Conditional Probability

What you will learn

• Understand probability given a condition.
• Use restricted totals.
• Read conditional information from tables.
• Use tree diagrams for conditional situations.
• Recognise conditional probability from tables, Venn diagrams, tree diagrams and written context.
• Avoid using the original total when the condition narrows the group.
• Handle reversed given-that questions by counting the restricted ordered pairs or Venn overlap carefully.
• Use complement regions carefully when the condition says not or outside.
• Translate conditional notation and reversed wording into a restricted denominator before simplifying.

Key facts before you start

Conditional probability infographic

Why this matters

Conditional probability stops you using the wrong total. Once a condition is known, the denominator is the restricted group, not the original population.

Prior knowledge

You should already be comfortable with:

• Fractions.
• Two-way tables.
• Probability trees.
• Dependent events.

Clear explanation

Main idea

Conditional probability means the total group has changed because you already know something. If you know a pupil is in Year 11, only Year 11 pupils belong in the denominator.

Method

In a two-way table, choose the row or column named by the condition first. In a tree diagram, move along the branch that has already happened, then use the probabilities from that point.

In a Venn diagram, the condition names the circle or region you are allowed to use as the denominator. The event asked for is the part of that restricted region that also matches the question.

Be careful with reversed wording. P(first red given second green) is not automatically the same as P(second green given first red); count the cases that match the stated condition.

With notation, P(A | B) means probability of A given B. The event after the vertical bar is the condition, so it decides the denominator.

Conditions can also use complement language such as not in a club, outside the circle or did not choose swimming. In those questions, build the restricted group from the people or outcomes that are left after excluding the named event.

Answer tip

The phrase given that is a strong signal for conditional probability. Read it as: restrict the sample space before calculating.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For conditional probability, marks usually come from stating the condition, using the restricted denominator, selecting the matching numerator, interpreting notation such as P(A | B) and explaining why the original total is no longer the right total.

Common mistakes

• Using the original total: given that narrows the group before you calculate.
• Mixing up the condition and the target event: “given that B” sets the denominator; the probability asked for is usually A within that group.
• Ignoring table direction: check whether the condition names a row, a column or a branch.
• Forgetting changed counts after an event: in without-replacement questions, the known first event changes what remains.

Extension challenge

In a group of 30 pupils, 18 play football, 12 play hockey and 7 play both. Given that a pupil plays football, find the probability that they also play hockey.

Exam-board guidance

Conditional Probability appears within the shared GCSE Maths probability content used by the supported routes. Exact wording, tiering and calculator expectations can vary, but the core skill is the same: restrict the sample space first, then calculate from the outcomes that are still possible.

Next lesson

Next, continue with Product Rule for Counting.