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ATAR Maths

1.1 Fundamentals

1. Basics

Probability is a branch of mathematics that deals with the likelihood or chance of events occurring. Here’s a breakdown of the fundamental concepts:

1.1 Experiment

An experiment is any process or action with an uncertain outcome. Examples include flipping a coin, rolling a die, or drawing a card from a deck.

1.2 Sample Space (S)

The sample space is the set of all possible outcomes of an experiment.

  • Example: When flipping a coin, the sample space is .

1.3 Event (E)

An event is a subset of the sample space. It is one or more outcomes that we are interested in.

  • Example: In rolling a six-sided die, an event could be rolling an odd number, .

1.4 Probability of an Event

The probability of an event , denoted , is a measure of how likely the event is to occur. The probability is calculated as: The probability of any event lies between 0 and 1:

  • means the event is impossible.
  • means the event is certain to occur.

1.5 Complement of an Event

The complement of an event , denoted , is the event that does not occur. The probability of the complement is:

1.6 Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time. If and are mutually exclusive, then: where denotes intersection (the event that both happen).

1.7 Addition Rule

For mutually exclusive events, the probability of either event occurring is the sum of their probabilities: For non-mutually exclusive events, the addition rule adjusts to avoid double-counting the overlap: where denotes union (the event that either or both happen).

1.8 Independent Events

Two events are independent if the occurrence of one does not affect the occurrence of the other. For independent events and , the probability of both occurring is:

1.9 Conditional Probability

Conditional probability is the probability of one event occurring given that another event has occurred. If , the conditional probability of given is:

1.10 Bayes’ Theorem

Bayes’ Theorem is a fundamental formula for finding conditional probabilities: It allows us to update probabilities based on new information.