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Grade 12 · Mathematics · Paper 1

Probability

Master counting principles, permutations, combinations and probability rules — with interactive worked examples.

Probability Fundamentals

Key vocabulary:
  • Sample space S — the set of all possible outcomes of an experiment.
  • Event — any subset of the sample space.
  • n(A) — the number of outcomes in event A.
  • n(S) — the total number of equally likely outcomes.
Theorem 1📌 Learn for exam
Classical Probability

When all outcomes are equally likely:

P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}

Probability is always a value between 0 and 1 inclusive: 0≤P(A)≤10 \le P(A) \le 1

Theorem 2📌 Learn for exam
Complement Rule
P(A′)=1−P(A)P(A') = 1 - P(A)

A′ (read "A complement" or "not A") is the event that A does NOT occur. Since A and A′ together make up all of S, their probabilities must sum to 1.

Theorem 3📌 Learn for exam
Addition Rule (General)

For any two events A and B:

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

We subtract P(A∩B)P(A \cap B) because the outcomes in both A and B get counted twice if we simply add.

Theorem 4📌 Learn for exam
Mutually Exclusive Events

Events A and B are mutually exclusive (cannot both occur) when:

P(A∩B)=0P(A \cap B) = 0

In that case the addition rule simplifies to:

P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
Tip
On a Venn diagram, mutually exclusive events are drawn as circles that do not overlap. Complementary events A and A′ are always mutually exclusive.
▸ Venn Diagram — drag dots to change P(A) and P(B)use the ± buttons to change the overlap
Loading diagram…
Reading the Venn diagram:
  • The rectangle represents S — everything inside it is a possible outcome.
  • The green circle is event A, the blue circle is event B.
  • The overlapping lens (yellow tint) is A∩BA \cap B — outcomes in both A and B.
  • P(A∪B)P(A \cup B) is the total shaded area (green + blue + overlap counted once).
  • Anything outside both circles but inside the rectangle is (A∪B)′(A \cup B)'.
Worked Example

Bag of balls

A bag contains 5 red, 3 blue and 2 green balls. One ball is drawn at random. Find: (a) P(red), (b) P(not green), (c) P(red or blue).

Worked Example

Addition rule with overlap

P(A) = 0.6, P(B) = 0.5, P(A and B) = 0.3. Find P(A or B).