17 questions arranged by DBE cognitive level — probability rules, Venn diagrams, tree diagrams, contingency tables, and the fundamental counting principle, including restricted arrangements and combined counting-and-probability problems. Work each one on paper first, then reveal the memo.
17
practice questions
4
cognitive levels
17
worked memos
100%
independently verified
How to use this bank.
Start at Level 1 and move up — don't jump to Level 4 first.
Before choosing a rule, always decide: are the events mutually exclusive, independent, or neither? That single decision determines which formula applies.
For any arrangement question, decide first whether repetition is allowed and whether order matters.
Reveal the memo only after a genuine attempt.
Accuracy note: every question below was independently solved from scratch before publication, cross-checked against its own working rather than assumed correct. Six questions are adapted from real DBE/provincial exam papers (labelled with their source below); the rest are original "Equation Station SA Practice Question" items written to match the exact CAPS scope taught in the Summary Notes for this topic.
L1 — Knowledge 4 Qs
L2 — Routine Procedures 5 Qs
L3 — Complex Procedures 4 Qs
L4 — Problem Solving 4 Qs
24%
Level 1 | Knowledge
Direct Rule Substitution
One direct application of the complementary, sum, or product rule, plus recognising when the general addition rule applies.
Events \(A\) and \(B\) are mutually exclusive, with \(P(A)=0{,}2\) and \(P(B)=0{,}5\). Calculate \(P(A\text{ or }B)\).
Memo
✓ Mutually exclusive: \(P(A\text{ or }B)=P(A)+P(B)=0{,}2+0{,}5\)✓ \(\boxed{0{,}7}\)
Q3Equation Station SA Practice Question2 marks
Independent Events
The Product Rule
Events \(A\) and \(B\) are independent, with \(P(A)=0{,}4\) and \(P(B)=0{,}3\). Calculate \(P(A\text{ and }B)\).
Memo
✓ Independent: \(P(A\text{ and }B)=P(A)\times P(B)=0{,}4\times0{,}3\)✓ \(\boxed{0{,}12}\)
Q4Equation Station SA Practice Question1 mark
General Addition Rule
When the General Rule Applies
For which events \(A\) and \(B\) does \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\) apply?
Memo
✓ This rule works for ANY two events, whether or not they overlap.✓ When mutually exclusive, \(P(A\text{ and }B)=0\) and it reduces to the sum rule.✓ \(\boxed{\text{Any two events }A\text{ and }B}\)
29%
Level 2 | Routine Procedures
One Established Method
The general addition rule with real numbers, the fundamental counting principle, arranging distinct objects, a tree diagram, and reading a contingency table.
Q5Equation Station SA Practice Question3 marks
General Addition Rule
Tea or Coffee
In a class of 50 learners, 28 like tea, 20 like coffee, and 12 like both. Calculate \(P(\text{tea or coffee})\) for a randomly chosen learner.
Memo
✓ \(P(\text{tea or coffee})=\dfrac{28+20-12}{50}=\dfrac{36}{50}\)✓ \(\boxed{0{,}72}\)
Q6Equation Station SA Practice Question2 marks
Fundamental Counting Principle
PIN Codes with Repetition
A 4-digit PIN code is formed using the digits 0-9, and digits may be repeated. How many different PIN codes are possible?
Memo
✓ Each of the 4 positions has 10 independent choices: \(10\times10\times10\times10\)✓ \(\boxed{10\,000}\)
Q7Equation Station SA Practice Question2 marks
Arrangements
Arranging Distinct Objects
In how many ways can 5 different books be arranged in a row on a shelf?
A bag of balls contains 7 green balls and 5 yellow balls. Sihle randomly selects two balls from the bag, one at a time, without replacing the first ball. Calculate \(P(\text{one yellow and one green ball, in any order})\).
Memo
✓ Two orders give one of each colour: \(P=\dfrac{5}{12}\times\dfrac{7}{11}+\dfrac{7}{12}\times\dfrac{5}{11}=\dfrac{70}{132}\)✓ \(\boxed{\dfrac{35}{66}}\approx0{,}53\)
Q9Equation Station SA Practice Question2 marks
Contingency Tables
Reading a Marginal Probability
200 learners were surveyed: 65 male learners own a cellphone and 15 do not; 85 female learners own a cellphone and 35 do not. Calculate \(P(\text{female})\) for a randomly selected learner.
Memo
✓ Total female \(=85+35=120\)✓ \(P(\text{female})=\dfrac{120}{200}\)✓ \(\boxed{0{,}6}\)
24%
Level 3 | Complex Procedures
Multi-Step Methods
A three-event Venn diagram, restricted arrangements in both directions, and testing statistical independence from a contingency table.
Q10Free State, September 20254 marks
Venn Diagrams
Three-Event Venn Diagram
A survey of 100 Grade 12 learners asked which of three subjects (Mathematics M, Physical Sciences P, Life Sciences L) they prefer: 8 prefer all three, 12 prefer M and P, 5 prefer P and L but not M, \(x\) prefer M and L but not P, 61 prefer M, 19 prefer P, 73 prefer L, and 14 prefer none of the three. Calculate the value of \(x\).
