Probability: the 0-1 scale, sample spaces and enumerating sets
Simple and combined events, theoretical probability from a sample space, and reading a 2-set Venn diagram
About three lessons of 45 to 60 minutes
From 'no chance' to 'certain', and everywhere in between
Weather forecasters, insurers and games designers all put a number on how likely something is, from 0 (impossible) to 1 (certain). Learning to read and place a probability on that scale, and to count outcomes systematically rather than guess, turns 'maybe' into a real, checkable number.
This unit builds 3 connected skills: finding a simple probability as a reduced fraction and placing it on the 0-1 scale, building a full sample space for 2 combined events (like 2 dice or 2 spinners) to find theoretical probabilities, and reading a 2-set Venn diagram to enumerate every combination systematically.
- P(picking a red counter from a bag of 12) = 5/12a reduced fraction, not a decimal guess
- 2 dice added together: 36 equally likely outcomesthe full sample space, listed systematically
- '70% chance of rain' sits between an even chance and certainthe 0-1 scale gives probability a common language
- A survey of tea and coffee drinkers, in a Venn diagram4 counts (both, only 1, only the other, neither) enumerate every person exactly once
What students will be able to do
Students will find simple event probabilities as reduced fractions and place them on the 0-1 probability scale, build the sample space for 2 combined events and use it to calculate theoretical probabilities, and use the 4 counts of a 2-set Venn diagram to enumerate outcomes systematically.
- I can find a simple probability as a fraction in its simplest form, using a real gcd reduction.
- I can find the probability of an event NOT happening, since it always adds to 1 with the event happening.
- I can place a probability on the 0-1 scale: impossible, unlikely, an even chance, likely, or certain.
- I can build the full sample space for 2 combined events and use it to find a theoretical probability.
- I can use the 4 counts of a 2-set Venn diagram (both, only 1 set, only the other, neither) to find a probability.
Standards this unit teaches
- KS3 Maths: ProbabilityUK National Curriculum (England)Probability
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Probability" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale"; "understand that the probabilities of all possible outcomes sum to 1"; "enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams"; "generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Probability in the glossarythe basic 0 to 1 scale this unit builds on
- Number: fractions & simplifying worksheetssimplifying a probability fraction uses the same skill as simplifying any fraction
- Probability worksheets (all grades)revisit basic single-event probability first if that foundation is shaky
Words to teach and display
- Probability scale
- a number line from 0 (impossible) to 1 (certain) that every probability sits on, with 1/2 marking an even chance
- Complementary events
- an event and its opposite (not happening), whose probabilities always add to exactly 1
- Sample space
- the complete list of every possible equally likely outcome of an experiment or combination of experiments
- Theoretical probability
- the probability calculated by reasoning about equally likely outcomes in the sample space, before any trial is run
- Venn diagram
- a diagram of overlapping circles showing which items belong to 1 set, both sets, or neither
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The probability scale and simple events
ConcreteEvery probability is a number between 0 and 1 (or 0% and 100%), where 0 means an event is impossible, 1 means it is certain, and 1/2 marks an even chance, exactly as likely to happen as not.
A simple event's probability is found as favourable outcomes / total outcomes, always reduced to its simplest form using a real gcd (greatest common divisor), never left as an unreduced or decimal-approximated fraction. Complementary events (an outcome and its opposite) always add to exactly 1, so P(not A) = 1 - P(A).
A bag contains 3 red counters, 4 blue counters and 5 green counters. One is picked at random. Find P(red) in its simplest form, then find P(not red).
- Total counters = 3 + 4 + 5 = 12.
- P(red) = 3/12. gcd(3, 12) = 3, so divide top and bottom by 3: 1/4.
- P(not red) = 1 - P(red) = 1 - 1/4 = 3/4.
Answer: P(red) = 1/4. P(not red) = 3/4.
- A spinner has 8 equal sections, 3 of them yellow. Find P(yellow) in its simplest form.
