Math resources Statistics and probability

Probability

Tree diagram probability

# Tree diagram probability

Here you will learn about tree diagrams in probability, including what they are and how to complete them. You will also look at calculating probabilities using tree diagrams.

Students will first learn about tree diagram probability as part of statistics and probability in 7 th grade and continue to learn about it in high school.

## What is tree diagram probability?

Tree diagram probability is a way of organizing the information for two or more probability events. Probability tree diagrams show all the possible outcomes of the events and can be used to solve probability questions.

A simple tree diagram has branches that match each outcome.

For example,

A coin is flipped and a dice is rolled.

What is the probability of getting a ‘tail’ and a ‘6’?

The first event is flipping the coin. The two possible outcomes are ‘heads’ and ‘tails’.

These are mutually exclusive events. They cannot happen at the same time.

The second event is rolling the dice. The possible outcomes are ‘1’, ‘2’, ‘3’, ‘4’, ‘5’ and ‘6’.

The tree diagram shows 12 possible outcomes and each has an equal chance. Only one of the outcomes is tails and then 6, so the probability is \cfrac{1}{12}.

To avoid writing out each outcome, you can create a more efficient tree diagram that utilizes probability calculations and rules.

For example,

A coin is flipped and a dice is rolled.

What is the probability of getting a ‘tail’ and a ‘6’?

Since the question is only interested in ‘6’, so you can have a ‘6’ branch and a ‘not a 6 ’ branch.

These outcomes can occur whether the coin landed on heads or tails, so add these outcomes to the end of both branches, with the probabilities written along the branches.

The probabilities of the events can be written as fractions or decimals. For individual events occurring the probabilities on each set of branches add up to \bf{1}.

The probability of getting a ‘6’ is \cfrac{1}{6}.

The probability of getting ‘not a 6 ’ will be 1-\cfrac{1}{6} = \cfrac{5}{6}.

Remember that the probabilities on each set of branches add up to \bf{1}.

You want the probability of getting a tail and a 6 , so follow the path that shows a tail and a 6.

The AND rule for probability states that for independent events,

\text{P(A and B) }=\text{P(A)} \times \text{P(B)}

Taking the probabilities from the corresponding branches of the tree diagram, you get:

The probability of getting a ‘tail’ and a ‘6’ is:

\cfrac{1}{2}\times \cfrac{1}{6}=\cfrac{1}{12}

## Common Core State Standards

How does this relate to 7 th grade math and high school math?

• Grade 7 – Statistics and Probability (7.SP.C.8)
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.6)
Find the conditional probability of A given B as the fraction of B’ s outcomes that also belong to A, and interpret the answer in terms of the model.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.7)
Apply the Addition Rule, P(A \text { or } B) = P(A) + P(B) \, – \, P(A \text { and } B), and interpret the answer in terms of the model.

• Statistics and Probability – Conditional Probability and the rules of Probability (HS.S-CP.B.8)
Apply the general Multiplication Rule in a uniform probability model, P(A \text { and } B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

## How to draw a simple tree diagram to find probability

In order to use a tree diagram to find probability:

1. Create a set of branches for the first event, showing all possible outcomes.
2. Continue to add sets of branches for each event, showing all possible outcomes.
3. Find the probability of those outcomes by creating a ratio.

## Probability tree diagrams examples

### Example 1: two independent events

The spinner above is spun twice.

Show a tree diagram for all possible outcomes.

Then state the probability for spinning red both times.

1. Create a set of branches for the first event, showing all possible outcomes.

The area of each color is the same on the spinner, so each outcome has the same probability of being spun.

2Continue to add sets of branches for each event, showing all possible outcomes.

3Find the probability of those outcomes by creating a ratio.

\text { Probability }=\cfrac{\text { desired outcome }}{\text { total possible outcomes }}=\cfrac{1}{16}

### Example 2: two independent events

A marble is removed at random from the bag above and the color noted.

The marble is replaced.

A second marble is removed at random and the color is noted.

