Probability Practice Test
Build probability skills from primary-school chance language through sample spaces, theoretical and experimental probability, multistage events, Venn diagrams, dependent events and Stage 5 further probability.
- 12 topic-based levels
- 102 practice questions
- Instant answer checks
- Hints and full working
Your answers and checked results are saved automatically in this browser.
Level 1
Primary Chance Language & Probability Scale
0/10
Level 2
Outcomes, Events & Sample Spaces
0/10
Level 3
Simple Theoretical Probability & Conversions
0/10
Level 4
Complementary Events
0/8
Level 5
Experimental Probability & Relative Frequency
0/8
Level 6
Organising Sample Spaces & Two-Stage Outcomes
0/8
Level 7
Independent Events
0/8
Level 8
Dependent Events & Without Replacement
0/8
Level 9
Multistage Events & Tree-Diagram Thinking
0/8
Level 10
Mutually Exclusive Events, OR & Overlap
0/8
Level 11
Venn Diagrams & Two-Way Tables
0/8
Level 12
Conditional & Further Probability
0/8
Probability in NSW Mathematics: What Students Need to Know
Probability is the mathematics of chance and uncertainty. Students learn how to describe possible outcomes, calculate how likely events are, compare theoretical results with experiments, and solve increasingly complex problems involving more than one event.
In the current NSW Mathematics K–10 syllabus, probability forms part of the Statistics and probability area. For Years 7–10, probability develops from fundamental ideas such as outcomes and sample spaces into multistage probability. Some Stage 5 pathways extend this further through more advanced probability concepts.
Schools do not necessarily teach every probability idea in exactly the same school year. Focus on understanding the concepts your class is currently studying rather than assuming that a topic belongs only to Year 7, Year 8, Year 9 or Year 10.
1. Probability Basics: Chance, Outcomes, Events and Sample Spaces
A random experiment is an activity where the exact result cannot be known in advance. Rolling a die, tossing a coin or drawing a card are common examples.
- Outcome: one possible result of an experiment.
- Sample space: the complete set of possible outcomes.
- Event: one outcome or a group of outcomes we are interested in.
- Equally likely outcomes: outcomes that have the same chance of occurring.
Sample space = {1, 2, 3, 4, 5, 6}
If event E is “roll an even number”, then: E = {2, 4, 6}.
The Probability Scale
Probability is measured from 0 to 1 inclusive. It can also be written as a fraction, decimal or percentage.
| Probability | Meaning | Example |
|---|---|---|
| 0 | Impossible | Rolling a 7 on a standard six-sided die |
| Close to 0 | Unlikely | Choosing one particular card from a full deck |
| 0.5 | Even chance | Getting heads on a fair coin |
| Close to 1 | Likely | Rolling a number less than 6 on a fair die |
| 1 | Certain | Rolling a number from 1 to 6 on a standard die |
2. Calculating Theoretical Probability
When all outcomes are equally likely, probability can be calculated using:
A fair die has 6 equally likely outcomes.
Even outcomes = {2, 4, 6}, so there are 3 favourable outcomes.
Probability with Playing Cards
A standard deck contains 52 cards: 4 suits, 13 cards in each suit and 4 cards of each rank.
3. Complementary Events
A complementary event describes an event not happening. The probability of an event and its complement must add to 1.
Therefore:
If P(rain) = 0.35:
Sometimes finding the probability of an event directly is difficult. It can be much easier to calculate the probability of the opposite event and subtract it from 1.
4. Experimental Probability and Relative Frequency
Theoretical probability tells us what we expect to happen. Experimental probability uses results collected from an actual experiment or simulation.
A coin is tossed 100 times and lands on heads 46 times.
The theoretical probability is 0.5. An experimental result does not have to match the theoretical probability exactly.
What Happens When the Number of Trials Increases?
With a fair random process, experimental results will often become more stable as the number of trials increases. This is why simulations with many trials can be useful when studying probability.
5. Organising Sample Spaces
For simple experiments, a list may be enough. For multistage experiments, students need systematic ways to make sure no outcomes are missed.
Two Coins
Possible outcomes:
There are 4 equally likely outcomes, so:
Two Dice
Two dice create 36 ordered outcomes. An outcome table is usually more reliable than trying to list all 36 outcomes mentally.
Each die has 3 even numbers out of 6.
6. Independent and Dependent Events
Independent Events
Events are independent when the result of one event does not change the probability of the other.
For independent events:
Dependent Events
Events are dependent when the first event changes the probability of the next event.
This commonly happens when an item is selected and not replaced.
A bag contains 3 red and 2 blue counters.
Probability the first counter is red = 3/5. After taking one red counter out, 2 red counters remain among 4 counters.
7. Multistage Probability and Tree Diagrams
A multistage event involves two or more stages. Examples include tossing several coins, rolling more than one die, or selecting several objects one after another.
Tree diagrams are useful because they display the possible pathways through an experiment.
- Draw a branch for each possible result of the first event.
- Add branches for the possible results of the next event.
- Write the relevant probability on each branch.
