Probability Trees

What you'll learn

How to label probability tree branches from fractions, decimals, or counts.
When to multiply along branches and when to add different routes.
How to use complements such as “not win” or “not late”.
How to handle questions like “both”, “different”, and “at least one”.

1. Probability basics

A probability tells you how likely something is. It can be written as a fraction, decimal, or percentage. A probability of 0 means impossible, and a probability of 1 means certain.

Definition

Event

An event is an outcome, or a group of outcomes, that you are interested in. For example, choosing a blue counter or winning a match.

Definition

Complement

The complement of an event means “not that event”. An event and its complement always add to 1, so $A)=1−P(A)P(\text{not } A) = 1 - P(A)$ .

Example

Finding a missing branch label

In a game, the probability of winning is 0.65. What probability should go on the “not win” branch?

A single split in a probability tree where the two branch probabilities must add to 1.

The two outcomes are “win” and “not win”, so their probabilities must add to 1.
Subtract the win probability from 1:
$win)=1−0.65=0.35P(\text{not win}) = 1 - 0.65 = 0.35$
Label the second branch 0.35.

2. Building a probability tree

A probability tree shows events happening in stages. For example, first choose from Bag A, then choose from Bag B.

Definition

Probability tree

A probability tree is a diagram with branches that show the possible outcomes of two or more events, with probabilities written on the branches.

At each split in the tree, the probabilities should add to 1.

Definition

Independent events

Two events are independent if the result of the first event does not change the probabilities for the second event.

For separate bags, separate games, or different days with the same chance each day, the second set of branches usually has the same labels every time.

Example

Labelling a tree from two bags

Sofia has one bag with 2 red counters and 6 blue counters. A second bag has 4 red counters and 1 blue counter. She takes one counter from each bag.

Two separate bags lead to a two-stage probability tree, with the second-bag probabilities repeated after each first-bag outcome.

For the first bag, there are 8 counters in total:
$bag)=68=34P(\text{red from first bag}) = \frac{2}{8} = \frac{1}{4}, \quad P(\text{blue from first bag}) = \frac{6}{8} = \frac{3}{4}$
For the second bag, there are 5 counters in total:
$bag)=15P(\text{red from second bag}) = \frac{4}{5}, \quad P(\text{blue from second bag}) = \frac{1}{5}$
Draw the first split for the first bag: red with $14\frac{1}{4}$ and blue with $34\frac{3}{4}$ .
From the end of both first-bag branches, draw the second-bag branches: red with $45\frac{4}{5}$ and blue with $15\frac{1}{5}$ .
Check each split adds to 1.

Common Mistake

When probabilities change

If you take two items from the same bag without replacing the first item, the second probabilities can change. The examples here use separate bags, games, or days, so the probabilities repeat.

3. Multiply for “and” / “both”

When a question asks for two things to happen together, such as “win both games” or “choose two blue counters”, you follow one complete path through the tree.

Definition

Route

A route is one complete path from the start of a probability tree to an end outcome.

Key Idea

AND means multiply

To find the probability of one complete route, multiply the branch probabilities along that route.

Example

Winning two matches

Amira plays one badminton match and one table tennis match. The probability she wins badminton is $35\frac{3}{5}$ , and the probability she wins table tennis is $710\frac{7}{10}$ . Find the probability she wins both.

The “wins both” outcome is one complete route through the tree: win badminton then win table tennis.

The “not win” branches would be $25\frac{2}{5}$ for badminton and $310\frac{3}{10}$ for table tennis.
“Wins both” means: win badminton and win table tennis.
Multiply the two win probabilities:
$wins)=35×710=2150P(\text{both wins}) = \frac{3}{5} \times \frac{7}{10} = \frac{21}{50}$
The probability she wins both is $2150\frac{21}{50}$ .

Common Mistake

Adding for ‘both’

Do not add probabilities when the question asks for both events. Adding would mean “one or the other”, not one full route.