- Why must P(A) and P(not A) always add to exactly 1, whatever the event A is?
2. Sample spaces for combined events
PictorialWhen 2 events happen together, like spinning 2 spinners or rolling 2 dice, the sample space is every possible PAIR of outcomes. If the first event has m equally likely outcomes and the second has n, the sample space has exactly m x n equally likely pairs.
For 2 fair 6-sided dice, the sample space has 6 x 6 = 36 equally likely pairs. To find P(a specific total), count how many of the 36 pairs actually add to that total, then write favourable / 36 in its simplest form.
2 fair 6-sided dice are rolled and their scores are added. Find P(the total is 4).
- List every pair (first die, second die) that adds to 4: (1, 3), (2, 2), (3, 1). That is 3 pairs.
- The full sample space has 6 x 6 = 36 equally likely pairs.
- P(total = 4) = 3/36. gcd(3, 36) = 3, so divide top and bottom by 3: 1/12.
Answer: P(total = 4) = 1/12.
- Spinner A has 5 equal sections and Spinner B has 4 equal sections. How many outcomes are in the sample space?
- Why does a total of 7 have MORE ways to happen on 2 dice than a total of 2 or 12?
3. Enumerating sets with a Venn diagram
AbstractWhen a group is classified by 2 overlapping categories at once, like liking tea and liking coffee, a Venn diagram (2 overlapping circles) organises every person into exactly 1 of 4 regions: both, only the first set, only the second set, or neither.
Since every person fits exactly 1 of the 4 regions, the 4 counts always add up to the total group size, a useful check before calculating any probability from the diagram.
A survey of 40 people found: 8 like both tea and coffee, 14 like only tea, 7 like only coffee, and 11 like neither. Find P(likes tea in total, including both) and P(likes neither).
- Check the total: 8 + 14 + 7 + 11 = 40, matching the survey size.
- Likes tea in total = only tea + both = 14 + 8 = 22, so P(likes tea) = 22/40. gcd(22, 40) = 2, so 11/20.
- P(likes neither) = 11/40 (already in simplest form, since gcd(11, 40) = 1).
Answer: P(likes tea) = 11/20. P(likes neither) = 11/40.
- A survey of 30 people found: 5 like both dogs and cats, 12 like only dogs, 9 like only cats, 4 like neither. Find P(likes exactly one of dogs or cats).
- Why must the 4 regions of a 2-set Venn diagram always add up to the total group size?
Common misconceptions and how to address them
MisconceptionA probability of 0.5 written as a fraction must be 1/2, but 0.5 itself is somehow a different, bigger number.
Why it happens: Students treat fractions, decimals and the word 'evens' as 3 unrelated ideas rather than 3 names for the same point on the scale.
How to address it: Mark the SAME point on the number line 3 ways at once: 1/2, 0.5, and 'an even chance', so students see they are identical, not competing answers.
MisconceptionThe sample space for 2 combined events is found by ADDING the number of outcomes for each event, not multiplying.
Why it happens: Students over-generalise from single-event counting, where you simply count a list, to combined events, where every pairing must be considered.
How to address it: List a small example by hand first (e.g. all 6 outcomes of a coin and a 3-sided spinner: H1, H2, H3, T1, T2, T3) so the multiplication m x n is SEEN, not just stated as a rule.
MisconceptionOn 2 dice, every total from 2 to 12 is equally likely, since there are 11 possible totals.
Why it happens: Students count the number of DIFFERENT totals rather than the number of WAYS each total can be made.
How to address it: Show the full 36-pair sample space: a total of 7 can be made 6 different ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), but a total of 2 can only be made 1 way (1+1), so they are NOT equally likely.
MisconceptionIn a Venn diagram, 'likes tea' only means the region that does not overlap with coffee.
Why it happens: Students read the non-overlapping region as the whole of 'likes tea', forgetting the overlapping middle region also counts.
How to address it: Always ask 'in total, including both?' out loud, and physically point to BOTH regions (only tea, and the overlap) before adding their counts together.
