Probability of Combined Events · 9.2.3
Probability: dependent & independent events
Students calculate the probability of two events happening together by first deciding whether the second event's outcome is affected by the first (dependent) or not (independent). Independent events use P(A) × P(B); dependent events require recalculating probability after the first outcome, often shown with a tree diagram.
The official learning standard (9.2.3)
“Determine the probability of combined events for dependent and independent events.”
What it means
Students calculate the probability of two events happening together by first deciding whether the second event's outcome is affected by the first (dependent) or not (independent). Independent events use P(A) × P(B); dependent events require recalculating probability after the first outcome, often shown with a tree diagram.
How it is examined
Paper 1 items ask students to compute the probability of two combined events quickly, often using coins, dice or drawing items with or without replacement. Paper 2 typically presents multi-step scenarios (e.g.
drawing balls without replacement) requiring a tree diagram or full working to justify whether events are dependent or independent before calculating.
Worked example
A box contains 5 red pens and 3 blue pens. A pen is drawn and then a second pen is drawn.
Find the probability that both pens are red if (a) the first pen is replaced before the second draw, (b) the first pen is not replaced.
- (a) With replacement: total pens stay at 8 for both draws, so P(red) = 5/8 each time; the two draws are independent.
- P(both red) = 5/8 × 5/8 = 25/64.
- (b) Without replacement: P(first red) = 5/8. After removing 1 red pen, 4 red and 3 blue pens remain (7 total), so P(second red | first red) = 4/7.
- P(both red) = 5/8 × 4/7 = 20/56 = 5/14.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I tell if two events are independent or dependent?
Check whether the outcome of the first event changes the possibilities for the second. If items are replaced or the events are truly separate trials, they're independent; if items are removed (drawn without replacement) so the sample space shrinks, they're dependent.
What formula do I use for independent events?
Use P(A and B) = P(A) × P(B), where P(A) and P(B) are found separately without adjusting for previous outcomes, since neither event affects the other's probability.
How do I handle dependent events in the exam?
Recalculate the probability of the second event based on what remains after the first outcome, then multiply: P(A and B) = P(A) × P(B after A). A tree diagram with updated numbers on each branch helps avoid mistakes.