Tree Diagram Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Tree Diagram.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

A tree diagram is a branching diagram that shows all possible outcomes of a multi-step random process. Each branch represents one choice or event, and complete paths show combined outcomes.

A tree diagram prevents you from losing cases when a probability problem unfolds in stages. Instead of guessing the outcomes, you build them step by step.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Tree Diagram starts by naming the possible outcomes and the event rule before assigning or combining probabilities.

Common stuck point: Students often know a procedure related to tree diagram but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

medium

A bag has

4

red and

6

blue. Two are drawn with replacement. Find

P(\text{red, then blue})

With replacement: P(Red then Blue) = 0.4 × 0.6 = 0.24

Answer

0.24

First step

Red probability:

4/10 = 0.4

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Example 2

medium

A bag has

4

red and

6

blue. Two are drawn WITHOUT replacement. Find

P(\text{red, then blue})

Without replacement: P(Red then Blue) = 2/5 × 2/3 = 4/15

Example 3

medium

A spinner lands red

\tfrac{1}{3}

, blue

\tfrac{2}{3}

. Spun twice, find

P(\text{different colors})

P(different colors) = P(RB) + P(BR) = 2/9 + 2/9 = 4/9

Example 4

medium

A bag has

2

red,

2

blue,

1

green. One ball is drawn, then another with replacement. Find

P(\text{both red})

Example 5

medium

A weather forecast says rain tomorrow has probability

0.4

. If it rains, traffic jam probability is

0.7

; if not, it's

0.2

. Find

P(\text{traffic jam})

P(traffic jam) = 0.4×0.7 + 0.6×0.2 = 0.28 + 0.12 = 0.40

Example 6

hard

A bag has

3

red,

2

blue. Two are drawn WITHOUT replacement. Find

P(\text{same color})

P(same color) = P(RR) + P(BB) = 6/20 + 2/20 = 0.4

Example 7

hard

From the previous question, given a positive test, what is the probability of disease?

Example 8

hard

A student takes two quizzes, each independently passed with probability

0.8

. Find

P(\text{exactly one pass})

P(exactly one pass) = P(PF) + P(FP) = 0.16 + 0.16 = 0.32

Example 9

hard

Box 1 has

1

red and

3

blue balls. Box 2 has

2

red and

2

blue. A box is chosen at random (each

\tfrac{1}{2}

), then one ball is drawn. Find

P(\text{red})

P(red) = 1/2×1/4 + 1/2×2/4 = 1/8 + 2/8 = 3/8

Example 10

challenge

A factory has machines A and B making widgets. A makes

60\%

of widgets with

5\%

defective; B makes

40\%

with

2\%

defective. A widget is found defective. What is the probability it came from A?

P(A | defective) = 0.030 / (0.030+0.008) ≈ 0.789

Example 11

challenge

A coin flips heads with probability

0.6

. Flipped

3

times, find

P(\text{exactly two heads})

using a tree.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A tree diagram is drawn for flipping two coins. How many complete paths (final outcomes) does it have?

Two-coin tree: 4 complete paths

Example 2

easy

On a tree diagram, a path goes left (probability

\frac{1}{2}

) then up (probability

\frac{1}{3}

). What is the path's probability?

Path Left→Up

Example 3

easy

A tree diagram has two stages with 3 branches then 2 branches. How many final outcomes?

Example 4

easy

A coin then a die is modeled by a tree. How many complete paths are there?

Example 5

easy

On a tree, the path HH has branches each of probability

\frac{1}{2}

. What is

P(HH)

P(HH)

Example 6

easy

A tree models drawing 2 marbles with replacement from a bag of 5. How many branches leave each node?

Example 7

easy

In a tree diagram, what operation combines branch probabilities along one full path?

Example 8

easy

A bag has 2 red and 1 blue. A tree models one draw. What are the two branch probabilities?

One draw from 2R + 1B bag

Example 9

medium

A bag has 3 red and 2 blue. Two are drawn with replacement. Using a tree, find

P(\text{red then red})

With replacement: P(RR)

Example 10

medium

A bag has 3 red and 2 blue. Two are drawn WITHOUT replacement. Find

P(\text{red then red})

Without replacement: P(RR)

Example 11

medium

Flipping two coins, use a tree to find

P(\text{exactly one head})

Example 12

medium

A spinner lands red

\frac{1}{4}

or blue

\frac{3}{4}

, spun twice. Find

P(\text{both same color})

P(both same) = P(RR) + P(BB)

Example 13

medium

A test has two questions, each true/false. Using a tree, find

P(\text{both correct})

by random guessing.

P(both correct by guessing)

Example 14

medium

A bag has 4 white and 1 black. Two drawn without replacement. Find

P(\text{first white, second black})

P(white then black)

Example 15

medium

Flipping three coins, use a tree to find

P(\text{at least one tail})

Example 16

medium

A coin then a spinner (win

\frac{1}{3}

, lose

\frac{2}{3}

). Using a tree, find

P(\text{heads and win})

P(heads and win)

Example 17

medium

A bag has 1 red and 3 blue. Two drawn without replacement. Using a tree, find

P(\text{blue then blue})

P(blue then blue)

Example 18

challenge

A box has 2 red and 3 green. Two drawn without replacement. Use a tree to find

P(\text{one of each color})

P(one of each) = P(RG) + P(GR)

Example 19

challenge

A spinner gives win

\frac{1}{3}

or lose

\frac{2}{3}

, spun three times. Find

P(\text{exactly one win})

Example 20

challenge

A bag has 3 red and 2 blue. Three drawn without replacement. Find

P(\text{all three red})

Example 21

easy

A tree models flipping a coin then rolling a die. How many complete paths does it have?

Example 22

easy

A spinner has

3

equally likely outcomes. A tree models spinning twice. How many complete paths?

Example 23

easy

A bag has

1

red and

3

blue balls. A tree shows one draw. What are the branch probabilities?

One draw: P(Red)

Example 24

easy

A path has branches with probabilities

\tfrac{1}{2}

\tfrac{1}{3}

\tfrac{1}{4}

. What is the path probability?

Example 25

easy

A coin is flipped three times. How many complete paths does the tree have?

Example 26

medium

A coin and a die are tossed. Use a tree to find

P(\text{heads and even number})

Example 27

medium

Flipping two coins, find

P(\text{at least one head})

P(at least one head) = P(HH)+P(HT)+P(TH)

Example 28

medium

A test has two yes/no questions, each answered correctly with probability

0.7

independently. Find

P(\text{both correct})

P(both correct)

Example 29

medium

Three children: each can be boy or girl with equal probability. How many complete paths and how many have at least two girls?

Example 30

hard

A disease has prevalence

0.01

. A test gives a positive result with probability

0.99

if diseased,

0.05

if not. Find

P(\text{positive test})

Example 31

hard

A bag has

5

red and

5

blue balls. Two are drawn without replacement. Find

P(\text{at least one red})

Example 32

hard

A jar contains

3

winning tickets and

7

losing tickets. Two tickets are drawn without replacement. Find

P(\text{exactly one winner})

P(exactly one winner)

Example 33

hard

From the previous problem, given the ball drawn was red, what is the probability it came from Box 2?

Example 34

challenge

A bag has

4

red and

6

blue balls. Three balls are drawn without replacement. Find

P(\text{all three red})

P(all three red)

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Related Concepts

Sample Space Basic Probability Compound Events Conditional Probability Multiplication Rule

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat sample spaceprobability basic