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Probability 9 min read

Probability for GCSE Maths — From Basics to Tree Diagrams

Master GCSE probability: basic probability, relative frequency, AND/OR rules, Venn diagrams, tree diagrams, and conditional probability. With worked examples.

probabilitytree diagramsVenn diagramsconditional probabilityGCSE maths

Basic Probability

Probability is the likelihood of an event happening, expressed as a fraction, decimal, or percentage between 0 (impossible) and 1 (certain). The formula is: P(event) = Number of favourable outcomes ÷ Total number of equally likely outcomes.

  • Probabilities always lie between 0 and 1 (inclusive)
  • P(event not happening) = 1 − P(event happening)
  • The probabilities of all mutually exclusive outcomes sum to 1

The AND Rule (Independent Events)

If two events are independent (one doesn't affect the other), multiply their probabilities to find the probability they BOTH happen.

Worked Example

A bag has 3 red and 7 blue balls. One is drawn, replaced, then another is drawn. P(both red)?

  1. 1.P(red) = 3/10 for each draw (independent, with replacement)
  2. 2.P(red AND red) = 3/10 × 3/10 = 9/100
Answer: 9/100 = 0.09

Tree Diagrams

Tree diagrams are the clearest way to list all possible outcomes and calculate combined probabilities. Draw branches for each event, write probabilities on the branches, and multiply along branches for AND, then add for OR.

Worked Example

A bag has 4 red and 6 blue balls. Draw two without replacement. P(different colours)?

  1. 1.P(R then B) = 4/10 × 6/9 = 24/90
  2. 2.P(B then R) = 6/10 × 4/9 = 24/90
  3. 3.P(different) = 24/90 + 24/90 = 48/90 = 8/15
Answer: 8/15

Venn Diagrams

Venn diagrams show the relationship between two (or three) events using overlapping circles. The overlap is the intersection (AND), the total region is the union (OR).

  • P(A ∩ B) = P(A AND B) = the intersection region
  • P(A ∪ B) = P(A OR B) = P(A) + P(B) − P(A ∩ B)
  • P(A | B) = P(A ∩ B) ÷ P(B) — conditional probability

Conditional Probability (Higher)

Conditional probability is the probability of event A given that event B has already happened. Use the formula P(A|B) = P(A ∩ B) ÷ P(B), or read directly from a two-way table or Venn diagram.

Worked Example

P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A|B).

  1. 1.P(A|B) = P(A ∩ B) ÷ P(B)
  2. 2.= 0.2 ÷ 0.5 = 0.4
Answer: 0.4

Frequently Asked Questions

What is the difference between experimental and theoretical probability?

Theoretical probability is calculated from equally likely outcomes (e.g., P(heads) = 1/2). Experimental probability is calculated from actual results of an experiment and approaches theoretical probability as the number of trials increases.

When do I add probabilities vs multiply?

Multiply for AND (both events happen). Add for OR (at least one event happens, for mutually exclusive events). This is the key rule to remember for tree diagrams.

Is conditional probability on Foundation or Higher?

Formal conditional probability using P(A|B) = P(A ∩ B)/P(B) is Higher tier. Foundation covers it informally through tree diagrams without replacement.

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