Key Stage 4 maths · Probability

GCSE conditional probability (higher): a guide for parents

Conditional probability asks how likely something is when you already know that something else has happened or is true. It is part of the higher-tier GCSE course. Children calculate and interpret conditional probabilities using two-way tables of expected frequencies, tree diagrams and Venn diagrams, learning that the question “given that” narrows down the group you are looking at before you count anything.

Conditional probability, explained

A conditional probability is the chance of one event given that another is known. It is written P(A | B), read “the probability of A given B”. The key idea is that the information changes the group being considered. If you are told a student walks to school, you no longer look at the whole year group: you look only at the students who walk. The new total is the number in that group, and the answer is the part of that group that also has the feature you want. Nearly every error in this topic comes from using the wrong total.

Two-way tables are the clearest place to start. They sort people or items by two features at once, such as gender and how they travel to school. Reading a conditional probability means finding the row or column that holds the condition, using its total as the denominator, and picking the right cell for the numerator. Children are often asked to work with “expected frequencies”, imagining a group of, say, 1,000 people and working out how many would be in each box. Imagining actual numbers is much easier than working with fractions of fractions.

Tree diagrams do the same job. The first branches show the chance of the first event, and the second set of branches are conditional probabilities because they depend on what already happened. Multiplying along a route gives P(A and B) = P(A) × P(B | A). Questions also run the tree backwards, such as “given that the two counters were different colours, what is the probability that the first was red?” These are solved by finding the probability of all the routes which fit the information, then dividing the route you want by that total.

Venn diagrams show the same idea with overlapping circles. P(A | B) is the number in the overlap of A and B divided by the total inside circle B, not the total of everybody. Children learn that P(A | B) and P(B | A) are different questions with different answers. If knowing B makes no difference to the chance of A, the two events are independent and P(A | B) = P(A). Seeing the difference between these matters in real life, in areas such as medical testing and legal evidence, where confusing the two can be seriously misleading.

Words your child will hear

Conditional probability: The probability of an event happening given that another event has already happened or is known.
P(A | B): “The probability of A given B”: look only at cases where B is true.
Two-way table: A table that sorts a group by two features at once, with totals for each row and column.
Expected frequency: How many you would expect to see in a group of a given size, such as out of 1,000 people.
Intersection: The overlap of two events in a Venn diagram: both A and B happening.
Independent: Two events where knowing one tells you nothing about the other, so P(A | B) = P(A).
Denominator: The bottom of a fraction. In this topic it is the total of the group given by the condition.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables
Calculate and interpret conditional probabilities through representation using tree diagrams and Venn diagrams

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A year group of 150 students is sorted by gender and how they travel to school. A student is picked at random. Given that the student walks, find the probability that they are a girl. Then find the probability that the student walks, given that they are a girl.

Put the information into a two-way table, filling in totals.

         Walk   Bus   Total
Girls      20    50      70
Boys       40    40      80
Total      60    90     150

“Given that the student walks” means only the Walk column matters. There are 60 students in that column, so 60 is the denominator.
Of those 60 who walk, 20 are girls. P(girl | walks) = 20/60 = 1/3.
For the second question the condition is “is a girl”, so only the Girls row matters, with a total of 70. Of those, 20 walk. P(walks | girl) = 20/70 = 2/7.
Compare: 1/3 is about 0.33 and 2/7 is about 0.29. The answers are different because the two questions look at different groups. Neither uses the whole 150.

P(girl | walks) = 1/3 and P(walks | girl) = 2/7.

Common mistakes and how to help

Swapping P(A | B) and P(B | A)

In everyday speech “girls who walk” and “walkers who are girls” sound the same. In maths they are different groups, and the answers differ: 1/3 and 2/7 in the example above.

What helps: Find the word “given” (or “of those who…”) and underline what follows it. That is the group, and its total is the denominator. Ask what fraction of this group has the other feature.

Dividing by the grand total

Up to now, probability has almost always meant “number of favourable outcomes out of everything”. 20/150 is the probability of being a girl who walks. It is a different quantity from the one asked for.

What helps: Cross out all the cells that don’t fit the condition. If a Venn diagram is used, the answer is the overlap divided by the whole of the circle that is the condition, not by everybody.

Not changing the numbers on the second branches of a tree

When items aren’t replaced, the second set of probabilities depends on what happened first. Copying the first branches as if the events were independent gives a wrong answer.

What helps: Before writing the second branches, ask what is left. If a red counter was taken from 5 red and 3 blue, there are now 4 red and 3 blue, out of 7.

Reading a tree backwards by quoting a single branch

Asked for the chance that the first counter was red, given that the two counters were different colours, children answer 5/8, the first branch they can see. That branch ignores the information. The correct working is 15/56 ÷ 30/56 = 1/2.

What helps: List every route that ends in the given result, add them to make the new total, then divide the probability of the route you want by that total. A table of expected numbers for a large number of trials makes this clear.

How to help at home

Cards on the table. Use a standard pack of 52 cards. Given that a card is red, what is the probability that it is a king? There are 26 red cards and 2 red kings, so 2/26 = 1/13. Now reverse it: given a king, the probability of red is 2/4 = 1/2. Different questions, different answers.
Run your own survey. Ask 20 family and friends two yes or no questions, such as “do you drink tea?” and “do you have a pet?”. Fill in a two-way table together, then make up “given that” questions and answer them from the table.
Imagine 1,000 people. When you hear a percentage in the news, pretend 1,000 people were involved and work out how many fit each box. Seeing the numbers laid out, instead of working with percentages of percentages, makes the order of the question much clearer.
Sort out the wording. Make up pairs of sentences such as “the chance a runner is tall” and “the chance a tall person is a runner”. Ask your child whether they mean the same, and which group each one starts from.

Builds on

Key Stage 3: Probability and Venn diagrams