“Add for or, multiply for and” is a useful starting reminder, but it leaves out two important questions. Can the events overlap? Does the probability of the second event depend on the first? Without those checks, a familiar rule can produce an incorrect answer.
In AS Level probability, addition combines alternative ways an event can occur. Multiplication finds the probability of events happening together, using a conditional probability when necessary. The difficult part is usually identifying those events accurately.
This guide uses original practice examples. Check the relevant component and examination-year syllabus when choosing papers. If you are building the topic from the beginning, Abeel Eschool’s AS Level Math S1 course is a relevant place to review available lessons.
Define the event before selecting an operation
Suppose a question asks for the probability that a selected card is red or has a particular symbol. Name the two events before calculating. Event A might mean “the card is red,” while event B means “the card carries a star.”
Now ask whether a single card can satisfy both conditions. If red star cards exist, A and B overlap. Adding P(A) and P(B) counts those cards twice. The issue is not your arithmetic; it is the way the outcomes have been grouped.
For several selections, define the position as well. “A red card on the first draw” and “a red card on the second draw” are different events. Clear labels make it easier to recognise whether the question asks about one selection, two selections or a choice between different possible sequences.
A short sentence can prevent several lines of incorrect working. Write what is selected, when it is selected and which outcomes qualify.
Add mutually exclusive alternatives
Events are mutually exclusive if they cannot happen together in the situation being considered. On a single roll of an ordinary six-sided die, obtaining a two and obtaining a five are mutually exclusive.
Assuming the die is fair, the probability of a two or a five is 1/6 + 1/6 = 2/6 = 1/3. There is no overlapping outcome to remove. The acceptable results form two separate parts of the same sample space.
Notice the wording “on a single roll.” Across two rolls, a two and a five can both occur. Whether events are mutually exclusive depends on how they are defined, not on the numbers or colours mentioned in the question.
During revision, test exclusivity with a concrete question: “Can I describe an outcome that belongs to both events?” If the answer is yes, simple addition requires an overlap adjustment.
Subtract the overlap when alternatives share outcomes
Consider twelve equally likely cards. Five are blue, four have stars, and two are both blue and starred. The probability of blue or starred is (5 + 4 − 2)/12 = 7/12.
The general addition rule is P(A or B) = P(A) + P(B) − P(A and B). The subtraction removes the second count of the overlap. It does not remove the overlapping outcomes entirely; they still belong in the event “A or B.”
A small two-way table can make the example clearer. Two cards are blue and starred; three are blue without stars; two are starred without being blue; the remaining five are neither. Adding the first three groups gives seven qualifying cards.
Do not automatically assume that “or” means “exactly one.” In ordinary probability notation, A or B includes the possibility that both occur. If the question asks for exactly one, exclude the overlap from both groups instead.
Multiplication follows a sequence of conditions
Imagine a bag containing three green and two white counters. Two counters are drawn without replacement. To find the probability that both are green, start with the first draw: its probability of being green is 3/5.
After a green counter has been drawn and not replaced, two green counters remain among four counters. The conditional probability of green on the second draw is 2/4. Multiplying gives 3/5 × 2/4 = 3/10.
You multiplied because both stages must happen. You changed the second fraction because the first stage changed the bag. Multiplication does not require the original events to be independent; it requires the correct conditional probability for each stage.
A useful general statement is P(A and B) = P(A) × P(B given A). Independence is the special case in which P(B given A) equals P(B).
Know what independence actually means
Events A and B are independent when learning that A occurred does not change the probability of B. For independent events, P(A and B) = P(A)P(B).
For example, under the model of independent rolls of a fair die, the probability of a six on each of two rolls is 1/6 × 1/6 = 1/36. The first result does not change the probabilities on the next roll.
Independence is different from mutual exclusivity. If two events each have positive probability and cannot occur together, learning that one happened rules out the other. They are therefore not independent.
Avoid deciding that events are independent because their names sound unrelated. Use the assumptions in the question, or check a relevant probability relationship. Sampling procedures, shared conditions and restrictions can create dependence that is easy to overlook.
Use a tree to separate paths from branches
A probability tree shows stages. Probabilities on branches from one point describe the alternatives available at that stage. These branch probabilities should total one when they cover all possible outcomes.
