Statistics & probability · Grade 8
Probability Models and Compound Events
Probability measures how likely an event is. A model helps list possible outcomes and calculate probabilities consistently.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Understand the key terms and formulas.
- Apply the method to worked and new examples.
- Check answers and explain the reasoning.
1. Probability scale
Probability ranges from 0 (impossible) to 1 (certain). A probability of ½ means equally likely to happen or not happen. For equally likely outcomes, P(event) = favourable outcomes ÷ total outcomes.
2. Sample spaces
A sample space lists every possible outcome. For a fair six-sided die it is {1, 2, 3, 4, 5, 6}. Listing outcomes prevents omissions and double counting.
3. Complementary events
An event either happens or does not happen. P(not A) = 1 − P(A). For example, if P(rain) = 0.3, P(no rain) = 0.7.
4. Independent events
Two events are independent when the result of one does not change the probability of the other. For independent events, P(A and B) = P(A) × P(B). Example: two separate fair coin tosses.
5. Tree diagrams and caution
A tree diagram shows stages and their probabilities. Multiply along a path for an 'and' outcome; add probabilities for separate non-overlapping paths. Without replacement, probabilities change after the first selection.
Understand compound events
For independent events, the outcome of the first event does not change the probability of the second, so multiply the probabilities to find the chance that both happen. For mutually exclusive events, which cannot happen together, add their probabilities to find the chance that either happens. When an item is not replaced, the total number of possible outcomes changes and the events are dependent.
Worked reasoning: A bag contains 3 red and 2 blue counters. Two counters are drawn without replacement. The probability of drawing red twice is 3/5 × 2/4 = 6/20 = 3/10. The second fraction is 2/4 because one red counter has already been removed.
For two independent coin tosses, the probability of two heads is 1/2 × 1/2 = 1/4. A tree diagram helps show each possible path and its probability.
- Check whether the first event changes the sample space.
- Use multiplication for “and” when the events are independent.
- Use addition for “either” only after checking whether the events overlap.
Worked examples
- A fair die: P(even) = 3/6 = 1/2.
- A fair coin: P(two heads in two tosses) = 1/2 × 1/2 = 1/4.
- If P(A) = 0.65, P(not A) = 1 − 0.65 = 0.35.
Independent practice
Attempt each problem before reviewing the worked answer.
- Find P(rolling a 5) on a fair die. Answer: 1/6.
- Find P(rolling an odd number). Answer: 3/6 = 1/2.
- If P(A) = 0.2, find P(not A). Answer: 0.8.
- Find P(two heads) in two independent fair coin tosses. Answer: 1/4.
- How many outcomes are in the sample space for two coin tosses? Answer: 4: HH, HT, TH, TT.
- Find P(sum is 7) when rolling two fair dice. Answer: 6/36 = 1/6.
- What does mutually exclusive mean? Answer: Events cannot happen at the same time.
- Why do probabilities change without replacement? Answer: The total and favourable items remaining may change.
Knowledge check
Choose one answer for each question, then check for explanations.
Printable worksheet and answer key
Use the independent-practice questions above as a worksheet. Show each step, include units where appropriate, and check your answers after attempting the full set.
Study reminder
Write down the formula or reasoning used for each answer.