Compound probability
Complementary events, independent 'AND' events, two-way tables, and comparing simulated results with theory
About four lessons of 45 to 60 minutes
What are the odds of flipping heads AND rolling a six?
A single coin flip or a single die roll is simple enough. But most real chance situations involve more than one thing happening at once: a coin AND a die, two spinners, or two survey questions about the same group of people. These are compound events, and they need their own rules.
Four ideas unlock compound probability: complementary events (an event and its opposite always add to 1), independent 'AND' events (multiply the individual probabilities), two-way tables (organise combinations of two categories), and simulation (running many trials to check theory against reality).
- Flipping heads AND rolling a 6two independent events combined, multiply their probabilities
- '70% chance of rain' means what chance of no rain?complementary events always add to 1 (or 100%)
- Students who play sport AND/OR an instrumenta two-way table organises all four combinations
- Spinning a 4-colour spinner twice, 50 times oversimulation checks whether theory matches real results
What students will be able to do
Students will calculate the probability of a complementary event, find the probability of two independent events both occurring, use a two-way table to find probabilities of combined categories, and compare a simulation's relative frequency with the theoretical probability.
- I can find the probability of an event NOT happening, given the probability that it does, since the two always add to 1.
- I can find the probability of two independent events both happening by multiplying their individual probabilities.
- I can read a two-way table to find the probability of a combination of two categories.
- I can compare a simulation's relative frequency to the theoretical probability, and explain why they are rarely identical.
Standards this unit teaches
- AC9M8P01Australian Curriculum v9 (ACARA)Complementary events
Recognise that complementary events have probabilities that add to one, and use this to calculate probabilities in real contexts.
- AC9M8P02Australian Curriculum v9 (ACARA)Combinations for two events
Find all combinations for two events using two-way tables, tree diagrams and Venn diagrams, and use them to work out probabilities.
- AC9M8P03Australian Curriculum v9 (ACARA)Simulate compound events
Run repeated chance experiments and simulations with digital tools to estimate probabilities for compound events, and describe the results.
- AC9M9P01Australian Curriculum v9 (ACARA)Compound events (Year 9 bridge)
List outcomes for compound events with and without replacement using lists, tree diagrams, tables or arrays, and assign probabilities. This unit's independent-events work also reaches toward this Year 9 descriptor.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 7 statistics: mean, median, mode and range teaching unitrelative frequency is itself a kind of average of repeated trials
- Probability in the glossarythe basic 0 to 1 scale this unit builds on
- Number: ratios & scale worksheetssimplifying a probability fraction uses the same skill as simplifying any ratio
Words to teach and display
- Compound event
- an event made of two or more simpler events happening together
- Complementary events
- an event and its opposite (not happening), whose probabilities always add to 1
- Independent events
- events where one happening does not change the probability of the other
- Two-way table
- a table organising data or outcomes by two categories at once, e.g. sport (yes/no) by instrument (yes/no)
- Theoretical probability
- the probability calculated from reasoning about equally likely outcomes, before any trial is run
- Relative frequency
- the number of times an outcome occurred divided by the total number of trials in an actual experiment
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. Complementary events: it always adds to 1
ConcreteEvery event has an opposite: rain or no rain, rolling a 6 or not rolling a 6, passing or failing. These two outcomes cover every possibility, so their probabilities always add to exactly 1 (or 100%).
This means you can always find one probability from the other by subtracting from 1: P(not A) = 1 - P(A). If there is a 35% chance of rain, there must be a 65% chance of no rain, since 35% + 65% = 100%.
A fair six-sided die is rolled. What is the probability of NOT rolling a multiple of 3?
- Multiples of 3 on a die: 3 and 6, so P(multiple of 3) = 2/6 = 1/3.
- Complementary events add to 1: P(not a multiple of 3) = 1 - 1/3.
- 1 - 1/3 = 2/3.
Answer: P(not a multiple of 3) = 2/3.
- Why must P(A) and P(not A) always add to exactly 1?
- If P(a spinner lands on red) = 0.4, what is P(it does not land on red)?
