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CIE A-Level Maths Study Notes

4.3.4 Conditional Probability

Conditional probability is playing a crucial role in statistics, decision-making, and risk assessment. It focuses on evaluating the probability of one event occurring, given that another event has already taken place. Understanding this interplay between events is vital for a deep comprehension of probability theory.

Basic Concept

  • Definition: Conditional probability is the chance of event A happening given that event B has already happened, denoted as P(A|B).
  • Formula: P(AB)=P(AB)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)} , with P(B) > 0.

Understanding Events

  • Interdependence: Knowing how one event affects another is crucial.
  • Sample Spaces: All possible outcomes in a probability scenario.
  • Tree Diagrams: Useful for visualizing probabilities in complex situations.

Example Scenarios

A. Dice and Coin Toss:

Coin and die

Image courtesy of wentzwu

  • Events: A = "heads on coin toss", B = "even number on die".
  • Calculations:
    • P(B)=36P(B) = \frac{3}{6} (3 even numbers on die).
    • P(A and B)=12×36=14P(A \text{ and } B) = \frac{1}{2} \times \frac{3}{6} = \frac{1}{4} (independent events).
    • P(AB)=(1/4)(3/6)=12P(A|B) = \frac{(1/4)}{(3/6)} = \frac{1}{2}.

B. Card Draw from a Deck:


Images courtesy of thoughtsco

  • Events: A = "drawing a king", B = "drawing a face card".
  • Calculations:
    • P(B)=1252P(B) = \frac{12}{52} (12 face cards in deck).
    • P(A and B)=452 P(A \text{ and } B) = \frac{4}{52} (4 kings, all face cards).
    • P(AB)=(4/52)(12/52)=13P(A|B) = \frac{(4/52)}{(12/52)} = \frac{1}{3}.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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