Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 4: Statistics and Probability | Topic 4.5
📋 Learning Intentions
- Define a sample space, an event, and a complementary event using correct set notation.
- Calculate the probability of an event and its complement using \( P(A') = 1 - P(A) \).
- Identify equally likely outcomes and apply the classical probability formula.
- Calculate the expected number of occurrences of an event over \( n \) trials using \( E = nP(A) \).
- Interpret probability values in context, distinguishing between theoretical and experimental probability.
- Apply Venn diagrams and sample space diagrams to organise outcomes and calculate probabilities.
Real-World Introduction
Every time a weather forecaster says "there is a 70% chance of rain tomorrow," they are assigning a number to uncertainty. That number — a probability — is not a guess. It is a mathematically precise statement about how likely an outcome is, based on historical data and modelling. Understanding how to define, calculate, and interpret such numbers is the foundation of all statistical reasoning.
Probability underpins decisions in medicine (what is the chance a drug causes side effects?), finance (what is the likelihood a portfolio loses value?), and engineering (what is the probability a component fails?). Before any of these questions can be answered, the possible outcomes must be listed systematically — this is the role of the sample space.
🌍 Why This Matters Insurance companies use probability to set premiums: if the probability of a car accident in a given year is 0.03, and there are 10 000 policyholders, the company expects approximately 300 claims. This is a direct application of expected occurrences — a concept central to this topic.
Core Concepts & Definitions
Before calculating any probability, the set of all possible outcomes must be identified. Without knowing what can happen, it is impossible to measure how likely any particular outcome is. This complete list of outcomes is called the sample space.
🔑 Key Definition — Sample Space The sample space , denoted \( U \) or \( S \), is the set of all possible outcomes of a random experiment. Each element of the sample space is called an outcome (or elementary event). The number of outcomes in the sample space is written \( n(U) \).
For example, when a fair six-sided die is rolled, the sample space is \( U = \{1, 2, 3, 4, 5, 6\} \) and \( n(U) = 6 \). When a coin is flipped twice, the sample space is \( U = \{HH, HT, TH, TT\} \) and \( n(U) = 4 \). Listing outcomes systematically — using a