Memo
✓ M-and-P-only (excluding the triple overlap) \(=12-8=4\)✓ Only M \(=61-4-x-8=49-x\); Only P \(=19-4-5-8=2\); Only L \(=73-x-5-8=60-x\)✓ All regions sum to 100: \((49-x)+2+(60-x)+4+x+5+8+14=100\Rightarrow142-x=100\)✓ \(\boxed{x=42}\)
Q11Gauteng Preparatory Exam, September 20224 marks
Fundamental Counting Principle
Two Separate Groups, Each Together
Four different Economics books and three different Life Sciences books are placed on a shelf. In how many ways can all 7 books be arranged if all the Economics books must be together AND all the Life Sciences books must be together?
Memo
✓ Glue each subject into its own block: 2 blocks can be ordered \(2!\) ways✓ Arrange inside the blocks: \(4!\) (Economics) \(\times\,3!\) (Life Sciences)✓ \(2!\times4!\times3!=2\times24\times6\)✓ \(\boxed{288}\)
Q12Equation Station SA Practice Question4 marks
Fundamental Counting Principle
Must NOT Sit Together
In how many ways can 7 people be arranged in a row so that two particular people, P and Q, are NOT seated next to each other?
Memo
✓ Total: \(7!=5\,040\)✓ Together: \(6!\times2!=720\times2=1\,440\)✓ Not together \(=5\,040-1\,440\)✓ \(\boxed{3\,600}\)
Q13Western Cape Metro South, September 20223 marks
Contingency Tables
Testing Independence
A survey of 3\,000 drivers found: of 1\,970 male drivers, 170 failed their driving test and 1\,800 passed; of 1\,030 female drivers, 30 failed and 1\,000 passed. Is passing/failing the test independent of gender?
Memo
✓ \(P(\text{male})=\dfrac{1\,970}{3\,000}\approx0{,}657\), \(P(\text{failed})=\dfrac{200}{3\,000}\approx0{,}067\)✓ Product \(\approx0{,}657\times0{,}067\approx0{,}0438\)✓ \(P(\text{male and failed})=\dfrac{170}{3\,000}\approx0{,}0567\)✓ \(0{,}0438\neq0{,}0567\), so \(\boxed{\text{NOT independent}}\)
23%
Level 4 | Problem Solving
Combined Skills
Arrangements combined with probability, a subtle counting restriction, three dependent draws, and ordered selection without repetition.
Q14Free State, September 20255 marks
Arrangements & Probability
Arrangements and Probability Combined
The letters of the word MATHEMATICIAN are arranged in a random order, with each distinct arrangement counted once. Calculate the probability that the arrangement ends with the letter M.
Memo
✓ MATHEMATICIAN has 13 letters: M×2, A×3, T×2, I×2, plus H, E, C, N once each: total \(=\dfrac{13!}{2!\,3!\,2!\,2!}=129\,729\,600\)✓ Fix one M at the end; arrange the remaining 12 letters (M×1, A×3, T×2, I×2, +4 singles): \(\dfrac{12!}{3!\,2!\,2!}=19\,958\,400\)✓ Probability \(=\dfrac{19\,958\,400}{129\,729\,600}\)✓ \(\boxed{\dfrac{2}{13}}\approx0{,}154\)
Q15Equation Station SA Practice Question4 marks
Fundamental Counting Principle
Read the Restriction Carefully
A 4-digit PIN code is formed using the digits 0-9, with NO digit repeated (a leading zero is allowed, e.g. 0123 is valid). How many different PIN codes are possible?
Memo
✓ 10 choices for the 1st digit, 9 remaining for the 2nd, 8 for the 3rd, 7 for the 4th✓ \(10\times9\times8\times7\)✓ \(\boxed{5\,040}\)
Q16Equation Station SA Practice Question5 marks
Tree Diagrams
Three Dependent Draws
A bag contains 5 red and 3 blue balls. Three balls are drawn one at a time, without replacement. Calculate the probability that exactly 2 of the 3 balls drawn are red.
Memo
✓ Exactly 2 red can occur in 3 orders: RRB, RBR, BRR✓ Each order: \(\dfrac{5}{8}\times\dfrac{4}{7}\times\dfrac{3}{6}=\dfrac{5}{28}\)✓ Total \(=3\times\dfrac{5}{28}\)✓ \(\boxed{\dfrac{15}{28}}\)
Q17Western Cape Metro South, September 20225 marks
Fundamental Counting Principle
Grouped by Type
A fruit basket contains 7 bananas, 4 apples, 3 oranges and 5 guavas, with every piece of fruit physically distinct even within its own type. In how many ways can the 19 pieces be arranged in a row if fruit of the same type must all be grouped together?
Memo
✓ Glue each of the 4 fruit types into its own block; the 4 blocks can be ordered \(4!\) ways✓ Arrange inside each block: \(7!\) (bananas) \(\times\,4!\) (apples) \(\times\,3!\) (oranges) \(\times\,5!\) (guavas)✓ \(4!\times7!\times4!\times3!\times5!\)✓ \(\boxed{2\,090\,188\,800}\)