Conditional probability and a restricted group
Use the information already given to choose the correct denominator in a table, Venn diagram or tree.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Count intersections of two events.
- Read totals and without-replacement branches.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which total is used for probability given French?
What you will learn
- Calculate a probability given that a condition holds.
- Read conditional probabilities from tables and tree branches.
Given information changes the outcome set
In a group of forty, eighteen study French and six of those also study Spanish. Given that a selected learner studies French, the probability they study Spanish is 6/18 = 1/3. The denominator is now the eighteen French learners, not the whole group of forty.
Draw or highlight the restricted group first, then count the requested event inside it. Extended covers this conditional reasoning using Venn diagrams, two-way tables and trees. Formal P(A|B) notation and the conditional-probability formula are not required by the syllabus; the counting interpretation is enough here.
Read a branch within its condition
In a bag with three red and two blue counters, after a red is drawn without replacement, probability that the second is blue is 2/4 = 1/2. That is already conditioned on the first red. The joint probability of first red and second blue is different: (3/5)(2/4) = 3/10.
A two-way table works the same way. If twenty students take music and eight of those take art, probability of art given music is 8/20. Reversing the condition can change the answer because the denominator becomes the art total. Keep the wording 'given' attached to the group whose total you use.
Pause and explain
18 study French; 6 of them study Spanish. Find P(Spanish given French).
Worked example
Eighteen learners study French and six of them also study Spanish. Find the probability of Spanish given French.
Show the worked solution
- Restrict the outcome set to eighteen French learners.
- Within that group, six satisfy the Spanish event.
- Probability is 6/18 = 1/3.
Answer 1/3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Of 20 music students, 8 take art. Find probability of art given music.
Give me a hint
Restrict to music before counting art.
Compare my reasoning
- The conditioned total is twenty.
- Eight within it take art.
- Probability is 8/20 = 2/5.
2/5.
Look for these in your work
- Used the music total.
- Counted the overlap within it.
A bag has three red and two blue. Given the first draw was red and not replaced, find P(second blue).
Give me a hint
Use the remaining bag directly, without multiplying by the first red probability.
Compare my reasoning
- Two red and two blue remain.
- The restricted second draw has four possible counters.
- Probability is 2/4 = 1/2.
1/2.
Look for these in your work
- Applied the given first result.
- Distinguished conditional from joint probability.
In 50 students, 20 study art, 15 music and 6 both. Compare probability of art given music with music given art.
Give me a hint
The two conditions have different totals.
Compare my reasoning
- Art given music is 6/15 = 2/5.
- Music given art is 6/20 = 3/10.
- The overlap is unchanged but the restricted denominators differ.
2/5 versus 3/10.
Look for these in your work
- Kept the condition order.
- Explained the denominator difference.
Common mistakes
- Using the universal total despite a condition.
- Reversing the conditioned group.
- Multiplying by the given event when only a conditional probability is requested.
What can you explain now?
Is a tree branch probability always a joint probability?
Compare with the explanation
No.
A branch after a first outcome is conditioned on reaching that node; multiply complete paths for joint probabilities.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, write two reversed 'given' questions from one table and explain their denominators.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.