probability of drawing a green ticket is 2/9. What is the probability of not drawing a green ticket?   ____________
3. A spinner lands on red with probability 35%. What is the probability of not landing on red?   ____________
4. Use Part B: what is the probability that the card is not a prime number?   ____________

D. Stretch

A fair 8-sided die is numbered 1 to 8. Write an event with each probability: 0, 1/8, 1/4, 1/2, 3/4, 1.

Outcomes not equally likely

An outcome’s share is its count over the total for a bag, or its angle over 360° for a spinner, and the shares still add to 1. Students measure and make spinners, use counts, and test claims about chance.

We are learning to

• find each outcome’s probability from counts or from sector angles,
• check that the probabilities of all the outcomes add to 1,
• explain why the number of outcomes alone does not give their probabilities.

Success criteria

• I can find the probability of each outcome when the outcomes are not equally likely.
• I can explain why three colours need not mean 1/3 each.

You need

Cut-out sheet 2 (one per pair), protractors, scissors, glue and light card if you have it, paper clips, pencils and the worksheet.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Giving each outcome 1 over the number of outcomes when they are not equally likely.
• Writing one count over another, such as 4/6 for 4 green and 6 yellow, instead of over the total.
• Dividing an angle by 100 or 180 instead of by 360.

Answers

• Part A: Spinner A: 1, 2, 3 and 4 are 90° = 1/4 each; Spinner B: yellow is 180° = 1/2, blue and green are 90° = 1/4 each; Spinner C: A is 120° = 1/3, B is 90° = 1/4, C and D are 60° = 1/6 each, E is 30° = 1/12.
• Part B: (a) P(red) = 5/10 = 1/2, P(blue) = 3/10, P(white) = 2/10 = 1/5; (b) the sample space is {W, O, L, M}, with P(W) = 1/13, P(O) = 8/13, P(L) = 3/13, P(M) = 1/13 (13 tiles); (c) walk 12/28 = 3/7, car 9/28, bus 4/28 = 1/7, bike 3/28.
• Part C: (a) false, yellow is half the circle, so P(yellow) = 1/2 and blue and green are 1/4 each; (b) true, red is 3 of the 6 counters; (c) false, green is 4 of 10 marbles, so P(green) = 4/10 = 2/5; (d) false, the two add to 0.8, not 1.
• Part D: red 180° and blue 90° leave 90°, so green and yellow are 45° each, with probability 1/8 each.

Outcomes not equally likely

A. The spinners

Measure each sector of the spinners on cut-out sheet 2 with a protractor. A sector’s probability is its angle over 360°.

B. From counts

1. A bag holds 5 red, 3 blue and 2 white counters. One is drawn at random. Give the probability of each colour.
2. The letter tiles of WOOLLOOMOOLOO, a suburb of Sydney, go into a bag and one is drawn at random. Write the sample space, and the probability of each letter.
3. In a class of 28, 12 walk to school, 9 come by car, 4 come by bus and 3 ride a bike. One student is chosen at random. Give the probability of each way of travelling.

C. True or false?

1. Spinner B has three colours, so each colour has probability 1/3.   True or false, and why? ______________________
2. A bag of 3 red and 3 blue counters gives P(red) = 1/2.   True or false, and why? ______________________
3. A bag of 4 green and 6 yellow marbles gives P(green) = 4/6.   True or false, and why? ______________________
4. A spinner with only red and blue sectors has P(red) = 0.45 and P(blue) = 0.35.   True or false, and why? ______________________

D. Stretch

Predicting

In n trials, an outcome of probability p happens about p × n times. Students predict, test and solve.

We are learning to

• predict how often an outcome, or a related event such as “not green”, should happen in a number of trials,
• explain why a result can differ from the prediction.

Success criteria

• I can predict a count as the probability times the number of trials.
• I can explain why results rarely match a prediction exactly.

You need

The spinners from Day 4 with paper clips and pencils, the worksheet, calculators, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Expecting the predicted count every time, and calling a spinner unfair when it misses by a few.
• Multiplying by the number of outcomes instead of by the probability.
• Giving a relative frequency as a count, such as 18 instead of 18/40.

