GCSE MATHS • HIGHER TIER

Conditional Probability Tree Diagrams Made Simple

Organise dependent probability events, update branches and calculate multi-step outcomes accurately.

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Worked proofs

Arc Length in Conditional Probability Tree Diagrams in Four Moves Four Moves

1

Draw

Set out every possible outcome as branches.

2

Update

Change probabilities when an outcome affects the next event.

3

Multiply

Multiply probabilities along each required branch.

4

Add

Add different successful branches when the question requires more than one route.

What You’ll Learn

Understand conditional probability.

Read and construct probability tree diagrams.

Apply the FormulaARecognise dependent events and no-replacement situations.

Update branch probabilities correctly.

Multiply probabilities along branches.

Add probabilities across successful branches.

Use the complement method for “at least one” questions.

Key Rules

Probabilities on branches must add to 1.

Multiply probabilities along branches.

Add probabilities across different successful branches when required.

Update probabilities when events are dependent.

Worked Proof: Conditional Probability Tree Diagrams

Step-by-Step Method

  • 1. Write the possible outcomes as separate branches for the first event.
  • 2. Write the correct probability on each branch.
  • 3. Decide whether the item is replaced after the first selection.
  • 4. If there is no replacement, adjust both the number of favourable outcomes and the total number of items.
  • 5. To find the probability of a particular sequence, multiply the probabilities along that branch.

More Worked Examples

1

Question: A bag contains 3 red counters and 2 blue counters. Two counters are chosen without replacement. Find the probability of choosing two red counters.

First red = 3/5

Second red = 2/4
Probability = 3/5 × 2/4
= 6/20
= 3/10
Answer : 3/10

2

Question: A bag contains 4 green and 6 yellow counters. Two counters are selected without replacement. Find the probability of choosing two green counters.

Probability = 4/10 × 3/9

= 12/90
= 2/15
Answer : 2/15

3

Question: A card is drawn from a deck and not replaced. Find the probability of selecting two kings.

First king = 4/52

Second king = 3/51
Probability = 12/2652
= 1/221
Answer : 1/221

4

Question: A bag contains 5 red and 3 blue counters. Find the probability of selecting a red then a blue without replacement.

Probability = 5/8 × 3/7

= 15/56
Answer : 15/56
  • Common GCSE Mistakes
  • Forgetting to change probabilities after an item is removed.
  • Adding probabilities when they should be multiplied.
  • Forgetting to include all successful branches.
  • Failing to check that branch probabilities add to 1.
  • Exam Tips
  • Check whether the question uses replacement or no replacement.
  • Update both the numerator and total when an item is not replaced.
  • Multiply along a route through the tree.
  • Add separate successful routes when needed.
  • Use the complement method for suitable “at least one” questions.
  • Check probabilities from the same split add to 1.

Skills Used in Conditional Probability Tree Diagrams

Conditional Probability

Tree Diagrams

Fractions

Dependent Events

Multiplication of Probabilities

Addition of Probabilities

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Frequently Asked Questions

It is the probability of an event occurring given that another event has already happened.

The selected object is not returned, so the number of objects and possible outcomes changes.

Multiply probabilities along branches to find the probability of a particular sequence of outcomes.

Add different successful branches when more than one route satisfies the question.

Probabilities from the same split must add to 1.

The source uses it to find the probability of at least one event by subtracting the probability of none from 1.