- Define and explain Combined outcomes in your own words
- Use key terms such as Probability accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on Combined outcomes: sample space diagrams and listing all outcomes for two-step experiments.
Sample space diagrams and listing all outcomes for two-step experiments.
Key ideas
Tree diagrams
A tree diagram shows every branch of a multi-step experiment with probabilities on each branch. Multiply along a branch for that route's probability; add the routes that satisfy the event. Always check all branches from one point sum to 1.
The addition rule
For mutually exclusive events, P(A or B) = P(A) + P(B). Rolling a 1 or a 6 on a fair die gives 1/6 + 1/6 = 2/6 = 1/3. This only works when the events cannot both happen.
Key term — Probability: A number between 0 and 1 measuring how likely an event is, where 0 means impossible and 1 means certain. For equally likely outcomes it is (favourable outcomes) ÷ (total outcomes).
A bag holds 3 red and 2 blue counters. Two are drawn without replacement. Find the probability both are red.
First draw: P(red) = 3/5. Second draw: one red is gone, so 2 red remain out of 4 counters: P(red given red) = 2/4 = 1/2. Multiply: 3/5 × 1/2 = 3/10.
Answer: 3/10 (0.3 or 30%).
- Multiplying when events are dependent Without replacement the second probability changes — use the reduced totals, or draw a tree diagram.
- Adding probabilities of events that overlap The addition rule needs mutually exclusive events; drawing a red card or a king double-counts the king of hearts.
Practice
P = 3/6 = 1/2.
P = 1/2 × 1/2 = 1/4.
P = 1 − 0.3 = 0.7.
P = 1/4 + 1/4 = 1/2.
Quick check
Which of these best defines "Probability"?
Two dice are rolled. Find P(total of 7).
A bag holds 4 green and 6 yellow sweets. One sweet is taken at random. Find P(green).
- Combined outcomes: sample space diagrams and listing all outcomes for two-step experiments.
- Tree diagrams: A tree diagram shows every branch of a multi-step experiment with probabilities on each branch.
- Mutually exclusive: Two events that cannot happen at the same time, such as rolling a 2 and rolling a 5 on one throw; their probabilities add.
- Watch out for: multiplying when events are dependent