Ch 3 · 5.3 Probability · A Level Mathematics 9709
Topic 3 of 5 in Probability & Statistics 1 — A Level Mathematics 9709 — 58 lessons.
Introduction
Weather forecasts, medical tests and game shows all deal in chance. This lesson gives the rules for combining probabilities and the precise meaning of two words that are easy to confuse: exclusive and independent. Notation: P(A), P(A ∪ B) ("A or B"), P(A ∩ B) ("A and B"), P(A mid B) ("A given B"), and A' for "not A". (Syllabus section 5.3, Paper 5.)
Key ideas — Equally likely outcomes
If all outcomes are equally likely,
🧑🏫 Teacher's explanation
Imagine you have a bag of 10 marbles: 7 are blue and 3 are red. If you reach in and grab one without looking, you have a 7 in 10 chance of getting blue. This works because every single marble has the exact same chance of being picked. In maths, we call these 'equally likely outcomes'. When every outcome is equally likely, finding a probability is just a simple game of counting: how many ways can I get what I want, divided by how many things could possibly happen.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled. Find the probability of rolling a prime number.
🔍 Worked example — Standard
A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. Find the probability that both balls are red.
🔍 Worked example — Exam-style
Three fair coins are tossed. Find the probability of getting at least two heads.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Adding and multiplying
Without replacement, the probabilities on later branches change (the bag has one fewer ball).
🧑🏫 Teacher's explanation
Think of probability as a way to measure how much 'space' an event takes up in a world of possibilities. When we talk about 'OR', we are combining different ways to win. If you want to win by rolling a 2 or a 5 on a die, you have more ways to win than if you only wanted a 2. So, we add the chances together. Just remember: this only works if the two things can't happen at the same time (like rolling a 2 and a 5 simultaneously).
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled. Find the probability of rolling a 2 or a 5.
🔍 Worked example — Standard
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. Find the probability that both marbles are red.
🔍 Worked example — Exam-style
In a group of 100 students, 60 study Maths, 40 study Physics, and 20 study both. A student is chosen at random. Given that the student studies Physics, find the probability they also study Maths.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Exclusive events
A and B are mutually exclusive if they cannot both happen: P(A ∩ B) = 0. Example: on one die, "a 2" and "an odd number".
🧑🏫 Teacher's explanation
Imagine you are standing at a fork in the road. You can either go left or you can go right. You cannot do both at the same time. In probability, we call these 'mutually exclusive' events. They are like two different paths that never cross; if you are on one, you cannot possibly be on the other.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled once. Let event A be 'rolling a 2' and event B be 'rolling an odd number'. Determine if A and B are mutually exclusive and find P(A ∪ B).
🔍 Worked example — Standard
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. Let event A be 'the first marble is red' and event B be 'the first marble is blue'. Are A and B mutually exclusive? Calculate P(A ∪ B).
🔍 Worked example — Exam-style
In a class of 30 students, 12 students study Biology, 15 study Chemistry, and 5 study both. Let B be 'studies Biology' and C be 'studies Chemistry'. Are B and C mutually exclusive? Calculate P(B ∪ C).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) List the outcomes in A and B. (b) Are A and B mutually exclusive? Justify your answer. (c) Calculate P(A ∪ B). [4 marks]
Key ideas — Independent events
A and B are independent if one happening does not change the probability of the other. The test:
🧑🏫 Teacher's explanation
Imagine you are playing a game where you flip a coin and roll a die at the same time. If the coin lands on Heads, does that change the likelihood of the die landing on a 6? Of course not. The outcome of the coin has zero influence on the outcome of the die. This is the essence of independence: knowing the result of one event tells you absolutely nothing about the likelihood of the other.
📖 Definitions
🔍 Worked example — Easy
A fair six-sided die is rolled. Let event A be 'rolling a 2' and event B be 'rolling an odd number'. Determine if A and B are independent.
🔍 Worked example — Standard
Two fair coins are tossed. Let event A be 'the first coin is Heads' and event B be 'the second coin is Heads'. Show that A and B are independent.
🔍 Worked example — Exam-style
In a bag, there are 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. Let A be 'the first marble is red' and B be 'the second marble is red'. Determine if A and B are independent.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Conditional probability
P(A mid B) = P(A ∩ B)/P(B)
🧑🏫 Teacher's explanation
Imagine you are looking at a large group of people. If I ask, "What is the probability that a person is wearing a red shirt?", you look at everyone. But if I say, "Given that the person is wearing glasses, what is the probability they are wearing a red shirt?", you suddenly ignore everyone without glasses. You have 'shrunk' your world down to only the people with glasses. This is the essence of conditional probability.
📖 Definitions
🔍 Worked example — Easy
A bag contains 5 red marbles and 3 blue marbles. A marble is drawn at random and not replaced. A second marble is drawn. Find the probability that the second marble is red, given that the first marble was blue.
🔍 Worked example — Standard
In a class, 60% of students study Biology, 40% study Chemistry, and 20% study both. A student is chosen at random. Find the probability that the student studies Biology, given that they study Chemistry.
🔍 Worked example — Exam-style
A tree diagram shows two stages. In stage 1, there is a 0.7 chance of event X and 0.3 chance of event Y. In stage 2, if X happened, there is a 0.4 chance of Z. If Y happened, there is a 0.8 chance of Z. Find P(X mid Z).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples
Two fair dice are thrown. P(total 8): outcomes (2,6), (3,5), (4,4), (5,3), (6,2), so 5/36.
🔍 Worked example — Easy
A bag contains 6 green marbles and 4 yellow marbles. Two marbles are drawn at random without replacement. Find the probability that both marbles are green.
🔍 Worked example — Standard
Two fair six-sided dice are rolled. Let event A be 'the sum of the dice is 7' and event B be 'the first die shows a 4'. Determine whether events A and B are independent.
🔍 Worked example — Exam-style
In a class, 60% of students study Biology, 40% study Chemistry, and 20% study both. A student is chosen at random. Given that the student studies Biology, find the probability they also study Chemistry.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) pairs (4,5),(5,4),(4,6),(6,4),(5,5),(5,6),(6,5),(6,6): 8/36 = tfrac29. (2) 0.4 × 0.5 = 0.2 = P(A ∩ B), so independent; not exclusive since P(A ∩ B) ≠ 0. (3) dfrac^6C_3 + ^4C_3^10C_3 = 20 + 4/120 = tfrac15. (4) 0.02 × 0.95/0.02 × 0.95 + 0.98 × 0.03 = 0.019/0.0484 = 0.393. (5) P(B) = 0.3 + 0.1 = 0.4; P(A mid B) = 0.3 / 0.4 = 0.75.