Ch 3 · E8.4 Conditional Probability · IGCSE Mathematics 0580
Topic 3 of 3 in Probability — IGCSE Mathematics 0580 — 18 lessons.
Introduction
"Given that it is raining, what is the chance the bus is late?" A conditional probability is a probability worked out when you already know something has happened. The trick is to shrink the sample space to only the outcomes that fit what you know. (Syllabus section E8.4, IGCSE 0580 Extended only. The notation P(A mid B) and formulas are not required.)
Key ideas
"Given that B has happened" means: ignore every outcome not in B. Then
🧑🏫 Teacher's explanation
Imagine you are looking at a large bag of mixed sweets. If I ask, "What is the probability of picking a red sweet?", you look at the whole bag. But if I say, "Given that the sweet is chocolate, what is the probability it is red?", the non-chocolate sweets no longer exist to you. You have shrunk your world down to only the chocolate sweets. This is the essence of conditional probability.
📖 Definitions
🔍 Worked example — Easy
In a class of 30 students, 18 play football and 12 play tennis. 5 students play both. A student is chosen at random. Find the probability that the student plays tennis, given that they play football.
🔍 Worked example — Standard
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. Find the probability that the second marble is blue, given that the first marble was red.
🔍 Worked example — Exam-style
A tree diagram shows the probability of rain on Monday and Tuesday. P(Rain Mon) = 0.4. If it rains on Monday, P(Rain Tue | Rain Mon) = 0.7. If it does not rain on Monday, P(Rain Tue | No Rain Mon) = 0.2. Find the probability that it rained on Monday, given that it rained on Tuesday.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Of 30 students, 12 study French, 15 study Spanish and 6 study both. A student who studies Spanish is chosen. The probability that they also study French is 6/15 = dfrac25 (only the 15 Spanish students count).
🔍 Worked example — Easy
Of 30 students, 12 study French, 15 study Spanish and 6 study both. A student who studies Spanish is chosen. The probability that they also study French is 6/15 = 2/5.
🔍 Worked example — Standard
Consider the following table:
🔍 Worked example — Exam-style
A bag has 5 red and 3 blue balls. Two are taken without replacement. Given that the second ball is red, find the probability that the first was red.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 18/40 = 9/20. (2) 8/20 = tfrac25. (3) 0.4 × 0.7/0.4 × 0.7 + 0.6 × 0.1 = 0.28/0.34 = 14/17.