Conditional Probability (Higher)
Conditional probability is the probability that one event occurs given that another event is already known to have happened, written P(A given B), and is typically found using tree diagrams, two-way tables or Venn diagrams. At Higher tier it extends earlier tree diagram work to situations, such as sampling without replacement, where the probabilities change after the first event.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify the two events and clarify what is known or given (the condition) and what is being asked (the event whose probability is required, given the condition).
- If using a tree diagram, draw branches for the first event, then for the second event adjust the probabilities to account for the condition, for example reducing the numerator and denominator when sampling without replacement.
- If using a two-way table, restrict attention to only the row or column matching the given condition, then find the required probability as (frequency satisfying both) divided by (total frequency in that row or column).
- Multiply along tree branches to find joint (AND) probabilities where needed.
- Apply the conditional probability formula where needed: P(A given B) = P(A and B) / P(B).
- Check the final probability lies between 0 and 1 and that the interpretation makes sense in context.
Worked example
A bag contains 5 red counters and 3 blue counters. Two counters are taken from the bag at random, without replacement. Find the probability that the second counter is blue, given that the first counter taken was red.
- After removing 1 red counter, the bag contains 4 red and 3 blue counters, 7 counters in total.
- P(second blue given first red) = number of blue counters remaining / total counters remaining = 3/7.
- Final answer: 3/7.
Practice questions
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Q1A bag has 4 red and 6 green balls. One is drawn and not replaced. Given the first is red, how many balls remain in total, and how many are green?Show answer
Answer: 9 balls remain, 6 of which are green
Q2Using the bag in the previous question, find P(second is green given first is red).Show answer
Answer: 6/9 = 2/3
Q3A class of 30 students has 18 who study French. Of those, 10 also study Spanish. Find P(studies Spanish given studies French).Show answer
Answer: 10/18 = 5/9
Q4In a survey of 40 people, 25 own a car. Of those, 15 also own a bike. Find P(owns a bike given owns a car).Show answer
Answer: 15/25 = 3/5
Q5P(A and B) = 0.12 and P(B) = 0.3. Find P(A given B).Show answer
Answer: 0.4 (0.12/0.3)
Q6A box has 10 pens: 6 blue and 4 black. Two pens are picked without replacement. Find the probability that both are blue.Show answer
Answer: 1/3 (6/10 x 5/9 = 30/90)
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A drawer contains 7 black socks and 5 white socks. Two socks are taken at random without replacement. Find the probability that the second sock is white, given that the first sock taken was white.
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In a survey of 60 students, 36 study Biology. Of the students who study Biology, 24 also study Chemistry. Of the 24 students who do not study Biology, 9 study Chemistry. (a) Find P(studies Chemistry given studies Biology). (b) Find P(studies Chemistry given does not study Biology). (c) Comment on whether studying Biology and studying Chemistry appear to be independent.
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Given that P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2, calculate P(B given A).
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the conditional probability (higher) worksheet pack - 19 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 20 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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