Complete the tree diagram.

Calculate the probability that there will be marbles of different colors.

Create a set of branches for the first event, showing all possible outcomes.

Continue to add sets of branches for each event, showing all possible outcomes.

Find the probability of those outcomes by creating a ratio.

## How to use a tree diagram to find probability using rules

In order to use a tree diagram to find probability using rules:

1. Fill in the probabilities on the branches.
2. Consider which outcomes are required to answer the question.
3. Find the probability of those outcomes by multiplying along the branches.
4. Use the probability/probabilities you have calculated to answer the question.

## Probability tree diagrams examples

### Example 3: two independent events

Mary has to catch 2 buses to work. The probability the first bus will be late is 0.1 and the probability the second bus will be late is 0.3.

Complete the tree diagram.

Calculate the probability that at least one bus will be late.

Fill in the probabilities on the branches.

Consider which outcomes are required to answer the question.

Find the probability of those outcomes by multiplying along the branches.

Use the probability/probabilities you have calculated to answer the question.

### Example 4: dependent events

There are 9 pencils in a bag. 3 of the pencils are blue and the remaining pencils are yellow. A pencil is taken out at random.

A second pencil is taken at random.

Complete the tree diagram.

Work out the probability that two blue pencils are chosen.

Fill in the probabilities on the branches.

Consider which outcomes are required to answer the question.

Find the probability of those outcomes by multiplying along the branches.

Use the probability/probabilities you have calculated to answer the question.

### Example 5: dependent events

In a bag there are 7 counters. There are 2 black counters and the remaining counters are white.

A counter is removed and the color noted.

The counter is NOT replaced.

A second counter is removed and the color is noted.

Complete the tree diagram.

Calculate the probability that there will be one counter of each color picked.

Fill in the probabilities on the branches.

Consider which outcomes are required to answer the question.

Find the probability of those outcomes by multiplying along the branches.

Use the probability/probabilities you have calculated to answer the question.

### Example 6: three independent events

Mark does a coin toss with three fair coins.

Complete the tree diagram.

Calculate the probability of all three coins landing on heads.

Fill in the probabilities on the branches.

Consider which outcomes are required to answer the question.

Find the probability of those outcomes by multiplying along the branches.

Use the probability/probabilities you have calculated to answer the question.

### Teaching tips for tree diagram probability

• Start with examples that are easy to replicate, such as flipping a coin twice or rolling two dice. This allows students to use a simple tree diagram or a more advanced one with rules. As students advance, choose worksheets that have a variety of contexts and complexity.

### Easy mistakes to make

• Not proportionally representing unfair events in a sample space
When using any type of tree diagram, the probability of each outcome should be clearly represented.

For example,
A bag has 1 yellow tennis ball and 2 green tennis balls. One ball is removed, replaced and then another ball is removed.

The first tree diagram shows each color proportionally. The second tree diagram labels each branch with the probability.

• Reducing fractions while solving
It is usually not worth reducing fractions when working within probability questions. This is because the numerator and denominator give information about the event, for example, the number of marbles in a bag.

Also, we often need to add fractions and they need a common denominator. Only reduce right at the end of a question.

• Dependent events
Remember that for a series of events that include dependent events, the probability of the second event changes depending on the outcome of the first event.

### Practice probability tree diagrams questions

1. A fair coin is flipped and then a marble is chosen from the bag.

Which tree diagram shows the sample space?

The first set of branches is for the coin. Since heads and tails have an equal chance of being flipped, they each appear once.

The second set of branches is for the marbles. There are two green, one blue and one red. The tree diagram should reflect this proportion.

2. A spinner has green sections and blue sections.

The probability of the spinner landing on green is \cfrac{2}{7}.

The spinner is spun twice.

Using the tree diagram, calculate the probability of the spinner landing on green twice.