4. Add for “or” / different routes

Sometimes the result you want can happen in more than one way. For example, “different coloured ties” could mean black then red, or red then black.

Key Idea

OR means add routes

If a result can happen by more than one route, multiply along each route first, then add the route probabilities.

Example

Different colours on two days

Eli wears either a navy tie or a grey tie. On any day, $P(navy)=59P(\text{navy}) = \frac{5}{9}$ . Find the probability he wears different colours on Monday and Tuesday.

Different colours can happen by two separate routes, which are found first and then added.

First find the grey probability:
$P(grey)=1−59=49P(\text{grey}) = 1 - \frac{5}{9} = \frac{4}{9}$
Different colours can happen in two routes: navy then grey, or grey then navy.
Multiply along each route:
$navy)=49×59=2081\begin{aligned} P(\text{navy then grey}) &= \frac{5}{9} \times \frac{4}{9} = \frac{20}{81} \\ P(\text{grey then navy}) &= \frac{4}{9} \times \frac{5}{9} = \frac{20}{81} \end{aligned}$
Add the two route probabilities:
$colours)=2081+2081=4081P(\text{different colours}) = \frac{20}{81} + \frac{20}{81} = \frac{40}{81}$
The probability he wears different colours is $4081\frac{40}{81}$ .

5. “At least one” questions

Definition

At least one

“At least one” means one or more. For two events, it includes first only, second only, and both.

You could list every successful route, but there is often a quicker method.

Tip

Shortcut for at least one

“At least one” is the opposite of “none”, so use $one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none})$ .

Example

Late on at least one day

Sam catches a train on Saturday and Sunday. The probability the train is late on any day is 0.3. Find the probability it is late on at least one of the two days.

For “at least one late”, the only unsuccessful route is not late on both days, so the complement shortcut can be used.

The probability the train is not late is 0.7.
“None late” means not late on Saturday and not late on Sunday:
$late)=0.7×0.7=0.49P(\text{none late}) = 0.7 \times 0.7 = 0.49$
“At least one late” is the opposite of “none late”:
$late)=1−0.49=0.51P(\text{at least one late}) = 1 - 0.49 = 0.51$
The probability it is late on at least one day is 0.51.

6. Trees with three outcomes

Not every event has only two outcomes. A game might have win, draw, or lose. The important rule is the same: all branch probabilities from the same point must add to 1.

Example

Win, draw or lose

A player plays two rounds of a game. The probability of a win is 0.5, and the probability of a draw is 0.2. Find the probability the player wins both rounds.

A probability tree can have three branches at each split, and the repeated win branch gives the “wins both” route.

Find the missing lose probability:
$P(lose)=1−0.5−0.2=0.3P(\text{lose}) = 1 - 0.5 - 0.2 = 0.3$
For each round, the tree has three branches: win 0.5, draw 0.2, lose 0.3.
Because the probabilities are the same for each round, draw the same three branches after each first-round outcome.
“Wins both rounds” is the win then win route:
$both)=0.5×0.5=0.25P(\text{win both}) = 0.5 \times 0.5 = 0.25$
The probability the player wins both rounds is 0.25.

Exam technique

In the exam

Write the missing branch probabilities first, and check every split adds to 1.
For “both” or “and”, follow one route and multiply along it.
For “or”, “different”, or “at least one”, find all suitable routes and add them, or use the complement shortcut.

Self review

Check yourself

Can you label a two-stage tree when given counts in two bags?
Can you explain why “navy then grey” and “grey then navy” are different routes?
When a question says “at least one”, what is the shortcut using “none”?

Probability Trees

What you'll learn

How to label probability tree branches from fractions, decimals, or counts.
When to multiply along branches and when to add different routes.
How to use complements such as “not win” or “not late”.
How to handle questions like “both”, “different”, and “at least one”.

1. Probability basics

A probability tells you how likely something is. It can be written as a fraction, decimal, or percentage. A probability of 0 means impossible, and a probability of 1 means certain.