2. Independent events: the 'AND' rule
PictorialWhen two events are independent, one does not affect the other, so the probability of BOTH happening is found by multiplying their individual probabilities.
Flipping a coin and rolling a die are independent: the coin does not know or care what the die shows. P(heads AND rolling a 4) = P(heads) x P(rolling a 4) = 1/2 x 1/6 = 1/12. Out of the 12 equally likely coin-die combinations, exactly 1 is 'heads and 4'.
Spinner A has 4 equal sections and Spinner B has 5 equal sections. Both are spun once. Find the probability of landing on one specific section on BOTH spinners.
- P(specific section on A) = 1/4.
- P(specific section on B) = 1/5.
- Independent events: multiply. 1/4 x 1/5 = 1/20.
Answer: 1/20.
- Why does 'independent' justify multiplying the two probabilities?
- How many total equally likely outcomes are there for the coin-and-die example, and how does that match 1/12?
3. Two-way tables: combinations of two categories
PictorialWhen a group is classified by two categories at once, like playing sport (yes/no) and playing an instrument (yes/no), a two-way table organises every combination so probabilities can be read off directly.
Of 40 students: 8 play both sport and an instrument, 14 play sport only, 7 play an instrument only, and 11 play neither. Every student fits exactly one of these four cells, so the four counts must add to the total.
Using the 40-student data (8 both, 14 sport only, 7 instrument only, 11 neither), find P(plays neither) and P(plays sport in total).
- Check the total: 8 + 14 + 7 + 11 = 40, matching the group size.
- P(plays neither) = 11/40.
- Total who play sport = both + sport only = 8 + 14 = 22, so P(plays sport) = 22/40 = 11/20.
Answer: P(plays neither) = 11/40. P(plays sport) = 11/20.
- Why must the four cell counts in a two-way table add up to the total group size?
- How do you find 'plays sport in total' from a two-way table, if only the four combination cells are given?
4. Simulating compound events and comparing with theory
AbstractTheoretical probability is calculated by reasoning; a simulation instead runs many real (or digital) trials and counts what actually happens. The two usually agree closely but rarely match exactly, because of chance variation.
A 4-colour spinner (red, blue, green, yellow, all equal) is spun twice. Theoretically, P(same colour both times) = 4 x (1/4 x 1/4) = 4/16 = 1/4, since there are 4 ways to match (RR, BB, GG, YY) out of 16 total pairs. In an actual simulation of 40 trials, 'same colour twice' happened 11 times: a relative frequency of 11/40 = 0.275, close to the theoretical 0.25 but not identical.
In a simulation, a fair coin was flipped twice, 50 times over, and 'two heads in a row' occurred 11 times. Find the relative frequency and compare it with the theoretical probability.
- Relative frequency = 11/50 = 0.22.
- Theoretical probability: P(heads) x P(heads) = 1/2 x 1/2 = 1/4 = 0.25.
- 0.22 is close to 0.25, but not exactly equal, which is expected chance variation with only 50 trials.
Answer: Relative frequency = 0.22, theoretical probability = 0.25. They are close, as expected.
- Why would running 500 trials instead of 50 usually bring the relative frequency closer to the theoretical probability?
- Does a simulation result different from the theoretical probability mean the theory is wrong?
Common misconceptions and how to address them
MisconceptionP(A) and P(not A) only add to 1 if the event is 'fair' (like a coin).
Why it happens: Students first meet complementary events with fair examples and assume the rule needs fairness.
How to address it: The complement rule works for ANY event, fair or not, because 'A happens' and 'A does not happen' cover every possibility no matter what A is. A 70% chance of rain always means a 30% chance of no rain.
MisconceptionFor independent events, you ADD the probabilities to find 'both happening'.
Why it happens: Students confuse the rule for 'either event' (which involves addition) with the rule for 'both events' (which involves multiplication).
How to address it: 'AND' (both happening) multiplies; a different rule ('OR', either happening) uses addition. Keep the two rules and their trigger words, AND versus OR, clearly separate.
