Tree Diagram Examples in Statistics

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.

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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 44 red and 66 blue. Two are drawn with replacement. Find P(red, then blue)P(\text{red, then blue}).

Answer

0.240.24

First step

1
Red probability: 4/10=0.44/10 = 0.4.

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Example 2

medium
A bag has 44 red and 66 blue. Two are drawn WITHOUT replacement. Find P(red, then blue)P(\text{red, then blue}).

Example 3

medium
A spinner lands red 13\tfrac{1}{3}, blue 23\tfrac{2}{3}. Spun twice, find P(different colors)P(\text{different colors}).

Example 4

medium
A bag has 22 red, 22 blue, 11 green. One ball is drawn, then another with replacement. Find P(both red)P(\text{both red}).

Example 5

medium
A weather forecast says rain tomorrow has probability 0.40.4. If it rains, traffic jam probability is 0.70.7; if not, it's 0.20.2. Find P(traffic jam)P(\text{traffic jam}).

Example 6

hard
A bag has 33 red, 22 blue. Two are drawn WITHOUT replacement. Find P(same color)P(\text{same color}).

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 0.80.8. Find P(exactly one pass)P(\text{exactly one pass}).

Example 9

hard
Box 1 has 11 red and 33 blue balls. Box 2 has 22 red and 22 blue. A box is chosen at random (each 12\tfrac{1}{2}), then one ball is drawn. Find P(red)P(\text{red}).

Example 10

challenge
A factory has machines A and B making widgets. A makes 60%60\% of widgets with 5%5\% defective; B makes 40%40\% with 2%2\% defective. A widget is found defective. What is the probability it came from A?

Example 11

challenge
A coin flips heads with probability 0.60.6. Flipped 33 times, find P(exactly two heads)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?

Example 2

easy
On a tree diagram, a path goes left (probability 12\frac{1}{2}) then up (probability 13\frac{1}{3}). What is the path's probability?

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 12\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?

Example 9

medium
A bag has 3 red and 2 blue. Two are drawn with replacement. Using a tree, find P(red then red)P(\text{red then red}).

Example 10

medium
A bag has 3 red and 2 blue. Two are drawn WITHOUT replacement. Find P(red then red)P(\text{red then red}).

Example 11

medium
Flipping two coins, use a tree to find P(exactly one head)P(\text{exactly one head}).

Example 12

medium
A spinner lands red 14\frac{1}{4} or blue 34\frac{3}{4}, spun twice. Find P(both same color)P(\text{both same color}).

Example 13

medium
A test has two questions, each true/false. Using a tree, find P(both correct)P(\text{both correct}) by random guessing.

Example 14

medium
A bag has 4 white and 1 black. Two drawn without replacement. Find P(first white, second black)P(\text{first white, second black}).

Example 15

medium
Flipping three coins, use a tree to find P(at least one tail)P(\text{at least one tail}).

Example 16

medium
A coin then a spinner (win 13\frac{1}{3}, lose 23\frac{2}{3}). Using a tree, find P(heads and win)P(\text{heads and win}).

Example 17

medium
A bag has 1 red and 3 blue. Two drawn without replacement. Using a tree, find P(blue then blue)P(\text{blue then blue}).

Example 18

challenge
A box has 2 red and 3 green. Two drawn without replacement. Use a tree to find P(one of each color)P(\text{one of each color}).

Example 19

challenge
A spinner gives win 13\frac{1}{3} or lose 23\frac{2}{3}, spun three times. Find P(exactly one win)P(\text{exactly one win}).

Example 20

challenge
A bag has 3 red and 2 blue. Three drawn without replacement. Find P(all three red)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 33 equally likely outcomes. A tree models spinning twice. How many complete paths?

Example 23

easy
A bag has 11 red and 33 blue balls. A tree shows one draw. What are the branch probabilities?

Example 24

easy
A path has branches with probabilities 12\tfrac{1}{2}, 13\tfrac{1}{3}, 14\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(heads and even number)P(\text{heads and even number}).

Example 27

medium
Flipping two coins, find P(at least one head)P(\text{at least one head}).

Example 28

medium
A test has two yes/no questions, each answered correctly with probability 0.70.7 independently. Find P(both correct)P(\text{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 0.010.01. A test gives a positive result with probability 0.990.99 if diseased, 0.050.05 if not. Find P(positive test)P(\text{positive test}).

Example 31

hard
A bag has 55 red and 55 blue balls. Two are drawn without replacement. Find P(at least one red)P(\text{at least one red}).

Example 32

hard
A jar contains 33 winning tickets and 77 losing tickets. Two tickets are drawn without replacement. Find P(exactly one winner)P(\text{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 44 red and 66 blue balls. Three balls are drawn without replacement. Find P(all three red)P(\text{all three red}).
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Background Knowledge

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

stat sample spaceprobability basic