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# Probability trees - dependent events
- URL: https://www.esheets.io/probability-trees-dependent-events/
- Published: 2026-07-08T18:00:54.000Z
- Updated: 2026-07-09T20:15:20.000Z
- Author: Richard Linnington 2
- Tags: Maths, Probability

Probability trees are an excellent tool for modelling dependent events, where the outcome of the first event changes the probability of the second. A common example is picking items from a bag without replacing them. This interactive worksheet lets you practise completing dependent probability trees using fractions, and then calculate combined outcomes. [Jump to the questions](#practise-now)

[Looking for questions on independent events?](https://www.esheets.io/probability-trees-independent-events/)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

This worksheet tests your ability to complete probability trees for dependent events.

You will be presented with a scenario where an object is chosen without replacement, altering the probabilities for the second choice.

#### What you’ll practise

- Identifying dependent events and selection without replacement.
- Completing branch probabilities as fractions.
- Calculating the changing second-stage probabilities.
- Multiplying fractions along branches to find path outcomes.
- Adding combined fraction outcomes where appropriate to answer specific questions.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Topic guide

**Dependent events** occur when the outcome of one event affects the probabilities of the next event. A classic example is picking items from a container **without replacement**.

### Why probabilities change without replacement

If you select an object and do not replace it:

- The total number of objects left in the container decreases by 1 (the denominator).
- The number of objects of the selected colour also decreases by 1 (the numerator).

Because the contents have changed, the probabilities for the second selection will be different from the first. This is unlike selection **with replacement**, where the contents return to their original state and the probabilities remain the same.

### How to complete the second branches of a probability tree

Consider a bag with **3 red** and **4 blue** counters. A counter is chosen at random and is not replaced. A second counter is then chosen.

For the first choice, there are 7 counters in total:

- P(red) = 3/7
- P(blue) = 4/7

If the first counter was **red**, the bag now contains **2 red** and **4 blue** counters (6 in total). The second set of branches after choosing red will be:

- P(red) = 2/6 = 1/3
- P(blue) = 4/6 = 2/3

If the first counter was **blue**, the bag now contains **3 red** and **3 blue** counters (6 in total). The second set of branches after choosing blue will be:

- P(red) = 3/6 = 1/2
- P(blue) = 3/6 = 1/2

*Note: Equivalent fractions represent the same probability. Answering 2/6 or 1/3 is mathematically correct.*

### Multiplying probabilities along a path

To find the probability of a combined outcome, **multiply** the fractions along the path that leads to it.

For example, the probability of choosing red then blue is:  
P(red, blue) = 3/7 × 4/6 = 12/42 = 2/7

### Adding suitable outcomes

To find the probability of multiple outcomes, **add** the probabilities of the valid paths.

For example, the probability of choosing one of each colour is the sum of P(red, blue) and P(blue, red).  
(3/7 × 4/6) + (4/7 × 3/6) = 12/42 + 12/42 = 24/42 = 4/7

### Common mistakes

- Forgetting to decrease the total number of items (the denominator) by 1 for the second choice.
- Copying the first-stage probabilities to the second-stage branches instead of calculating the new values.
- Adding along paths instead of multiplying.

### Things to remember

- Always check the wording carefully to see if the item is replaced or not.
- Without replacement, both the total count and the count of the selected item decrease by 1 for the second choice.
- Probabilities on each pair of branches must always sum to 1.

[Dependent probability visualisation tool ->](https://www.esheets.io/probability-visualisation-tool/)