Definition

Event

An event is an outcome, or a group of outcomes, that you are interested in. For example, choosing a blue counter or winning a match.

Definition

Complement

The complement of an event means “not that event”. An event and its complement always add to 1, so $A)=1−P(A)P(\text{not } A) = 1 - P(A)$ .

Example

Finding a missing branch label

In a game, the probability of winning is 0.65. What probability should go on the “not win” branch?

A single split in a probability tree where the two branch probabilities must add to 1.

The two outcomes are “win” and “not win”, so their probabilities must add to 1.
Subtract the win probability from 1:
$win)=1−0.65=0.35P(\text{not win}) = 1 - 0.65 = 0.35$
Label the second branch 0.35.

2. Building a probability tree

A probability tree shows events happening in stages. For example, first choose from Bag A, then choose from Bag B.

Definition

Probability tree

A probability tree is a diagram with branches that show the possible outcomes of two or more events, with probabilities written on the branches.

At each split in the tree, the probabilities should add to 1.

Definition

Independent events

Two events are independent if the result of the first event does not change the probabilities for the second event.

For separate bags, separate games, or different days with the same chance each day, the second set of branches usually has the same labels every time.

Example

Labelling a tree from two bags

Sofia has one bag with 2 red counters and 6 blue counters. A second bag has 4 red counters and 1 blue counter. She takes one counter from each bag.

Two separate bags lead to a two-stage probability tree, with the second-bag probabilities repeated after each first-bag outcome.

For the first bag, there are 8 counters in total:
$bag)=68=34P(\text{red from first bag}) = \frac{2}{8} = \frac{1}{4}, \quad P(\text{blue from first bag}) = \frac{6}{8} = \frac{3}{4}$
For the second bag, there are 5 counters in total:
$bag)=15P(\text{red from second bag}) = \frac{4}{5}, \quad P(\text{blue from second bag}) = \frac{1}{5}$
Draw the first split for the first bag: red with $14\frac{1}{4}$ and blue with $34\frac{3}{4}$ .
From the end of both first-bag branches, draw the second-bag branches: red with $45\frac{4}{5}$ and blue with $15\frac{1}{5}$ .
Check each split adds to 1.

Common Mistake

When probabilities change

If you take two items from the same bag without replacing the first item, the second probabilities can change. The examples here use separate bags, games, or days, so the probabilities repeat.

3. Multiply for “and” / “both”

When a question asks for two things to happen together, such as “win both games” or “choose two blue counters”, you follow one complete path through the tree.

Definition

Route

A route is one complete path from the start of a probability tree to an end outcome.

Key Idea

AND means multiply

To find the probability of one complete route, multiply the branch probabilities along that route.

Example

Winning two matches

The “wins both” outcome is one complete route through the tree: win badminton then win table tennis.

The “not win” branches would be $25\frac{2}{5}$ for badminton and $310\frac{3}{10}$ for table tennis.
“Wins both” means: win badminton and win table tennis.
Multiply the two win probabilities:
$wins)=35×710=2150P(\text{both wins}) = \frac{3}{5} \times \frac{7}{10} = \frac{21}{50}$
The probability she wins both is $2150\frac{21}{50}$ .

Common Mistake

Adding for ‘both’

Do not add probabilities when the question asks for both events. Adding would mean “one or the other”, not one full route.

4. Add for “or” / different routes

Sometimes the result you want can happen in more than one way. For example, “different coloured ties” could mean black then red, or red then black.

Key Idea

OR means add routes

If a result can happen by more than one route, multiply along each route first, then add the route probabilities.

Example

Different colours on two days

Eli wears either a navy tie or a grey tie. On any day, $P(navy)=59P(\text{navy}) = \frac{5}{9}$ . Find the probability he wears different colours on Monday and Tuesday.

Different colours can happen by two separate routes, which are found first and then added.

