Probability of Combined Events · 9.2.2
Formula for combined event probability
Through exploration, usually by listing a full sample space in a table or tree diagram, students propose that the probability of two combined events equals the product of their individual probabilities, P(A and B) = P(A) × P(B), then check this conjecture by comparing it against outcomes counted directly.
The official learning standard (9.2.2)
“Make and verify conjecture about the formula of probability of combined events.”
What it means
Through exploration, usually by listing a full sample space in a table or tree diagram, students propose that the probability of two combined events equals the product of their individual probabilities, P(A and B) = P(A) × P(B), then check this conjecture by comparing it against outcomes counted directly.
How it is examined
This standard is mostly assessed through exploratory or investigative-style tasks in Paper 2, where students list a sample space, calculate probability using direct counting and using the multiplication formula, and confirm both match. The multiplication rule itself is then applied directly in later Paper 1 and Paper 2 questions.
Worked example
A fair coin is tossed and a fair die is rolled together. By listing all the outcomes, verify that P(Head and a number greater than 4) = P(Head) × P(a number greater than 4).
- List the sample space: coin {H, T} combined with die {1, 2, 3, 4, 5, 6} gives 12 equally likely outcomes in total.
- Outcomes that satisfy 'Head and a number greater than 4': (H, 5) and (H, 6), 2 outcomes.
- By direct counting, P(Head and a number greater than 4) = 2 ÷ 12 = 1/6.
- From the individual sample spaces, P(Head) = 1/2 and P(a number greater than 4) = 2/6 = 1/3.
- Multiply: P(Head) × P(a number greater than 4) = 1/2 × 1/3 = 1/6.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why do we need to 'verify' the formula instead of just using it?
This standard asks you to explore and confirm the multiplication rule for yourself by comparing it to a fully listed sample space, rather than simply memorising it. This builds real understanding of why the formula works, not just how to use it.
Does P(A and B) = P(A) × P(B) work for dependent events too?
Not directly. This formula applies to independent combined events only.
For dependent events, the probability of the second event must be adjusted to reflect the outcome of the first, which is covered in a later standard.
What tools are best for setting up the sample space to verify the formula?
A two-way table works well for two separate objects like a coin and a die, while a tree diagram is useful when there are more than two stages or the events happen one after another.