Probability

Overview

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title: “Summary: Chapter S1B Probability” format: html: toc: true toc-depth: 3 page-layout: full —

Probability (Chapter S1B)

Comprehensive Summary of A-Level H2 Mathematics (9758) Core Concepts

1. Key Definitions & Concepts

A random experiment is an experiment whose outcome cannot be predicted exactly.

  • Sample Space (\(\Omega\)): The set of all possible outcomes.
  • Event: A subset of the sample space.
  • Impossible Event (\(\emptyset\)): An event that contains no outcomes.

Theoretical Probability

If the sample space \(\Omega\) consists of a finite number of equally likely outcomes, the probability of an event \(E\) is given by: \[ \mathrm{P}(E) = \frac{n(E)}{n(\Omega)} \]


2. Set Operations & Properties

Understanding set operations is crucial for Venn probability problems.

Operation Notation Meaning
Complement \(A'\) Event \(A\) does not occur.
Intersection \(A \cap B\) Both event \(A\) and event \(B\) occur.
Union \(A \cup B\) Event \(A\) occurs, or event \(B\) occurs, or both occur (at least one occurs).

Important Probability Rules

Fundamental Properties:

  • Bounds: \(0 \le \mathrm{P}(A) \le 1\)
  • Certainty: \(\mathrm{P}(\Omega) = 1\), \(\mathrm{P}(\emptyset) = 0\)
  • Subset: If \(A \subseteq B\), then \(\mathrm{P}(A) \le \mathrm{P}(B)\)

Golden Formulae (Always check if these apply!):

  1. Complementary Law: \(\mathrm{P}(A) + \mathrm{P}(A') = 1\)
  2. Addition Law: \(\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)\)
  3. Difference: \(\mathrm{P}(A \cap B') = \mathrm{P}(A) - \mathrm{P}(A \cap B)\)
  4. De Morgan’s Laws:
    • \(\mathrm{P}(A' \cap B') = \mathrm{P}((A \cup B)') = 1 - \mathrm{P}(A \cup B)\)
    • \(\mathrm{P}(A' \cup B') = \mathrm{P}((A \cap B)') = 1 - \mathrm{P}(A \cap B)\)


3. Mutually Exclusive vs. Independent Events

A frequent source of A-Level exam questions requires distinguishing between Mutually Exclusive and Independent events.

Two events \(A\) and \(B\) are mutually exclusive if they cannot happen at the same time.

  • Condition: \(\mathrm{P}(A \cap B) = 0\)
  • Addition Law Simplifies To: \(\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B)\)
  • Venn Diagram: Two disjoint circles (no overlap).

Two events \(A\) and \(B\) are independent if the occurrence of one does not affect the probability of the other.

  • Condition: \(\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B)\)
  • Conditional form: \(\mathrm{P}(A | B) = \mathrm{P}(A)\) and \(\mathrm{P}(B | A) = \mathrm{P}(B)\)
  • Note: If \(A\) and \(B\) are independent, then \((A \text{ and } B')\), \((A' \text{ and } B)\), and \((A' \text{ and } B')\) are also independent pairs.

⚠️ Warning: Non-Empty Mutually Exclusive Events (\(\mathrm{P}(A)>0\), \(\mathrm{P}(B)>0\)) cannot be Independent. Since \(\mathrm{P}(A \cap B) = 0 \neq \mathrm{P}(A)\mathrm{P}(B)\).


4. Conditional Probability

The probability of an event \(A\) happening, given that event \(B\) has already occurred. This “given” information essentially creates a reduced sample space.

Important

Conditional Probability Formula \[ \mathrm{P}(A | B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}, \quad \text{provided } \mathrm{P}(B) \neq 0 \]

Rearranging this gives the Multiplication Law: \[ \mathrm{P}(A \cap B) = \mathrm{P}(B) \times \mathrm{P}(A | B) \]

Problem-Solving Strategies

  1. Venn Diagrams: Use them to display intersections, unions and complements visually. They are extremely effective when combining conditional statements with set formula rules.
  2. Probability Trees: Useful for events happening in succession.
    • Multiply along the branches to find the probability of a combined specific outcome.
    • Add the probabilities of mutually exclusive end-outcomes.
  3. Combinatorics (P&C in Probability): If the problem involves selecting without replacement or grouping over large sample spaces, compute the ratio of the number of favorable combinations/permutations over the total combinations/permutations.

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title: “Selected School Past Year Questions”

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title: “A-Level Past Year Questions”