Sheet № 190 · Foundation + Higher · AQA · Edexcel · OCR
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. F
§Key definitions
Question:
A fair coin is flipped and a fair dice is rolled. Find the probability of getting heads and a 6.
Answer:
The probability of heads and a 6 is 1/12.
(a)
P(1st red) = 5/8. After removing a red bead: P(2nd red) = 4/7.
(b)
There are two ways to get one of each colour:
(c)
P(theory only) = 0.7 x 0.4 = 0.28. P(practical only) = 0.3 x 0.6 = 0.18.
§Formulas to memorise
P(A and B) = P(A) x P(B) (for independent events)
P(A and B) = P(A) x P(B given A has occurred) (for 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.
Worked example
A fair coin is flipped and a fair dice is rolled. Find the probability of getting heads and a 6.
Working:
⚠ 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.