MisconceptionA simulation result that differs from the theoretical probability means the simulation was done wrong.
Why it happens: Students expect experimental results to match theory exactly, not accounting for natural chance variation.
How to address it: Small differences between a simulation's relative frequency and the theoretical probability are normal and expected, especially with fewer trials. More trials generally bring the two closer together, but they rarely match exactly.
Guided practice (with answers)
1. P(a spinner lands on blue) = 3/8. Find P(it does not land on blue).
Answer: 5/8, because 1 - 3/8 = 5/8.
2. A coin is flipped and a 6-sided die is rolled. Find P(tails AND rolling a 2).
Answer: 1/12, because P(tails) x P(rolling a 2) = 1/2 x 1/6 = 1/12.
3. Spinner A has 3 equal sections and Spinner B has 6 equal sections. Find P(a specific section on both).
Answer: 1/18, because 1/3 x 1/6 = 1/18.
4. A two-way table of 50 people shows: 12 like tea and coffee, 18 like tea only, 9 like coffee only, 11 like neither. Find P(likes neither).
Answer: 11/50, because 12 + 18 + 9 + 11 = 50, and 11 people like neither out of 50.
5. In 100 simulated trials, an event predicted to have theoretical probability 0.2 occurred 24 times. Find the relative frequency and compare it with the theoretical probability.
Answer: Relative frequency = 24/100 = 0.24, close to the theoretical 0.2, which is the expected kind of chance variation.
Independent practice worksheets
Practise complementary and compound events with computed, never-wrong answer keys.
Differentiation
- Start with complementary events using simple, familiar contexts (a coin, a single die) before compound 'AND' events.
- List every outcome of a small compound event by hand (e.g. all 12 coin-die pairs) before relying on the multiplication rule.
- Provide a blank two-way table template with the four cells and both totals labelled, ready to fill in.
- Use small trial counts (10 or 20) for simulation examples so counting relative frequency is quick and concrete.
- Introduce dependent (without replacement) compound events, where the first outcome changes the probability of the second.
- Ask students to design and run their own simple simulation (e.g. 30 coin-die trials) and compare their relative frequency with theory.
- Explore three-category two-way (or three-way) breakdowns for a richer real data set.
- Investigate how many trials are needed, roughly, before a simulation's relative frequency reliably lands within 5 percentage points of the theoretical probability.
Assessment: exit ticket
A three-question exit ticket sampling complementary events, the 'AND' rule, and a two-way table.
1. P(a bus arrives late) = 0.15. Find P(it does not arrive late).
Answer: 0.85, because 1 - 0.15 = 0.85.
2. A coin is flipped and an 8-sided die is rolled. Find P(heads AND rolling an 8).
Answer: 1/16, because P(heads) x P(rolling an 8) = 1/2 x 1/8 = 1/16.
3. A two-way table of 60 students shows: 20 both walk to school and bring lunch, 15 walk only, 10 bring lunch only, 15 neither. Find P(neither).
Answer: 1/4, because 20 + 15 + 10 + 15 = 60, and 15/60 = 1/4.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 complementary events (section 1), Lesson 2 independent 'AND' events (section 2), Lesson 3 two-way tables (section 3), Lesson 4 simulation versus theory plus the exit ticket (section 4 and assessment).
- This unit assumes comfort with basic single-event probability as a fraction (Year 7 sample space unit). Revisit that first if fractions-as-probabilities feel shaky.
- Language to repeat: complementary events add to 1; 'AND' for independent events MULTIPLIES; a two-way table's four cells always sum to the total; a simulation's relative frequency approaches, but rarely exactly equals, the theoretical probability.
- Curriculum note: AC9M8P01-03 (Australian Curriculum v9) cover complementary events, two-way-table combinations, and simulation at Year 8; AC9M9P01 extends compound events (with and without replacement) at Year 9. This unit's independent-events work (section 2) reaches toward that Year 9 descriptor, though without-replacement (dependent) events are flagged as extension work, not core content here.
- Present and print both work: use the Print button for a clean handout, or build the spinner sample-space grid live on the board with the class before calculating the probability.