Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.6
📋 Learning Intentions
- Calculate combined probabilities using the addition rule \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) for non-mutually exclusive events.
- Find conditional probabilities using \( P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} \) and interpret the result in context.
- Construct Venn diagrams and tree diagrams to represent sample spaces and calculate probabilities systematically.
- Determine whether two events are independent using the condition \( P(A \cap B) = P(A) \times P(B) \).
- Apply the multiplication rule for independent and dependent events in multi-stage probability problems.
- Interpret probability results in real-world contexts, distinguishing between mutually exclusive and independent events.
Real-World Introduction
A medical screening test for a disease does not simply return "positive" or "negative" — the result depends on whether the patient actually has the disease. Knowing that a test is positive changes the probability that the patient is truly ill. This is conditional probability in action: the likelihood of one event shifts when we know another event has already occurred.
Insurance companies, weather forecasters, and machine-learning algorithms all rely on combined and conditional probability to make decisions under uncertainty. When two risk factors are present simultaneously — say, smoking and a family history of heart disease — the combined probability of illness is not simply the sum of the two individual risks. Understanding how probabilities combine, and when events influence each other, is the mathematical foundation of modern data science.
🌍 Why This Matters In medical diagnostics, Bayes' theorem — built directly on conditional probability — is used to calculate the probability that a patient has a disease given a positive test result. Misunderstanding conditional probability leads to the "prosecutor's fallacy," a documented cause of wrongful convictions in legal cases worldwide.
Core Concepts & Definitions
Before combining probabilities, it is essential to be precise about what constitutes an event and how events relate to each other within a sample space. The sample space \( U \) (or \( S \)) is the set of all possible outcomes. Events are subsets of this space, and their relationships — overlap, containment, or mutual exclusion — determine which probability rules apply.
🔑 Key Definition — Sample Space and Events
- The sample space \( U \) is the set of all possible outcomes of an experiment.
- An event \( A \) is any subset of \( U \). Its probability satisfies \( 0 \leq P(A) \leq 1 \).
- The complement of \( A \) is \( A' \) (or \(