Probability of Combined Events · Form 4
Probability of Combined Events: Common Mistakes
The mistakes that quietly cost marks in Probability of Combined Events, and how to avoid each one in the SPM exam.
In our experience teaching Probability of Combined Events, most lost marks come from a small set of repeated slips, not from a lack of understanding. Here they are, with the fix for each.
Mistakes to avoid
- Adding when you should multiply along a branch
- Not reducing the probability on the second draw of a dependent event
- Forgetting to subtract the overlap for non-mutually-exclusive events
Six more slips that quietly cost marks
- What students write: for two greens without replacement, 6/10 × 6/9. → Why it loses marks: the denominator was reduced but the numerator was not, so the green already drawn is counted twice. → Correct working: one green is gone too, so 6/10 × 5/9 = 30/90 = 1/3.
- What students write: P(at least one yellow) = 4/10 + 4/9. → Why it loses marks: adding the single-draw chances double-counts and can exceed 1. → Correct working: use the complement, P(at least one yellow) = 1 − P(no yellow) = 1 − (6/10 × 5/9) = 1 − 30/90 = 60/90 = 2/3.
- What students write: P(exactly one yellow) = 6/10 × 4/9 only. → Why it loses marks: only the green-then-yellow order is counted; yellow-then-green is missed. → Correct working: add both orders, (6/10 × 4/9) + (4/10 × 6/9) = 24/90 + 24/90 = 48/90 = 8/15.
- What students write: P(even or greater than 3 on a die) = 3/6 + 3/6 = 1. → Why it loses marks: the events overlap (4 and 6 are in both), so they are not mutually exclusive. → Correct working: P = 3/6 + 3/6 − 2/6 = 4/6 = 2/3.
- What students write: a machine makes a good part with probability 2/3 on each independent run; for two runs, P(both good) by adding 2/3 + 2/3. → Why it loses marks: 'both … and …' means multiply along the branch, not add across it. → Correct working: P(both good) = 2/3 × 2/3 = 4/9.
- What students write: on a first draw, P(green) = 6/10 and P(not green) = 6/10. → Why it loses marks: the two branches at one point must total 1, and 6/10 + 6/10 ≠ 1. → Correct working: P(not green) = 1 − 6/10 = 4/10, so the branches read 6/10 and 4/10.
Catch these before you hand in
Run three quick tests on any combined-events answer. Is every probability between 0 and 1?
Do the branches at each node of your tree add to 1? Did you multiply along a branch for 'and' and add across branches for 'or', subtracting the overlap only when the events can co-occur?
Any 'no' points straight at one of the six slips above.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
My probabilities added to more than 1, what went wrong?
A probability above 1 is impossible, so it is a useful alarm. Usually you added two branch probabilities that should have been multiplied, or you handled 'at least one' by summing instead of using 1 − P(none).
Go back, check every 'and' is a multiply, and use the complement for 'at least'.
Why is 'exactly one' different from 'at least one'?
'Exactly one' means one success and one failure, so you add the two orders and stop there. 'At least one' means one or two successes, that is everything except zero, so the quickest route is 1 − P(none).
Confusing them makes you count the both-success case in the wrong place.
How can I quickly check my tree diagram is correct?
Two fast checks catch most errors. First, the branches leaving any single point must add to 1, for example 5/8 and 3/8.
Second, the probabilities at all the branch tips must add to 1 as well. If either total is off, a probability is wrong before you even combine them.