Guided practice (with answers)
1. A bag has 6 red and 4 blue counters. Find P(blue) in its simplest form.
Answer: 2/5, because 4/10 reduces by gcd 2 to 2/5.
2. P(a spinner lands on green) = 3/8. Find P(it does not land on green).
Answer: 5/8, because 1 - 3/8 = 5/8.
3. A probability is 9/10. Where does it sit on the 0-1 scale?
Answer: Likely, because 9/10 is more than 1/2 but not equal to 1.
4. A coin is flipped and a 6-sided die is rolled. How many outcomes are in the sample space?
Answer: 12, because 2 x 6 = 12.
5. 2 fair 6-sided dice are rolled and added. Find P(the total is 12).
Answer: 1/36, because only (6, 6) makes 12, and there are 36 pairs in total.
6. A survey of 50 people found 10 like both walking and cycling, 20 like only walking, 12 like only cycling, and 8 like neither. Find P(likes neither).
Answer: 4/25, because 8/50 reduces by gcd 2 to 4/25.
Independent practice worksheets
Practise probability with computed, never-wrong answer keys.
Differentiation
- Keep a printed 0-1 number line with 'impossible', 'evens' and 'certain' labelled, for quick reference when classifying a probability.
- For combined events, physically list every pair for small sample spaces (up to about 12 outcomes) before trusting the m x n shortcut.
- Provide a blank 2-set Venn diagram template with the 4 regions already outlined, ready to fill in from worded data.
- Practise fraction simplification (finding a gcd) in isolation first if that arithmetic step is shaky.
- Investigate 3-event combined sample spaces (e.g. 3 coins), and find the pattern for how many total outcomes m x n x p gives.
- Explore a 3-set Venn diagram (8 regions) for a richer real data set.
- Compare theoretical probability (calculated from the sample space) with an actual simulation of many trials, and discuss why they are rarely identical.
- Research how a games designer uses probability to balance a dice-based board game, using real sample-space reasoning.
Assessment: exit ticket
A three-question exit ticket sampling simple-event probability, a combined-event sample space, and a Venn diagram.
1. A bag has 4 red and 8 blue counters. Find P(red) in its simplest form.
Answer: 1/3, because 4/12 reduces by gcd 4 to 1/3.
2. 2 fair 6-sided dice are rolled and added. Find P(the total is 9).
Answer: 1/9, because 4 pairs make 9 (3+6, 4+5, 5+4, 6+3) out of 36, and 4/36 reduces by gcd 4 to 1/9.
3. A survey of 20 people found 4 like both jazz and rock, 8 like only jazz, 5 like only rock, and 3 like neither. Find P(likes jazz in total).
Answer: 3/5, because (4 + 8)/20 = 12/20 reduces by gcd 4 to 3/5.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 the probability scale and simple events (section 1), Lesson 2 sample spaces for combined events (section 2), Lesson 3 Venn diagrams plus the exit ticket (section 3).
- Every probability in the matching worksheets is an exact reduced fraction from a real gcd() reduction, never a decimal approximation of a fraction, matching this site's 'never wrong' answer-key policy.
- This unit sits alongside the existing generic compound-probability unit (year8Year9CompoundProbability, Common Core/ACARA-cited): that unit's focus is complementary events, the independent 'AND' rule, 2-way tables and simulation-versus-theory; this unit's focus, cited to the UK National Curriculum, is the 0-1 scale, systematic sample-space enumeration for single and combined events, and Venn diagrams. The 2 units share some underlying ideas (as any 2 probability units inevitably will) but teach genuinely distinct worked examples and worksheets.
- Language to keep repeating: probabilities are fractions in SIMPLEST form; a sample space for 2 combined events multiplies, it does not add; a Venn diagram's 4 regions always sum to the total group size.
- Present and print both work: build the dice sample-space grid and the probability-scale number line live on the board with the class, then print the 3 linked worksheets for independent practice.