Answers with working

1. yellow 1/2 × 40 = 20, blue 1/4 × 40 = 10, green 1/4 × 40 = 10; counts near these fit the prediction.
2. P(7 or 8) = 2/8 = 1/4, so about 50; the other 150 are not.
3. A 20, B 15, C 10, D 10, E 5; a vowel (A or E) is 5/12 of the circle, so about 25 times.
4. blue is 5/10 = 1/2, so about 75; not green is 8/10 = 4/5, so about 120.
5. P(0) = 1/10 = 0.1 = 10%, so about 50 zeros; an even digit is 5 of 10, so about 250.
6. not necessarily: 10 is an average and results vary; 6 in 60 is not unusual, so roll many more times.
7. red 20/80 = 0.25, blue 41/80 ≈ 0.51, green 19/80 ≈ 0.24; likely red 1/4 (90°), blue 1/2 (180°), green 1/4 (90°).
8. 0.15 × 50 = 7.5: about 7 or 8 days late and 42 or 43 on time; a prediction is an average, so it need not be whole.

Predicting

For each problem: find the probability, then multiply by the number of trials.

Problems (7 and 8 are the stretch)

1. Predict how many times spinner B lands on each colour in 40 spins. Then spin it 40 times and tally.

   Colour	Probability	Predicted in 40	Tally	Count	Relative frequency
   yellow
   blue
   green

2. A fair 8-sided die numbered 1 to 8 is rolled 200 times. About how many rolls are greater than 6, and how many are not?
3. Spinner C is spun 60 times. Predict how often it lands on each letter, and how often on a vowel.
4. A bag holds 3 red, 5 blue and 2 green counters. A counter is drawn and put back, 150 times. About how many draws are blue, and how many are not green?
5. A fair 10-sided die numbered 0 to 9 is rolled 500 times. Predict the relative frequency of a 0 as a fraction, a decimal and a percentage. About how many zeros, and how many even digits?
6. Ari rolled a fair 6-sided die 60 times and got a six only 6 times. Ari says the die is unfair, because 10 sixes were expected. Do you agree? Explain.
7. Stretch. A spinner was spun 80 times: red 20, blue 41, green 19. Sketch a spinner that could have given these results, and give its angles.
8. Stretch. The probability that the school bus is late on a given day is 0.15. Over 50 school days, predict how many days it is late and how many it is on time. Why need a prediction not be a whole number?

Chance cards

Day 3. Cut out the 12 cards. Lay them in a line on the desk, from impossible to certain.

Spinners to make

Days 4 and 5. Measure the sector angles before you cut. Then cut out each spinner and glue it to light card if you can. To spin, put the point of a pencil through a paper clip at the centre dot, hold the pencil upright and still, and flick the clip. The sector that the end of the clip stops on is the result.

What we know: Sample space and probability

Work on your own. Show your working, and give fractions in lowest terms.

1. A spinner has 6 equal sectors coloured red, red, blue, blue, blue and white. Write the sample space for the colour it lands on.   ____________
2. A spinner has 9 equal sectors numbered 1 to 9. Give the probability of each outcome as a fraction and as a percentage to one decimal place.   ____________
3. A card is drawn at random from cards numbered 1 to 15. List the outcomes that are multiples of 4, and give the probability of a multiple of 4.   ____________
4. The probability that a spinner lands on blue is 3/11. What is the probability that it does not land on blue?   ____________

5. A bag holds 4 red, 7 yellow and 9 green counters. Give the probability of drawing each colour.   ____________
6. A spinner has three sectors of 160°, 120° and 80°. Give the probability of each sector, and check that they add to 1.   ____________
7. A spinner lands on green with probability 1/5. In 250 spins, about how many greens do you expect, and how many spins that are not green?   ____________
8. A bag holds 2 red and 8 white counters. Lin draws a counter and puts it back, 50 times, and gets red 13 times. Lin expected 10. Is something wrong? Explain.

Answers and marking guide

Answers with working

1. {red, blue, white}: each colour once, though red and blue cover more sectors than white.
2. 1/9 for each, which is 11.1% (about 0.111); the 9 shares add to 1.
3. {4, 8, 12}, so 3/15 = 1/5.
4. 1 − 3/11 = 8/11.
5. 20 counters: P(red) = 4/20 = 1/5, P(yellow) = 7/20, P(green) = 9/20.
6. 160/360 = 4/9, 120/360 = 1/3, 80/360 = 2/9, and 4/9 + 3/9 + 2/9 = 9/9 = 1.
7. 1/5 × 250 = 50 greens, and 200 spins that are not green.
8. No. P(red) = 2/10 = 1/5, so about 10 reds is the average to expect, but results vary; 13 is close. The relative frequency 13/50 = 0.26 should settle nearer 0.2 over many more draws.

A quick three-level guide

Eight questions, four ideas. A student at standard lists a sample space with each outcome once, gives probabilities from equal shares, counts and angles, and predicts a count as the probability times the number of trials.

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: sample spaces, equally likely outcomes, events, outcomes not equally likely, and predicting.