- Define and explain Tree diagrams and conditional probability in your own words
- Use key terms such as Conditional 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 Tree diagrams and conditional probability: building tree diagrams for events with and without replacement, and finding conditional probabilities.
Building tree diagrams for events with and without replacement, and finding conditional probabilities.
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 multiplication rule
For independent events, P(A and B) = P(A) × P(B). Two heads on two fair coin tosses gives 1/2 × 1/2 = 1/4. For dependent events such as drawing without replacement, use the new totals after each draw.
Key term — Conditional probability: The probability of an event given that another event has already happened, written P(A given B), used when events are not independent.
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 "Conditional 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).
- Tree diagrams and conditional probability: building tree diagrams for events with and without replacement, and finding conditional probabilities.
- 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