---
title: "Tree Diagrams Without Replacement"
description: "Draw tree diagrams for two or three stages where the first outcome changes the second, multiply along a branch, and add the routes that fit the question."
canonical: https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ
source: https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ.md
retrieved: 2026-09-12
---

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# Tree Diagrams Without Replacement

Draw tree diagrams for two or three stages where the first outcome changes the second, multiply along a branch, and add the routes that fit the question.

Subject: Mathematics · Area: Probability · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ

## Ready when they can

- Adjust the second set of branch probabilities when an item is not replaced
- Multiply along a branch and add the branches that satisfy the condition
- Check that the branches at each stage add to 1

## Lesson: Trees when nothing goes back

Some questions run in stages, and a tree draws every route for you. Each branch carries its probability, and the branches leaving one point always add to 1. When the item is not put back, the second stage must be rebuilt from what is left.

**Example.** A bag holds cards 1 to 5: three odd, two even. First draw: P(even) = 2/5. Set one even card aside and rebuild: 4 cards left with 1 even, so P(even second) = 1/4. Both totals dropped by one because the removed card is gone.

A route's chance is the product along its branches. Even then even gives 2/5 times 1/4 = 2/20 = 0.1. Odd then even gives 3/5 times 2/4 = 6/20 = 0.3. When several routes fit the question, add them: one odd and one even in either order is 0.3 plus 0.3 = 0.6.

**Tip.** At every branching point, check the probabilities add to 1. If they do not, a fraction was built from the old totals. Rebuild each stage from the cards that are actually left.

**Recap.** Rebuild the second stage from what is left, multiply along a route, add the routes that fit.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Tree diagrams](https://lightmysky.com/learn/mathematics/tree-diagrams-mt_GDtFU5fyUv)
- [Independent Events and the Multiplication Rule](https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN)

## Opens up

- [Conditional Probability from Tables and Venn Diagrams](https://lightmysky.com/learn/mathematics/conditional-probability-from-tables-and-venn-diagrams-mt_Vx88ugViYP)