Module 19 of 30Lesson 1 of 12 · P-C1 Single Events and the Complement Rule
Probability strand · Unit P-C1 · Syllabus C8.1

Single Events & the Complement Rule

🔵 Core levelP(not A) = 1 − P(A)

1 Learn it
🔑 The exam writes probability with brackets: P(A) means “the probability of event A”. The complement rule: \(P(\text{not } A) = 1 – P(A)\), because A either happens or it does not — nothing else is possible.

All the probabilities of every possible outcome of an experiment add to 1. That gives you missing values for free: if a spinner lands red, blue or green with P(red) = 0.35 and P(blue) = 0.4, then P(green) = 1 − 0.35 − 0.4 = 0.25.

Watch the wording: “a card is chosen at random” means every card is equally likely — that phrase is your licence to count outcomes.

2 Watch it work
EXAMPLE 1

The probability that a bus is late is 0.15. Find P(not late).

1 − 0.15 = 0.85.

EXAMPLE 2

A spinner lands on red, blue or green. P(red) = 0.35, P(blue) = 0.4. Find P(green).

1 − 0.35 − 0.4 = 0.25.

EXAMPLE 3

The letters of the word KALAHARI are written on cards and one is picked at random. Find P(A) and P(not A).

KALAHARI has 8 letters with three A’s → P(A) = \(\dfrac{3}{8}\); P(not A) = 1 − \(\dfrac{3}{8}\) = \(\dfrac{5}{8}\).

EXAMPLE 4

A box holds only 2-pula and 5-pula coins. A coin is taken at random; P(5-pula) = \(\dfrac{7}{12}\). There are 36 coins. How many are 2-pula?

P(2-pula) = 1 − \(\dfrac{7}{12}\) = \(\dfrac{5}{12}\);   \(\dfrac{5}{12}\) × 36 = 15 coins.

3 Your turn — try first, then reveal
🚫 No calculator[1 mark]

Q1. P(a seed grows) = 0.7. Find P(the seed does not grow).

👀 Show answer
✍️ How your working should look in the exam
1 − 0.7 = 0.3   ✓ B1
Why this works
Complement rule.

🚫 No calculator[2 marks]

Q2. A spinner lands on 1, 2, 3 or 4. P(1) = 0.2, P(2) = 0.3, P(3) = 0.1. Find P(4).

👀 Show answer
✍️ How your working should look in the exam
0.2 + 0.3 + 0.1 = 0.6   ✓ M1
P(4) = 1 − 0.6 = 0.4   ✓ A1
Why this works
All outcomes together must total 1.

🚫 No calculator[2 marks]

Q3. One letter of GABORONE is picked at random. Find P(O) and P(not O).

👀 Show answer
✍️ How your working should look in the exam
P(O) = 28 = 14   ✓ B1
P(not O) = 34   ✓ B1
Why this works
GABORONE has 8 letters, two of them O.

🔢 Calculator[3 marks]

Q4. A drawer holds only black and white socks. A sock is taken at random; P(black) = \(\dfrac{5}{9}\). There are 45 socks. How many are white?

👀 Show answer
✍️ How your working should look in the exam
P(white) = 1 − 59 = 49   ✓ M1
49 × 45   ✓ M1
= 20 socks   ✓ A1
Why this works
Complement first, then fraction OF the total.

🚫 No calculator[2 marks]

Q5. P(a team wins) = 0.55 and P(the match is drawn) = 0.2. Find P(the team loses).

👀 Show answer
✍️ How your working should look in the exam
1 − 0.55 − 0.2   ✓ M1
= 0.25   ✓ A1
Why this works
Win, draw and lose cover every possibility, so they sum to 1.

4 Common mistakes — don’t lose easy marks
  • Subtracting from 100 instead of 1 when working in decimals (0.7 → 0.3, not 99.3).
  • Forgetting repeated letters: KALAHARI has THREE A’s, not one.
  • Complement of a fraction done wrong: 1 − \(\dfrac{5}{9}\) = \(\dfrac{4}{9}\), not \(\dfrac{5}{9}\) flipped.
  • Not simplifying when the answer line asks for simplest form.
5 How the exam asks this

Missing-probability tables are a guaranteed 2-marker. Next: what happens when you actually run the experiment — relative frequency in P-C2.

✅ Take the Check Yourself quiz below, then continue to P-C2 Relative Frequency.

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