Probability of Combined Events · 9.2.1
Dependent vs independent events
Students learn to decide whether the outcome of one event affects the probability of another. If the first event changes what is possible or the chances for the second (such as drawing without replacement), the events are dependent; if not (such as drawing with replacement or separate trials), they are independent.
The official learning standard (9.2.1)
“Differentiate between dependent and independent events.”
What it means
Students learn to decide whether the outcome of one event affects the probability of another. If the first event changes what is possible or the chances for the second (such as drawing without replacement), the events are dependent; if not (such as drawing with replacement or separate trials), they are independent.
How it is examined
This standard is usually tested as part of a larger probability question in Paper 2, where students must first classify events as dependent or independent, commonly in with-replacement versus without-replacement scenarios, before choosing the correct method to calculate the combined probability. Paper 1 may test classification alone.
Worked example
A bag contains 4 red and 3 blue marbles. State whether the two draws described are dependent or independent, with a reason: (a) Two marbles are drawn one after another, with the first marble replaced before the second draw.
(b) Two marbles are drawn one after another, without replacing the first marble.
- (a) With replacement: after the first marble is drawn, it is returned to the bag, so the bag still contains 4 red and 3 blue marbles (7 total) for the second draw.
- Since the composition of the bag is unchanged, the probability of the second draw does not depend on the outcome of the first draw, the events are independent.
- (b) Without replacement: after the first marble is drawn, it is not returned, so only 6 marbles remain and the number of red or blue marbles left depends on what was drawn first.
- Since the probability of the second draw changes based on the first outcome, the events are dependent.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Are rolling a die and tossing a coin dependent or independent events?
Independent. The die and the coin are separate objects with their own outcomes, what the die shows has no effect at all on what the coin shows, or on the coin's probabilities, so the two events do not depend on each other.
Does 'without replacement' always mean the events are dependent?
At this level, yes. Removing an item without replacing it changes the total number of items and possibly the number of a particular type remaining, so the probability of the second event always depends on the first outcome.
Why does it matter whether events are dependent or independent?
It determines how you calculate the probability of both events happening together. Independent events use P(A) × P(B) directly, while dependent events require the probability of the second event to be adjusted based on what happened in the first.