Probability of Combined Events · Form 4
Probability of Combined Events: Revision Notes
A tight revision summary of Probability of Combined Events for SPM Mathematics, the key ideas, the formulae the exam gives you, and what to focus on before the paper.
The big idea
This chapter builds on basic probability to handle two or more events together. You learn the difference between independent and dependent events, between mutually exclusive and non-mutually-exclusive ones, and how to use tree diagrams and the given probability formulae to combine them.
Key ideas to revise
- Independent vs dependent. For independent events one outcome does not change the next; for dependent events (like drawing without replacing) it does.
- Tree diagrams. A tree lays out every path and its probability. Multiply along a branch, add across branches.
- Mutually exclusive or not. Mutually exclusive events cannot happen together, so P(A or B) = P(A) + P(B); otherwise you subtract the overlap.
One worked mini-example per key idea
- Independent events (multiply): Two fair coins are tossed. P(both heads) = 1/2 × 1/2 = 1/4.
- Dependent events (without replacement): A box holds 4 good and 2 faulty pens. Two are taken without replacement. P(both good) = 4/6 × 3/5 = 12/30 = 2/5.
- Tree diagram (multiply along a branch, add across branches): A bag has 2 red and 1 green marble; two draws with replacement. P(one of each colour) = (2/3 × 1/3) + (1/3 × 2/3) = 2/9 + 2/9 = 4/9.
- Mutually exclusive or not (add): On one die, P(2 or 5) = 1/6 + 1/6 = 1/3 (they cannot both happen). From cards numbered 1–10, P(even or multiple of 3) = 5/10 + 3/10 − 1/10 = 7/10 (the overlap at 6 is subtracted).
A pre-paper checklist for this chapter
- Draw the tree first and check the branches leaving each point add to 1, for example 5/8 and 3/8, not 5/8 and 5/8.
- Decide with or without replacement before you write the second-draw denominators; without replacement means the total drops by 1.
- Underline the joining words: 'and' → multiply, 'or' → add, 'at least' → use 1 − P(none), 'exactly' → count every order.
- The formula sheet gives P(A∪B) = P(A) + P(B) − P(A∩B); you apply it, you do not memorise it, so subtract the overlap only when the two events can happen together.
- Keep fractions over a common denominator and simplify only on the last line, so an early rounding never spoils the final mark.
- The one habit that saves marks: check the final probability lies between 0 and 1, anything above 1 means you added when you should have multiplied.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Should I draw a tree diagram even in Paper 1?
Paper 1 is multiple choice and tightly timed, so a full tree is often too slow. But a quick two-branch sketch in the margin still pays off for any 'and/or' or 'without replacement' item, it turns the multiply-or-add decision into something you can see rather than guess under pressure.
How many decimal places should a probability answer have?
Give the exact fraction whenever you can, because it is never rounded and never wrong. If the question asks for a decimal, follow that and usually keep three or four significant figures.
Never round partway through a tree calculation, carry the fraction and convert only on the final line.
Can I leave my answer as an unsimplified fraction?
An equivalent fraction such as 20/56 is normally accepted, so you rarely lose the mark for not simplifying. Even so, reducing to 5/14 is safer and neater, and it makes checking against another part of the question far easier.
Simplify on the last line, not during the working.