Tree Diagram Formula

A tree diagram is a branching diagram that shows all possible outcomes of a multi-step random process.

The Formula

P(path)=∏branch probabilities on that pathP(\text{path}) = \prod \text{branch probabilities on that path}

When to use: 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.

Quick Example

If you flip a coin and then roll a die, the tree diagram starts with H and T, and each of those branches splits into 1 through 6, giving 12 outcomes in all.

Notation

Each full path from start to finish represents one combined outcome.

What This Formula Means

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.

Formal View

A tree diagram represents a sequential sample space. The probability of a terminal outcome is the product of the conditional branch probabilities along its path.

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

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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

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

Common Mistakes

  • Forgetting branches and missing valid outcomes - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Adding branch probabilities when the path requires multiplication - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Treating different stages as if they happen at the same time - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing tree diagram from a keyword alone - Keywords like chance, probability, outcome are only clues; the data structure must match the concept.

Why This Formula Matters

Tree Diagram helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

Frequently Asked Questions

What is the Tree Diagram formula?

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.

How do you use the Tree Diagram formula?

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.

What do the symbols mean in the Tree Diagram formula?

Each full path from start to finish represents one combined outcome.

Why is the Tree Diagram formula important in Statistics?

Tree Diagram helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

What do students get wrong about Tree Diagram?

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.

What should I learn before the Tree Diagram formula?

Before studying the Tree Diagram formula, you should understand: stat sample space, probability basic.

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