Probability is a number between 0 and 1 that says how likely something is. Calculate it by counting outcomes: favourable ÷ total. This guide walks the KS3 probability syllabus with worked examples and the two error patterns that confuse most Year 8 students.

The probability scale

Probabilities live between 0 and 1:

0 ─────────────── 0.5 ─────────────── 1
impossible    even chance         certain
  • 0 = won't happen (a regular six-sided die rolling a 7)
  • 0.5 = 50/50 (a fair coin showing heads)
  • 1 = will definitely happen (rolling a number between 1 and 6 on a six-sided die)

Can be written as fractions (1/2), decimals (0.5), or percentages (50%). All three mean the same thing.

The core formula

P(event) = number of favourable outcomes ÷ total number of outcomes

This assumes all outcomes are equally likely (a fair coin, a fair die, a well-shuffled deck).

Worked example 1: Fair coin

Probability of heads:

  • Favourable outcomes: heads (1)
  • Total outcomes: heads, tails (2)
  • P(heads) = 1/2 = 0.5 = 50%

Worked example 2: Six-sided die

Probability of rolling a 3:

  • Favourable: 3 (just one)
  • Total: 1, 2, 3, 4, 5, 6 (six outcomes)
  • P(3) = 1/6

Probability of rolling an even number:

  • Favourable: 2, 4, 6 (three outcomes)
  • Total: 6 outcomes
  • P(even) = 3/6 = 1/2

Worked example 3: Card deck

A standard pack has 52 cards: 4 suits × 13 ranks.

Probability of drawing a king:

  • Favourable: 4 kings
  • Total: 52
  • P(king) = 4/52 = 1/13

Probability of drawing a heart:

  • Favourable: 13 hearts
  • Total: 52
  • P(heart) = 13/52 = 1/4

Sample space diagrams

A sample space is the complete list of outcomes. For tossing two coins:

      Coin 2 = H    Coin 2 = T
Coin 1 = H    HH         HT
Coin 1 = T    TH         TT

Four equally likely outcomes. Probability of exactly one head: 2/4 = 1/2 (HT or TH).

The "and" rule (independent events)

Two events are independent if one doesn't affect the other (two coin tosses, two die rolls).

P(A and B) = P(A) × P(B)

Example: roll a die twice. Probability of two sixes?

  • P(six on first) = 1/6
  • P(six on second) = 1/6
  • P(both) = 1/6 × 1/6 = 1/36

The "or" rule (mutually exclusive events)

Two events are mutually exclusive if they can't both happen (rolling a 3 OR a 5 — same die roll can't be both).

P(A or B) = P(A) + P(B)

Example: rolling a 3 or a 5 on one die:

  • P(3) = 1/6, P(5) = 1/6
  • P(3 or 5) = 1/6 + 1/6 = 2/6 = 1/3

The complement (the "not" trick)

P(not A) = 1 − P(A)

Example: probability of NOT rolling a six:

  • P(six) = 1/6
  • P(not six) = 1 − 1/6 = 5/6

Useful when calculating "at least one" or "none" — often faster than enumerating outcomes.

The two confusing errors

Error 1: Adding probabilities of independent events

❌ "Two coins both heads = 1/2 + 1/2 = 1"

That can't be right — 1 means certain, but two coins both heads is far from certain.

✅ Independent events → multiply. 1/2 × 1/2 = 1/4.

The "or" rule adds; the "and" rule multiplies. Different keywords.

Error 2: Forgetting equally likely

The formula assumes equal outcomes. For a weighted die (cheating), the formula breaks. KS3 questions almost always specify "fair" — but watch for trick questions where outcomes aren't equally likely.

Practice problems

  1. P(heads on a fair coin) = 1/2
  2. P(rolling a prime number on a 6-sided die — 2, 3, 5) = 3/6 = 1/2
  3. P(drawing a red card from a deck) = 26/52 = 1/2
  4. P(two heads in two coin flips) = 1/2 × 1/2 = 1/4
  5. P(NOT rolling a 1 on a die) = 1 − 1/6 = 5/6
  6. P(rolling a 2 or 4 on one die) = 1/6 + 1/6 = 1/3

How Professor Pi teaches this

For "Probability of rolling a 3 on a 6-sided die":

  • L1: "How many outcomes does a die have?"
  • L2: "6 outcomes. How many of them are 'rolling a 3'?"
  • L3: "Just one. So what's the probability?"
  • L4: worked twin for "P(rolling a 5) = 1/6", then you do P(3).

FAQ

How do I calculate probability in KS3?

Number of favourable outcomes ÷ total outcomes. For a fair coin landing heads: 1 ÷ 2 = 1/2 = 50%.

What's the probability scale in maths?

0 to 1. 0 impossible, 1 certain, 0.5 even chance. Can be fractions, decimals, or percentages.

What's a sample space?

The complete list of all possible outcomes. For a six-sided die: {1, 2, 3, 4, 5, 6}.

Pedagogy from Professor Pi at aitutors.me. Updated 20 May 2026.