166. Tree Diagrams for Multi-Step Probability

Learning Intentions

• Understand that a tree can be used to list the outcomes of experiments involving two or more steps
• Use a tree diagram to determine the probability of events in multi-step events

Pre-requisite Summary

• A probability is a number between $0$ and $1$
• An experiment is a chance process made up of one or more steps
• An outcome is one possible result, and an event is a set of outcomes
• The sample space is the set of all possible outcomes
• For a multi-step experiment, each step can lead to several new branches
• A tree diagram can be used to organise all possible outcomes clearly
• To find the probability of an event, compare favourable outcomes with total outcomes
• In ordered outcomes, the position of each result matters, for example $HT$ is different from $TH$

Worked Examples

Worked Example 1
A coin is tossed twice. Draw a tree diagram and list the sample space.

Worked Example 2
A coin is tossed twice. Use a tree diagram to find the probability of getting exactly one head.

Worked Example 3
A spinner with equal sections labelled $R$, $G$ and $B$ is spun and then a fair coin is tossed. Use a tree diagram to find the probability of getting $G$ and tails.

Worked Example 4
A fair die is rolled and then a coin is tossed. Use a tree diagram to find the probability of rolling an even number and then getting heads.

Worked Example 5
A fair coin is tossed three times. Use a tree diagram to find the probability of getting exactly two heads.

Worked Example 6
A bag contains cards labelled $A$ and $B$. One card is chosen, replaced, and then another card is chosen. Use a tree diagram to find the probability of getting $AB$.

Problems

Problem 1
A coin is tossed twice. Draw a tree diagram and list the sample space.

Problem 2
A coin is tossed twice. Use a tree diagram to find the probability of getting exactly one tail.

Problem 3
A spinner with equal sections labelled $R$, $G$ and $B$ is spun and then a fair coin is tossed. Use a tree diagram to find the probability of getting $B$ and heads.

Problem 4
A fair die is rolled and then a coin is tossed. Use a tree diagram to find the probability of rolling an odd number and then getting tails.

Problem 5
A fair coin is tossed three times. Use a tree diagram to find the probability of getting exactly two tails.

Problem 6
A bag contains cards labelled $A$ and $B$. One card is chosen, replaced, and then another card is chosen. Use a tree diagram to find the probability of getting $BA$.

Exercises

Understanding and Fluency

1. For each experiment, state how many branches come from the first step.
   a) tossing a coin
   b) rolling a fair six-sided die
   c) spinning a spinner with sections $R$, $G$, $B$

2. Draw a tree diagram and list the sample space for each experiment.
   a) toss a coin twice
   b) toss a coin three times
   c) choose $A$ or $B$, then toss a coin

3. A coin is tossed twice. Find each probability using a tree diagram.
   a) $P(HH)$
   b) $P(\text{exactly one head})$
   c) $P(\text{at least one tail})$

4. A coin is tossed three times. Find each probability using a tree diagram.
   a) $P(HHH)$
   b) $P(\text{exactly two heads})$
   c) $P(\text{no heads})$

5. A spinner with equal sections labelled $R$, $G$ and $B$ is spun, then a fair coin is tossed. Find each probability using a tree diagram.
   a) $P(R\text{ and H})$
   b) $P(B\text{ and T})$
   c) $P(G\text{ and H})$

6. A fair die is rolled and then a fair coin is tossed. Find each probability using a tree diagram.
   a) $P(4\text{ and H})$
   b) $P(\text{even and T})$
   c) $P(\text{number greater than }4\text{ and H})$

7. A fair die is rolled twice. Find each probability using a tree diagram.
   a) doubles
   b) a total of $6$
   c) first roll $=2$ and second roll $=5$

8. A bag contains cards labelled $A$, $B$ and $C$. One card is chosen, replaced, and then another card is chosen. Find each probability using a tree diagram.
   a) $AA$
   b) one $B$ then one $C$
   c) two matching letters

Reasoning

9. Explain why a tree diagram is useful for experiments with two or more steps.

10. A student draws a tree for tossing a coin twice but lists only $HH$ and $TT$. Explain the error.

11. Explain why $HT$ and $TH$ are different outcomes in a tree diagram for two coin tosses.

12. A student says that three coin tosses have $6$ outcomes because there are $3$ tosses and $2$ results each time. Explain why this is incorrect.

13. Explain why the total number of outcomes in a multi-step experiment can often be found by multiplying the number of outcomes at each step.

Problem-solving

14. A game uses a fair coin tossed three times.

    Use a tree diagram to find the probability of getting exactly one head.

15. A fair die is rolled and then a spinner with equal sections labelled $A$, $B$, $C$ is spun.

    Use a tree diagram to find the probability of rolling a number less than $3$ and then landing on $B$.

16. A bag contains two colours of counters, red and blue. One counter is chosen, replaced, and then another is chosen.

    Use a tree diagram to find the probability of getting one red and one blue in any order.

17. A fair coin is tossed three times.

    Use a tree diagram to find the probability of getting at least two heads.

18. A spinner with equal sections labelled red, yellow and green is spun twice.

    Use a tree diagram to find the probability of getting the same colour both times.

Potential Misunderstandings

• Thinking a tree diagram is only for two-step experiments and cannot be extended to more steps
• Forgetting to include all branches at each stage of the tree
• Treating outcomes such as $HT$ and $TH$ as the same when order matters
• Counting outcomes incorrectly by adding branches instead of following complete paths
• Missing some final outcomes because the tree is unfinished
• Confusing a single path in the tree with the whole event
• Using the number of favourable outcomes as the denominator instead of the total number of complete outcomes
• Forgetting that replacement changes whether the branches stay the same from one step to the next
• Believing that a tree diagram changes the probability, rather than simply organising the outcomes clearly