\cfrac{2}{7}

\cfrac{4}{49}

\cfrac{5}{7}

\cfrac{4}{14}

\begin{aligned}P(\text{2 Greens})&=P(\text{Green and green}) \\\\ &=P(\text{Green})\times P(\text{Green}) \\\\ &=\cfrac{2}{7}\times \cfrac{2}{7} \\\\ &=\cfrac{4}{49} \end{aligned}

3. In a bag there are 10 balls. There are 2 red balls and the remaining balls are yellow.

A ball is removed at random and the color noted.

The first ball is replaced.

A second ball is removed at random and the color is noted.

Using the tree diagram, work out the probability that there will be a ball of each color chosen.

0.64

0.2

0.16

0.32

\begin{aligned}P(\text{One of each color})&=P(\text{R,Y}) \, or \, P(\text{Y,R}) \\\\ &=P(\text{R,Y}) + P(\text{Y,R}) \\\\ &=(0.2\times 0.8)+(0.8\times 0.2) \\\\ &=0.16+0.16 \\\\ &=0.32 \end{aligned}

4. A football team has a probability of 0.7 of winning games.

Using a tree diagram, find the probability that they win at least 1 of their next two games.

0.49

0.3

0.42

0.91

\begin{aligned}P(\text{At least one win})&=P(\text{Win, Win}) \, or \, P(\text{Win, Not win}) \, or \, P(\text{Not win, Win}) \\\\ &=P(\text{Win, Win}) + P(\text{Win, Not win})+P(\text{Not win, Win}) \\\\ &=(0.7\times 0.7)+(0.7\times 0.3)+P(0.3\times 0.7) \\\\ &=0.49+0.21+0.21 \\\\ &=0.91 \end{aligned}

5. There are 6 chocolates in a box. 4 of the chocolates are milk chocolates and the remaining chocolates are plain chocolates. A chocolate is taken out at random and is eaten.

A second chocolate is taken at random and is also eaten.

Calculate the probability that two milk chocolates are eaten.

\cfrac{2}{6}

\cfrac{3}{5}

\cfrac{2}{5}

\cfrac{13}{36}

Since the first chocolate is eaten, the probabilities on the second set of branches are different.

\begin{aligned}P(\text{Two Milks})&=P(\text{Milk and Milk}) \\\\ &=P(\text{Milk}) \times P(\text{Milk}) \\\\ &=\cfrac{4}{6}\times \cfrac{3}{5} \\\\ &=\cfrac{12}{30} \\\\ &=\cfrac{2}{5} \end{aligned}

6. In a bag there are 8 counters. There are 3 blue counters and the remaining counters are yellow.

A counter is removed at random and the color noted.

The counter is NOT replaced.

A second counter is removed at random and the color is noted.

Using the tree diagram, work out the probability that there will be at least one blue counter picked.

\cfrac{9}{14}

\cfrac{15}{28}

\cfrac{5}{14}

\cfrac{3}{28}

\begin{aligned}P(\text{At least one blue})&=P(\text{B,B}) \, or \, P(\text{B,Y}) \, or \, P(\text{Y,B}) \\\\ &=P(\text{B,B}) + P(\text{B,Y})+P(\text{Y,B}) \\\\ &=(\cfrac{3}{8}\times\cfrac{2}{7})+(\cfrac{3}{8}\times \cfrac{5}{7})+P(\cfrac{5}{8}\times \cfrac{3}{7}) \\\\ &=\cfrac{6}{56}+\cfrac{15}{56}+\cfrac{15}{56} \\\\ &=\cfrac{36}{56} \\\\ &=\cfrac{9}{14} \end{aligned}

## Tree diagram probability FAQs

What is a tree diagram probability?

It is a visual representation that shows the outcomes of an event and then the outcomes of at least one other event. Each outcome is represented by a branch and each event is represented by a set of branches.

Can tree diagram probability be used for independent and dependent events?

Tree diagrams can be used for both independent and dependent events. The events ‘flipping a coin’ and ‘rolling a dice’ are independent events – where the outcome of one event does not affect the outcome of the other event.

Events can also be dependent events – where the outcome of one event depends upon what has happened before.

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