First find the grey probability:
$P(grey)=1−59=49P(\text{grey}) = 1 - \frac{5}{9} = \frac{4}{9}$
Different colours can happen in two routes: navy then grey, or grey then navy.
Multiply along each route:
$navy)=49×59=2081\begin{aligned} P(\text{navy then grey}) &= \frac{5}{9} \times \frac{4}{9} = \frac{20}{81} \\ P(\text{grey then navy}) &= \frac{4}{9} \times \frac{5}{9} = \frac{20}{81} \end{aligned}$
Add the two route probabilities:
$colours)=2081+2081=4081P(\text{different colours}) = \frac{20}{81} + \frac{20}{81} = \frac{40}{81}$
The probability he wears different colours is $4081\frac{40}{81}$ .

5. “At least one” questions

Definition

At least one

“At least one” means one or more. For two events, it includes first only, second only, and both.

You could list every successful route, but there is often a quicker method.

Tip

Shortcut for at least one

“At least one” is the opposite of “none”, so use $one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none})$ .

Example

Late on at least one day

Sam catches a train on Saturday and Sunday. The probability the train is late on any day is 0.3. Find the probability it is late on at least one of the two days.

For “at least one late”, the only unsuccessful route is not late on both days, so the complement shortcut can be used.

The probability the train is not late is 0.7.
“None late” means not late on Saturday and not late on Sunday:
$late)=0.7×0.7=0.49P(\text{none late}) = 0.7 \times 0.7 = 0.49$
“At least one late” is the opposite of “none late”:
$late)=1−0.49=0.51P(\text{at least one late}) = 1 - 0.49 = 0.51$
The probability it is late on at least one day is 0.51.

6. Trees with three outcomes

Not every event has only two outcomes. A game might have win, draw, or lose. The important rule is the same: all branch probabilities from the same point must add to 1.

Example

Win, draw or lose

A player plays two rounds of a game. The probability of a win is 0.5, and the probability of a draw is 0.2. Find the probability the player wins both rounds.

A probability tree can have three branches at each split, and the repeated win branch gives the “wins both” route.

Find the missing lose probability:
$P(lose)=1−0.5−0.2=0.3P(\text{lose}) = 1 - 0.5 - 0.2 = 0.3$
For each round, the tree has three branches: win 0.5, draw 0.2, lose 0.3.
Because the probabilities are the same for each round, draw the same three branches after each first-round outcome.
“Wins both rounds” is the win then win route:
$both)=0.5×0.5=0.25P(\text{win both}) = 0.5 \times 0.5 = 0.25$
The probability the player wins both rounds is 0.25.

Exam technique

In the exam

Write the missing branch probabilities first, and check every split adds to 1.
For “both” or “and”, follow one route and multiply along it.
For “or”, “different”, or “at least one”, find all suitable routes and add them, or use the complement shortcut.

Self review

Check yourself

Can you label a two-stage tree when given counts in two bags?
Can you explain why “navy then grey” and “grey then navy” are different routes?
When a question says “at least one”, what is the shortcut using “none”?

Probability Trees

What you'll learn

1. Probability basics

2. Building a probability tree

3. Multiply for “and” / “both”

4. Add for “or” / different routes

5. “At least one” questions

6. Trees with three outcomes

Recap questions

You've reached the end

Probability Trees

What you'll learn

1. Probability basics

2. Building a probability tree

3. Multiply for “and” / “both”

4. Add for “or” / different routes

5. “At least one” questions

6. Trees with three outcomes

Recap questions

You've reached the end

Probability Trees

What you'll learn

1. Probability basics

2. Building a probability tree

3. Multiply for “and” / “both”

4. Add for “or” / different routes

5. “At least one” questions

6. Trees with three outcomes

Recap questions

You've reached the end

Probability Trees Revision Guide

Probability Trees

What you'll learn

1. Probability basics

2. Building a probability tree

3. Multiply for “and” / “both”

4. Add for “or” / different routes

5. “At least one” questions

6. Trees with three outcomes

Recap questions

You've reached the end

Probability Trees Revision Guide