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 and
• 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 Solve the probability of an event, compare favourable outcomes with total outcomes
• In ordered outcomes, the position of each result matters, for example is different from

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 , and is spun and then a fair coin is tossed. Use a tree diagram to find the probability of getting 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 and . One card is chosen, replaced, and then another card is chosen. Use a tree diagram to find the probability of getting .

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 , and is spun and then a fair coin is tossed. Use a tree diagram to find the probability of getting 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 and . One card is chosen, replaced, and then another card is chosen. Use a tree diagram to find the probability of getting .

Exercises

Understanding and Fluency

Exercise 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 , ,

Exercise 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 or , then toss a coin

Exercise 3.

A coin is tossed twice. Find each probability Use a tree diagram.

a)

b)

c)

Exercise 4.

A coin is tossed three times. Find each probability using a tree diagram.

a)

b)

c)

Exercise 5.

A spinner with equal sections labelled , and is spun, then a fair coin is tossed. Find each probability using a tree diagram.

a)

b)

c)

Exercise 6.

A fair die is rolled and then a fair coin is tossed. Find each probability using a tree diagram.

a)

b)

c)

Exercise 7.

A fair die is rolled twice. Find each probability using a tree diagram.

a) doubles

b) a total of

c) first roll and second roll

Exercise 8.

A bag contains cards labelled , and . One card is chosen, replaced, and then another card is chosen. Find each probability using a tree diagram.

a)

b) one then one

c) two matching letters

Reasoning

Exercise 9.

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

Exercise 10.

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

Exercise 11.

Explain why and are different outcomes in a tree diagram for two coin tosses.

Exercise 12.

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

Exercise 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

Exercise 14.

A game uses a fair coin tossed three times.

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

Exercise 15.

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

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

Exercise 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.

Exercise 17.

A fair coin is tossed three times.

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

Exercise 17.

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 and 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

Next: 167. Two-Way Tables and Venn Diagrams