Probability of Combined Events · 9.4.1
Combined events problem solving
Students apply everything learned about combined events, independence, dependence, mutual exclusivity and the addition/multiplication rules, to multi-step real-life situations. This requires planning a solution strategy, interpreting the context correctly, and presenting full, logical working to reach a final probability.
The official learning standard (9.4.1)
“Solve problems involving probability of combined events.”
What it means
Students apply everything learned about combined events, independence, dependence, mutual exclusivity and the addition/multiplication rules, to multi-step real-life situations. This requires planning a solution strategy, interpreting the context correctly, and presenting full, logical working to reach a final probability.
How it is examined
This standard is assessed mainly in Paper 2, contributing higher-order-thinking questions worth several marks that combine real-life context (games, factories, surveys) with two or more events. Paper 1 may include shorter combined-event word problems that test a single concept quickly.
Worked example
A factory produces light bulbs using two machines, A and B. Machine A produces 60% of the bulbs, and 5% of Machine A's bulbs are defective.
Machine B produces the remaining 40%, and 8% of Machine B's bulbs are defective. A bulb is chosen at random from the factory's output.
Find the probability that it is defective.
- P(from A) = 0.6 and P(defective | A) = 0.05, so P(A and defective) = 0.6 × 0.05 = 0.03.
- P(from B) = 0.4 and P(defective | B) = 0.08, so P(B and defective) = 0.4 × 0.08 = 0.032.
- A bulb can only come from one machine, so 'defective from A' and 'defective from B' are mutually exclusive outcomes.
- Total P(defective) = 0.03 + 0.032 = 0.062 = 31/500.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I start a combined-events word problem?
Draw a tree diagram or list the stages of the situation, labelling each branch or stage with the given probability. Then decide whether to multiply (for 'and') or add (for 'or') based on how the events relate.
What's a common mistake in these problems?
Forgetting to check whether the situation is 'with replacement' or 'without replacement', or whether a bulb/item can only belong to one branch, this changes probabilities for later, dependent stages of the tree diagram.
Do I need to show full working for this type of question?
Yes, Paper 2 awards method marks for correct steps such as a tree diagram or formula substitution, even if the final numerical answer contains a small arithmetic slip, so always show every stage clearly.