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Grade 12 Probability — How to Identify the Question Type

Grade 12 probability extends Venn diagrams to counting principles: the fundamental counting principle, permutations, and combinations. Read the question carefully — 'how many arrangements' signals counting, while 'what is the probability' signals a fraction.

Type 1: Venn diagrams and probability rules (revision and extension)

Trigger words: "Draw a Venn diagram", "P(A and B)", "P(A or B)", "are the events independent"

Trigger structure: Two or three overlapping events. The question tests the addition rule, complementary events, or independence.

Method (no numbers — just the steps)

  1. Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
  2. Complementary: P(A') = 1 − P(A)
  3. Mutually exclusive: P(A ∩ B) = 0
  4. Independent: P(A ∩ B) = P(A) × P(B) — check this numerically
Grade 12Paper 1Level 2 — Routine Procedures
11.1 P(A) = 0.4, P(B) = 0.3, and P(A ∩ B) = 0.12. Determine whether A and B are independent.

Source: DBE Mathematics Paper 1, November 2022, Question 11.1

See the progression — same type, increasing difficulty

Easy
P(A) = 1/3 and P(B) = 1/4. If A and B are independent, find P(A ∩ B) and P(A ∪ B).

Practice question — not sourced from a past paper.

Medium
P(A ∪ B) = 0.75, P(A) = 0.5, and A and B are mutually exclusive. Find P(B).

Practice question — not sourced from a past paper.

Hard
In a class of 40, 25 play sport, 18 play music, and 8 play both. Draw a Venn diagram and find the probability that a learner selected at random plays sport but NOT music.

Practice question — not sourced from a past paper.

Type 2: The fundamental counting principle

Trigger words: "How many ways", "In how many different ways can", "arrangements"

Trigger structure: Objects fill positions one at a time, and the number of choices for each position is known or can be calculated.

Do not confuse with: Combinations — the counting principle counts arrangements where ORDER matters. If order doesn't matter, use combinations.

Method (no numbers — just the steps)

  1. Identify the positions (seats, digits, slots)
  2. Determine how many options are available for each position
  3. Multiply all the options together
  4. If repetition is NOT allowed, reduce by 1 for each position filled
Grade 12Paper 1Level 2 — Routine Procedures
12.1 How many 4-digit codes can be formed using the digits 1 to 9 if no digit may be repeated?

Source: DBE Mathematics Paper 1, November 2023, Question 12.1

Common mistake

Forgetting to reduce the number of choices after each position when repetition is not allowed.

See the progression — same type, increasing difficulty

Easy
How many 3-digit numbers can be formed from {1, 2, 3, 4, 5} if repetition is allowed?

Practice question — not sourced from a past paper.

Medium
How many arrangements of the letters A, B, C, D, E, F are possible if the arrangement must start with A and end with F?

Practice question — not sourced from a past paper.

Hard
How many even 4-digit numbers greater than 5 000 can be formed from {2, 3, 5, 6, 7, 8} without repetition?

Practice question — not sourced from a past paper.

Type 3: Permutations

Trigger words: "Permutations", "arrangements where order matters", "P(n, r)" or "nPr"

Trigger structure: Selecting r items from n where the ORDER of selection matters.

Do not confuse with: Combinations — in combinations, {A, B, C} and {C, A, B} are the same selection. In permutations, they are different arrangements.

Method (no numbers — just the steps)

  1. Use: P(n, r) = n! / (n − r)!
  2. If ALL n items are arranged: n! (factorial)
  3. For identical items: n! / (a! × b! × ...) where a, b, ... are the counts of each repeated item
Grade 12Paper 1Level 2 — Routine Procedures
12.2 In how many ways can the letters of the word MATHS be arranged?

Source: DBE Mathematics Paper 1, November 2021, Question 12.2

See the progression — same type, increasing difficulty

Easy
In how many ways can 4 people be seated in a row of 4 chairs?

Practice question — not sourced from a past paper.

Medium
In how many ways can the letters of the word MISSISSIPPI be arranged?

Practice question — not sourced from a past paper.

Hard
In how many ways can 6 people be seated in a row if two specific people must NOT sit next to each other?

Practice question — not sourced from a past paper.

Type 4: Combinations

Trigger words: "Combinations", "how many ways can ... be chosen", "selected", "C(n, r)" or "nCr"

Trigger structure: Selecting r items from n where ORDER does NOT matter — only which items are selected counts.

Do not confuse with: Permutations — if 'arrange' or 'order' appears, use permutations. If 'choose', 'select', or 'committee' appears, use combinations.

Method (no numbers — just the steps)

  1. Use: C(n, r) = n! / (r! × (n − r)!)
  2. For multi-group selections: multiply the combinations for each group
Grade 12Paper 1Level 2 — Routine Procedures
12.2 A committee of 3 must be chosen from 8 people. In how many ways can the committee be selected?

Source: DBE Mathematics Paper 1, November 2022, Question 12.2

Common mistake

Using permutations when the question says 'choose' — selecting 3 people from 10 for a committee is a combination, not a permutation.

See the progression — same type, increasing difficulty

Easy
How many ways can 2 books be chosen from a shelf of 7?

Practice question — not sourced from a past paper.

Medium
A team of 4 is chosen from 5 men and 6 women. The team must include exactly 2 women. In how many ways can the team be chosen?

Practice question — not sourced from a past paper.

Hard
From a group of 10 people (4 teachers and 6 learners), a committee of 5 is chosen. How many committees contain at least one teacher?

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.