A complete path describes a sequence. Multiply along that path, using the probabilities attached to its own branches. If several different complete paths satisfy the question, add their probabilities after checking that those paths are mutually exclusive.
Return to the bag with three green and two white counters. Exactly one green in two draws without replacement can occur as green then white, or white then green. The probabilities are 3/5 × 2/4 and 2/5 × 3/4. Adding gives 3/10 + 3/10 = 3/5.
The two paths cannot both describe the same ordered pair of draws. That is why their probabilities can be added directly. This example uses multiplication and addition in one solution without treating them as competing choices.
Compare replacement and no replacement explicitly
Suppose the first counter is returned to the bag before the second draw and the bag is mixed appropriately. Under the usual equally likely random-draw model, the probability of green on each draw is again 3/5.
The probability of two greens is then 3/5 × 3/5 = 9/25. Without replacement it was 3/10. Both calculations multiply along the required sequence, but they use different second-stage probabilities.
Write “with replacement” or “without replacement” beside your working during practice. Then check the counts after each possible first result. With no replacement, the second denominator is smaller; the second numerator depends on which colour was removed.
Do not rely only on a reduced denominator as a memory rule. If the question involves a different process, the actual conditional probability must come from that process.
Use a complement when it makes the event simpler
“At least one” often suggests considering the opposite event. For two independent rolls of a fair die, the probability of at least one six is one minus the probability of no sixes.
The no-six probability on each roll is 5/6, so the required probability is 1 − (5/6 × 5/6) = 11/36. This avoids separately counting a six on the first roll, a six on the second, and two sixes while correcting their overlap.
A complement is not a separate rule that replaces careful interpretation. You must identify the opposite event correctly. The complement of “at least two” is “zero or one,” not simply “none.”
Check that your original event and its complement are mutually exclusive and together cover every possible outcome. Their probabilities should add to one.
Diagnose the first wrong line in a solution
Suppose your answer exceeds one. Check whether overlapping events were added without subtraction, whether a conditional denominator was incorrect, or whether you added probabilities from incompatible stages. A probability outside the interval from zero to one signals a problem, but a result inside that interval can still be wrong.
If your answer differs from a mark scheme, locate the first point where your reasoning diverges. Do not begin by copying the final fraction. Ask whether the error was event definition, overlap, replacement, independence or arithmetic.
Keep a short error log with the question reference and one corrective action. “Before adding, list the overlap” is more useful than “revise probability.” To organise the wider routine, use Abeel’s guide to practising Cambridge past papers.
Practise deciding the method without a calculator
Take several short questions and write only the event, a sketch or table, and the proposed calculation. Delay the numerical evaluation. This tests whether you recognise the structure without letting calculator work hide uncertainty.
Next, complete the calculations and explain why the result is plausible. If an event becomes harder to satisfy because it requires two successes, a larger answer than either individual event deserves a second look.
Finally, mix topics. A worksheet labelled “independent events” tells you a major part of the method before you start. Mixed practice asks you to decide whether independence is appropriate.
At Abeel Eschool, students can explore subject support and recorded learning options. Check the course catalogue for current choices, or ask a specific question through admissions and FAQs. Bring your attempted event definition so the discussion starts with the actual difficulty.
Frequently asked questions
Do I always add when the question says “or”?
Use the addition rule, but account for overlap. If the events are mutually exclusive, the overlap is zero. If the question means exactly one event, the required outcome is different from an inclusive “or.”
Do I always multiply when the question says “and”?
A joint event can be calculated by multiplying one probability by the appropriate conditional probability. Use the simpler product of two unconditional probabilities only when independence is justified.
Are mutually exclusive events independent?
Two mutually exclusive events with positive probabilities are not independent. If one occurs, the other cannot occur, so its probability changes.
Why do I add after multiplying on a tree diagram?
Multiplication gives the probability of a complete path. Addition combines different qualifying paths when those paths are mutually exclusive.
How can I check an answer quickly?
Check that the result lies between zero and one, test whether the event is more or less restrictive than a related event, and compare with a small outcome list where practical. These checks support, but do not replace, correct reasoning.
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