Conditional probability is the probability of an event given that another event has happened. The words “given that” restrict the group you are choosing from, so the denominator becomes the size of that group rather than the overall total. Two-way tables and Venn diagrams make this easy to read off, and on a tree diagram each second-stage branch is already a conditional probability.
Conditional probability
- P(A | B) means the probability of A given B
- From a table or Venn diagram: P(A | B) = number in A and B ÷ number in B
- From probabilities: P(A | B) = P(A and B) ÷ P(B)
- Independent events: P(A and B) = P(A) × P(B)
“Given that” restricts the total
In a question such as “find the probability that a student plays netball, given that she is a girl”, you are only choosing from the girls. So the denominator is the number of girls, not the number of students. P(A | B) is read as “the probability of A given B”.
Using a two-way table
Find the row or column for the given condition and use its total as the denominator. If 26 of 50 students are girls and 13 of the girls play netball, P(netball | girl) = 13/26. Notice that P(girl | netball) is different: it uses the number of netball players as the denominator.
Conditional probability in a tree
On a tree diagram for picks without replacement, the second-stage branches are conditional probabilities. With 5 red and 3 blue counters, if the first is red then 7 remain, 3 of them blue, so P(second blue | first red) = 3/7.
Using a Venn diagram
Restrict to the circle for the given condition. If 25 people cycle and 9 of them also swim, P(swims | cycles) = 9/25.
Independence
Two events are independent if one does not affect the other, which means P(A and B) = P(A) × P(B), or equivalently P(A | B) = P(A). To test independence, compare P(A and B) with P(A) × P(B). For example, with P(A) = 0.6, P(B) = 0.25 and P(A and B) = 0.15, the product 0.6 × 0.25 is also 0.15, so A and B are independent.
Worked examples
Example 1. A sports survey: football has 14 boys and 9 girls, netball 4 boys and 13 girls, tennis 6 boys and 4 girls. A girl is chosen at random. Find P(she plays netball).
- Total girls: 9 + 13 + 4 = 26.
- 13 of them play netball.
Answer: 13/26 = 1/2.
Example 2. Using the same table, a netball player is chosen at random. Find P(they are a girl).
- Total netball players: 4 + 13 = 17.
- 13 are girls.
Answer: 13/17.
Example 3. A bag has 5 red and 3 blue counters. Two are taken without replacement. Find P(second is blue | first is red).
- After a red is taken: 7 counters, 3 of them blue.
Answer: 3/7.
Example 4. In a Venn diagram of 60 people, 25 cycle and 9 of those also swim. A cyclist is chosen. Find P(they swim).
- Restrict to the 25 cyclists.
- 9 of them swim.
Answer: 9/25.
Example 5. P(A) = 0.6, P(B) = 0.25 and P(A and B) = 0.15. Are A and B independent?
- P(A) × P(B) = 0.6 × 0.25 = 0.15.
- This equals P(A and B).
Answer: Yes, A and B are independent.
Common mistakes
- Dividing by the overall total. “Given that” means divide by the size of the given group.
- Confusing P(A | B) with P(B | A). They usually have different denominators.
- Ignoring the change after a pick. Without replacement, second-stage branches are conditional.
- Assuming events are independent. Check by comparing P(A and B) with P(A) × P(B).
- Using the wrong row or column. Find the row or column for the given condition first.
How conditional probability is examined
“Given that” questions are reserved for Higher papers, where they appear in tables, Venn diagrams and trees. Questions ask you to:
- find a conditional probability from a two-way table
- find a conditional probability from a Venn diagram
- use the second-stage branches of a tree
- decide whether two events are independent
The Venn diagrams worksheet and the tree diagrams worksheet include conditional probability questions with answers. The GCSE probability engine gives more.
Conditional probability uses two-way tables and without-replacement trees.
Frequently asked questions
What does “given that” mean in probability?
That you only consider the outcomes where the given event has happened, so the total becomes the size of that group.
How do you work out conditional probability from a table?
Divide the number in the relevant cell by the total for the given condition.
What does it mean for events to be independent?
One event does not affect the other, so P(A and B) = P(A) × P(B).