Independent and Dependent Events –

Independent and dependent events are tested on both Foundation and Higher GCSE Maths papers across AQA, Edexcel and OCR. You need to know when to multiply probabilities, how "without replacement" changes the probabilities, and how tree diagrams help organise combined events. This guide explains each concept clearly with worked examples. For an overview of every topic, see our complete GCSE Maths topics list.

What Are Independent and Dependent Events?

• Independent events — the outcome of one event does not affect the outcome of another. Example: flipping a coin and rolling a dice.
• Dependent events — the outcome of one event changes the probabilities of the next. Example: picking two cards from a deck without replacement.

Key Formulas

For independent events you multiply the original probabilities. For dependent events the second probability changes because the first event has altered the sample space.

Tree Diagrams

A tree diagram is a visual tool that shows all possible outcomes for two or more successive events. Each branch shows the probability, and you multiply along branches to find the probability of a specific combined outcome. To find P(A or B), add the probabilities of the relevant combined outcomes.

Step-by-Step Method

1. Determine whether the events are independent or dependent.
2. For independent events, multiply the probabilities directly.
3. For dependent events (e.g. without replacement), adjust the second probability to reflect the changed total.
4. Draw a tree diagram if there are multiple stages — list all outcomes at each branch.
5. Multiply along each path for combined probabilities. Add paths if the question asks for "or".

Worked Example 1 — Foundation Level

Question: A fair coin is flipped and a fair dice is rolled. Find the probability of getting heads and a 6.

Working:

These are independent events — the coin does not affect the dice.

P(heads) = 1/2. P(6) = 1/6.

P(heads and 6) = 1/2 x 1/6 = 1/12.

Answer: The probability of heads and a 6 is 1/12.

Worked Example 2 — Higher Level

Question: A bag contains 5 red and 3 blue beads. Two beads are picked without replacement. Find the probability of picking (a) two red beads, (b) one of each colour.

Working:

(a) P(1st red) = 5/8. After removing a red bead: P(2nd red) = 4/7.

P(two red) = 5/8 x 4/7 = 20/56 = 5/14.

(b) There are two ways to get one of each colour:

Red then Blue: P = 5/8 x 3/7 = 15/56.

Blue then Red: P = 3/8 x 5/7 = 15/56.

P(one of each) = 15/56 + 15/56 = 30/56 = 15/28.

Answer: (a) 5/14. (b) 15/28.

Worked Example 3 — Exam Style

Question: The probability that Amy passes her driving theory test is 0.7. The probability she passes her practical test is 0.6. The two tests are independent. (a) Draw a tree diagram. (b) Find the probability she passes both. (c) Find the probability she passes exactly one test.

Working:

(a) First branch: Theory — Pass (0.7) or Fail (0.3). From each, second branch: Practical — Pass (0.6) or Fail (0.4).

(b) P(both pass) = 0.7 x 0.6 = 0.42.

(c) P(theory only) = 0.7 x 0.4 = 0.28. P(practical only) = 0.3 x 0.6 = 0.18.

P(exactly one) = 0.28 + 0.18 = 0.46.

Answer: (b) 0.42. (c) 0.46.

Common Mistakes

• Multiplying when you should add. "And" means multiply along branches. "Or" means add the results of separate paths.
• Not adjusting for without replacement. When items are removed, the total decreases and so do the favourable outcomes for the next pick.
• Forgetting a path. When asked for "one of each", remember there are two orders (e.g. red-blue and blue-red). A tree diagram helps you avoid missing paths.
• Treating dependent events as independent. If items are not replaced, probabilities change. Always check whether replacement occurs.

Exam Tips

• Draw a tree diagram for any two-stage or three-stage probability question — it organises your working clearly.
• Label every branch with its probability. Multiply along branches, add between branches.
• For "at least one" questions, it is often easier to calculate 1 − P(none) rather than adding up every other possibility.
• For related topics, see mutually exclusive events and probability tree diagrams. For key formulas, visit our GCSE Maths formulas page.

Practice Questions

Q1 (Foundation): Two fair coins are flipped. Find the probability of getting two tails.

Q2 (Foundation): A box has 6 white and 4 black balls. One ball is picked, replaced, then another is picked. Find P(both white).

Q3 (Higher): A bag has 4 green and 6 yellow counters. Two are picked without replacement. Find the probability of picking at least one green.

Practise independent and dependent events free on GCSEMathsAI.

Related Topics

• Mutually Exclusive Events
• Probability Tree Diagrams
• Conditional Probability

Summary

• Independent events do not affect each other: P(A and B) = P(A) x P(B).
• Dependent events change probabilities: adjust the second probability after the first event occurs (e.g. without replacement reduces the total).
• Use tree diagrams to organise multi-stage probability problems: multiply along branches, add between branches.
• For "at least one" questions, consider using 1 − P(none) as a shortcut.
• Always check whether replacement occurs — this determines whether events are independent or dependent.