Determining Probability of Independent Events Using Tree Diagrams

By the end of this lesson, learners should be able to

1. Construct two-stage tree diagrams to represent sequential independent events.
2. Calculate combined probabilities along tree diagram branches using the multiplication rule.
3. Determine probabilities of compound events by adding probabilities of alternate successful branch paths.

Key vocabulary

Tree diagram

A visual diagram showing sequential event steps along branching lines with assigned probabilities.

We construct a tree diagram to display outcomes when flipping a coin twice.

Independent events

Events where the occurrence of one outcome does not affect another trial's probability.

Getting heads on a second coin flip is independent of the first flip.

Replacement

Putting an item back into a set before making a subsequent random selection.

Returning a drawn counter before the next pick keeps the total count constant.

Outcome

A possible single result obtained from a random experiment or multi-stage process.

Obtaining two heads is a single outcome when flipping a coin twice.

Lesson preview

Slide 1 · Introduction to Tree Diagrams in Independent TrialsOpening · 12 min

A tree diagram displays sequential random events as branching lines starting from a single node. Each stage represents one trial, and every branch displays its associated probability. When events are independent, such as repeated coin flips, the probability of an outcome at any step does not depend on previous outcomes. To evaluate the probability of a specific sequence, we multiply the branch probabilities along that path from start to finish. See Figure 1 for an example of a tree diagram showing two independent coin flips.

• A tree diagram branches out from a single starting point to visually represent sequential event outcomes.
• Each branch carries the individual probability of that specific outcome occurring in that trial stage.
• For independent events, the probability along a path is found by multiplying branch probabilities together.
• All final branch probabilities originating from a single node must sum up to exactly one.

Worked example. Step 1: Construct the first stage with two branches from origin $O$: $H$ with probability $\frac{1}{2}$ and $T$ with probability $\frac{1}{2}$. Step 2: From branch $H$, draw two second-stage branches: $H$ ($\frac{1}{2}$) and $T$ ($\frac{1}{2}$). Repeat for branch $T$. Step 3: Identify the path for outcome $HH$, which follows the upper branch twice. Step 4: Multiply probabilities along the $HH$ path: $P(HH)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$. Answer: $P(HH)=\frac{1}{4}$.

Check question: A fair coin is flipped twice. What is the probability of getting one head and one tail in any order?

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $\frac{3}{4}$
4. $1$

Show answer

$\frac{1}{2}$. The paths on a tree diagram that give one head and one tail are $HT$ and $TH$. The probability of each path is $\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$. Adding the probabilities of these mutually exclusive paths yields $\frac{1}{4}+\frac{1}{4}=\frac{2}{4}=\frac{1}{2}$.

Slide 2 · Calculating Probabilities for Compound Independent EventsModel · 14 min

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• Homework and the other 11 lessons in this sub-strand

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All 12 lessons in Probability 1

1. 1Performing Probability Experiments and Finding Experimental ProbabilityPerform experiments involving probabilities in different situations
2. 2Empirical Probabilities and Experimental InvestigationsPerform experiments involving probabilities in different situations
3. 3Identifying Range of Probability and Complementary EventsIdentify the range of probability in different situations
4. 4Generating Probability Space of EventsGenerate probability space of different events
5. 5Generating Probability Space for Compound EventsGenerate probability space of different events
6. 6Determining Probability of Mutually Exclusive Events and Introduction to Independent EventsDetermine the probability of mutually exclusive events and independent events
7. 7Determining Probability of Independent Events and Multi-Stage SelectionsDetermine the probability of mutually exclusive events and independent events
8. 8Probability Bounds, Complementary Events, and the Addition LawApply the laws of probability in different situations
9. 9Applying Laws of ProbabilityApply the laws of probability in different situations
10. 10Determining Probability of Independent Events Using Tree DiagramsDetermine the probability of independent events using tree diagrams
11. 11Tree Diagrams for Multi-Stage Independent EventsDetermine the probability of independent events using tree diagrams
12. 12Applications of Probability in Real-Life ContextsAppreciate the application of probability in real-life situations