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Chapter 1   Probability

1.1   Ways to define probability

1.1.1   Uncertainty and Probability

In many situations, we are uncertain about what will happen — the outcome is random. Probability describes this uncertainty on a scale of 0 to 1.

1.1.2   Terminology

The terminology used to describe randomness is closely related to set theory.

1.1.3   Classical definition of probability

If all outcomes from an experiment are equally likely, the probability of any event is the proportion of outcomes that are part of the event.

1.1.4   Applications of classical definition

In many games of chance such as cards or dice, outcomes are equally likely, and finding probabilities involves counting them.

1.1.5   Probability from relative frequency

In experiments without equally likely outcomes, we can often imagine repeating the experiment many times. The probability of an event can be defined as the limiting proportion of times it occurs with more and more repetitions.

1.1.6   Estimating probabilities

Defining the probability of an event in terms of the proportion of times that it occurs in an infinite number of repetitions of the experiment does not provide a numerical value. The proportion of times it happens in a finite number of repetitions provides an estimate.

1.1.7   Subjective probability

It is impossible to imagine some random experiments being repeated. A personal or subjective assessment of the probabiity of an event must be made, but different people often have different subjective probabilities.

1.2   Probability calculations

1.2.1   Core rules of probability

All definitions of probability satisfy three main properties called axioms. Basing probability on these axioms means that we rarely need to distinguish between the different definitions.

1.2.2   A few properties of probability

This page shows a few important results about probability that can be derived from the three axioms.

1.2.3   Examples

Two examples show how probabilities can be calculated from the axioms.

1.3   Conditional probability

1.3.1   Conditional probability

Partial knowledge about a random experiment can be described with conditional probabilities.

1.3.2   Graphical display of probabilities

Marginal and conditional probabilities can be displayed graphically in a Venn diagram whose areas are proportional to probability.

1.3.3   Partitions of the sample space

A set of events that are mutually exclusive and cover the complete sample space is called a partition of the sample space.

1.3.4   Tree diagrams

A tree diagram describes probabilities for events that happen in sequence.

1.3.5   More about tree diagrams

An example shows a tree diagram for a longer sequences of events.

1.3.6   Independence

If knowledge about whether one event has happened does not affect the probability of another event, the two events are called independent.

1.3.7   Law of total probability

If {B₁, B₂, ..., } is a partition of the sample space, the probability of another event A is the sum of the probabilities that it occurs with the various B.

1.3.8   Bayes theorem

The conditional probability of an event A, given B, P(A | B) can be found from P(A), P(B | A) and P(B | not A).