Sheet № 247 · Foundation + Higher · AQA · Edexcel · OCR
Combined Events Probability –
Combined events probability is a core topic on GCSE Maths papers at both Foundation and Higher tier. Questions ask you to find the probability of two or more events occurring together ("and") or at least one occurring ("or"). At Higher level, you must handle events with and without replacement, recognise whether events are independent, an
§Key definitions
Combined events
involve two or more outcomes happening. The key question is whether you need "and" (both events happen) or "or" (at least one happens).
Question:
A fair coin is flipped and a fair six-sided dice is rolled. Find the probability of getting a head and a 6.
(a)
Green and white are mutually exclusive.
(b)
P(yellow) = 6/12 = 1/2. P(not yellow) = 1 − 1/2 = 1/2.
(c)
P(1st yellow) = 6/12 = 1/2. After removing one yellow: 5 yellow out of 11 remain.
§Formulas to memorise
P(A and B) = P(A) × P(B)
P(A and B) = P(A) × P(B given A)
P(A or B) = P(A) + P(B)
P(A or B) = P(A) + P(B) − P(A and B)
Combined events: involve two or more outcomes happening. The key question is whether you need "and" (both events happen) or "or" (at least one happens).
The AND rule (multiplication rule):: For independent events (the outcome of one does not affect the other):
The OR rule (addition rule):: For mutually exclusive events (cannot happen at the same time):
Identify the events — what are A and B?
Decide "and" or "or" — does the question ask for both events or at least one?
Check independence — does the first event affect the second? (e.g. is there replacement?)
Worked example
A fair coin is flipped and a fair six-sided dice is rolled. Find the probability of getting a head and a 6.
Working:
⚠ Common mistakes
- ✗Using "and" when the question means "or" (or vice versa). "And" means both happen (multiply). "Or" means at least one happens (add).
- ✗Not adjusting for without replacement. When items are not replaced, the total decreases and the count of the relevant items may decrease — adjust both the numerator and denominator for the second event.
- ✗Adding probabilities for "and" questions. Multiplying is correct for "and"; adding is for "or."
- ✗Forgetting to subtract the overlap. For non-mutually exclusive "or" questions, use P(A) + P(B) − P(A and B).
✦ Exam tips
- →Draw a tree diagram to visualise combined events — it helps organise the multiplication and addition clearly. See probability tree diagrams.
- →"Without replacement" always means dependent events — adjust the second probability.
- →"With replacement" means independent events — probabilities stay the same for each draw.
- →At Foundation, most combined events questions use independent events or simple mutually exclusive situations.
- →At Higher, expect without-replacement and "not mutually exclusive" questions.