- Multiply probabilities along a pathway when all events on that pathway must occur.
- Add separate successful pathways when more than one pathway produces the required result.
The successful pathways are HT and TH.
Each pathway has probability:
There are two successful pathways:
8. Mutually Exclusive Events and the Addition Rule
Two events are mutually exclusive if they cannot happen at the same time in one trial.
When rolling one die, “roll a 2” and “roll a 5” are mutually exclusive. You cannot roll both with the same die at the same time.
For mutually exclusive events:
When Events Can Overlap
If events A and B can both occur, their overlap must not be counted twice.
On a die:
- Even numbers = {2, 4, 6}
- Multiples of 3 = {3, 6}
- The overlap is {6}
The combined event is {2, 3, 4, 6}.
9. Venn Diagrams, Tables and Probability
As probability becomes more complex, diagrams and tables become important tools for organising information.
| Representation | Best used for |
|---|---|
| Outcome list | Small sample spaces |
| Outcome table | Two-stage experiments such as two dice |
| Tree diagram | Sequential or multistage events |
| Venn diagram | Events that overlap or are mutually exclusive |
| Two-way table | Comparing two categories and calculating probabilities from data |
Students should learn to choose a representation that makes the structure of the problem easier to see rather than trying to memorise one method for every question.
10. Conditional Probability and Further Probability
In more advanced probability work, students may consider how knowing that one event has occurred changes the probability of another event. This idea is called conditional probability.
Suppose a card is known to be a picture card. The possible cards are now restricted to the jacks, queens and kings rather than all 52 cards.
Probability is therefore calculated using the new relevant sample space.
Not every student will study the most advanced probability concepts at exactly the same point. NSW Stage 5 pathways can extend Core learning into Further probability, so follow the sequence used by your school and teacher.
11. Theoretical Probability vs Experimental Probability
| Theoretical Probability | Experimental Probability |
|---|---|
| Based on mathematical reasoning | Based on observed results |
| Uses known possible outcomes | Uses data from trials or simulations |
| For a fair coin, P(heads) = 0.5 | A real experiment might produce 48 heads from 100 tosses |
| Does not change from trial to trial when the model stays the same | Can vary between experiments |
12. Common Probability Mistakes
Assuming All Outcomes Are Equally Likely
Before using favourable outcomes divided by total outcomes, check that each basic outcome really has the same probability.
Forgetting the Overlap in an “OR” Question
If A and B can happen together, subtract the overlap so it is not counted twice.
Treating Dependent Events as Independent
If an object is not replaced, the total number of objects and possibly the number of favourable outcomes will change.
Missing Outcomes
Use a table, list or tree diagram for multistage experiments instead of trying to keep every possibility in your head.
Confusing “AND” with “OR”
AND normally means both conditions must be satisfied. OR means at least one of the stated conditions is satisfied.
Giving an Impossible Probability
Every probability must lie between 0 and 1 inclusive. An answer such as 1.25 or −0.3 cannot be a valid probability.
13. A Reliable Method for Solving Probability Questions
- Identify the experiment. What exactly is happening?
- Identify the event. What outcome or group of outcomes does the question want?
- Find or organise the sample space. Use a list, table, Venn diagram or tree diagram if necessary.
- Decide whether the events are independent, dependent or overlapping.
- Choose the appropriate probability rule.
- Simplify the answer and check that it is between 0 and 1.
14. Probability in the NSW Mathematics Syllabus
The NSW Mathematics K–10 syllabus groups Years 7–10 learning into Stage 4 and Stage 5. Probability is included in the Stage 4/5 Core, and the course description specifically includes multistage probability. Stage 5 Paths provide extension through Further probability.
Probability learning should also develop the broader NSW Working mathematically processes: communicating, understanding and fluency, reasoning, and problem solving.
For the official syllabus, see the NSW Mathematics K–10 Syllabus .
Do not study probability as a list of disconnected formulas. NSW mathematics increasingly expects students to reason, select an appropriate method, interpret information and explain solutions.
15. What Should Students Practise?
| Foundation | Core Development | More Advanced |
|---|---|---|
| Probability scale | Complementary events | Dependent events |
| Outcomes and events | Experimental probability | Without replacement |
| Sample spaces | Two-stage experiments | Complex tree diagrams |
| Simple theoretical probability | Independent events | Overlapping events |
| Coins, dice, cards and spinners | Outcome tables and tree diagrams | Conditional and further probability ideas |
Start with the foundation concepts and move forward only when the sample space and basic probability ideas are secure. Most difficult probability questions are built from these same foundations.
Probability Help for NSW Students
Aussie Math Tutor NSW supports students with probability and other Mathematics topics through step-by-step teaching, worked examples and targeted practice.
Face-to-face support is available around Telopea, Oatlands, North Rocks, Dundas, Dundas Valley, Rydalmere, Ermington and Carlingford, with online tutoring available for students across NSW.
If probability feels confusing, the first step is usually to identify whether the difficulty comes from fractions, sample spaces, interpreting the wording of the question, or choosing the correct probability rule.
