Probability of Combined Events · Form 4
Independent vs dependent events, in plain terms
Two events are independent when one happening does not change the probability of the other, and dependent when it does. In this chapter the giveaway is usually whether the first item is put back before the second is chosen.
Independent: the second event doesn't 'feel' the first
Events are independent when the outcome of the first leaves the probabilities of the second exactly as they were. Tossing a coin and then rolling a die is independent, the coin has no memory, so the die is still a fair 1 in 6.
Drawing a ball, noting it, and putting it back before the next draw ('with replacement') is independent for the same reason.
Dependent: the first outcome rewrites the second
Events are dependent when the first outcome changes what is left, so the second probability must be recalculated. Taking two sweets from a bag of 5 without putting the first back ('without replacement') leaves only 4 sweets, so the second draw is out of 4, not 5.
This matters because it decides whether the second-stage numbers stay the same or shift, a difference that runs right through any tree diagram you build.
Reading the question, and the trap
Scan for words like 'replaced', 'returned', or 'put back' (independent) versus 'not replaced', 'without replacement', or a second item simply removed from what remains (dependent). The common trap is assuming every pair of events is independent and reusing the first probability twice; the second draw's total has often shrunk.
Note too that 'dependent' is about the probability changing, not about the events being related in meaning.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How can I tell whether two events in a question are independent or dependent?
Check whether the first outcome changes what's left for the second. Phrases like "without replacement" or "not returned to the bag" signal dependent events, because the sample space shrinks.
"With replacement" or choosing from separate groups keeps probabilities unchanged, so the events are independent.
What's the most common mistake students make with dependent events?
Using the same total for both draws even after an item is removed. For dependent events, the second probability's denominator (and sometimes numerator) must reflect what remains after the first outcome, forgetting this adjustment is the easiest mark to lose.
How do exam questions usually test independent and dependent events?
Through a tree diagram or word problem asking for something like P(both red) or P(different colours). You must identify replacement or non-replacement, label each branch's probability correctly, then multiply along the paths, marks reward each correctly adjusted stage, not just the final answer.