Learning Objectives
- Understand the theoretical (classical) approach to probability
- Find the probability of an event
- Understand complementary events and their probabilities
- Solve problems involving coins, dice, cards, and real-life situations
Key Concepts
Basic Definitions
An experiment is a trial that produces a well-defined outcome. A random experiment is one where the outcome cannot be predicted with certainty.
The sample space is the set of all possible outcomes of a random experiment.
An event is a subset of the sample space. An event occurs if the actual outcome belongs to it.
Theoretical Probability
P(E) = Number of outcomes favourable to E / Total number of outcomes
This assumes all outcomes are equally likely.
Important Properties
- 0 ≤ P(E) ≤ 1 for any event E
- P(sure event) = 1 (the event that always occurs)
- P(impossible event) = 0 (the event that never occurs)
- P(E) + P(E') = 1, where E' (or E-bar) is the complement of E
- The sum of probabilities of all elementary events = 1
Common Experiments
Tossing a coin: Sample space = {H, T}; P(H) = P(T) = 1/2.
Throwing a die: Sample space = {1, 2, 3, 4, 5, 6}; each P = 1/6.
Deck of 52 cards: 4 suits (hearts, diamonds, clubs, spades), each with 13 cards (A, 2-10, J, Q, K). 26 red cards, 26 black cards, 12 face cards (J, Q, K of each suit).
Complementary Events
If P(getting an even number on a die) = 3/6 = 1/2, then P(not getting an even number) = 1 - 1/2 = 1/2.
This is useful when it is easier to calculate the probability of the complementary event.
Summary
Probability measures how likely an event is to occur. The classical definition assumes equally likely outcomes. The probability of any event ranges from 0 to 1. Complementary events are key to simplifying many calculations. Understanding sample spaces for standard experiments (coins, dice, cards) is essential for board exams.
Important Terms
- Random Experiment
- An experiment whose outcome is uncertain
- Sample Space
- The set of all possible outcomes of a random experiment
- Event
- A subset of the sample space; a collection of outcomes
- Complementary Event
- The event that E does not occur, denoted E'; P(E') = 1 - P(E)
- Equally Likely Outcomes
- Outcomes that have the same probability of occurring
Quick Revision
- P(E) = Favourable outcomes / Total outcomes
- 0 ≤ P(E) ≤ 1; P(sure event) = 1; P(impossible event) = 0
- P(E) + P(not E) = 1
- A standard die has 6 outcomes; a coin has 2 outcomes
- A deck has 52 cards: 4 suits of 13 each; 12 face cards; 4 aces