Difference between revisions of "Probability"

Revision as of 20:55, 13 June 2008

Probability is traditionally considered one of the most difficult areas of mathematics, since probabilistic arguments often come up with apparently paradoxical or counterintuitive results. Examples include the Monty Haul paradox and the birthday problem. Probability can be loosely defined as the chance that an event will happen.

Introductory Probability

Before reading about the following topics, a student learning about probability should learn about introductory counting techniques.

Formal Definition of Probability

The foundations of probability reside in an area of analysis known as measure theory. Measure theory in general deals with integration, in particular, how to define and extend the notion of "area" or "volume." Intuitively, therefore, probability could be said to consider how much "volume" an event takes up in a space of outcomes.

In the language of measure theory, probability is formally defined as a triple known as a measure space, denoted $(\Omega, \mathfrak{a}, \mathit{P})$ . Here $\Omega$ is a set called the sample space, and $\mathfrak{a}$ is a class of events given by certain subsets of $\Omega$ . $\mathfrak{a}$ must satisfy certain properties (it must be a [[ $\sigma$ -algebra]]) to qualify as a class of events. Together, $\Omega$ and $\mathfrak{a}$ form what is known as a measurable space, $(\Omega, \mathfrak{a}$ . $\mathit{P}:\mathfrak{a}\to [0,1]$ is an assignment with certain properties (it is a special kind of measure), called the probability function, or probability measure. It assigns a "volume" to each possible event.

As a simple example, consider a single coin-toss. In this case, $\Omega = \{H, T\}$ , $\mathfrak{a} = \{\emptyset, \{H\}, \{T\}, \{H, T\}\}$ , and $\mathit{P}$ assigns the following probabilities to events in $\mathfrak{a}$ : $\mathit{P}(\emptyset)=0$ ,

$\mathit{P}(\{H\})=0.5$ ,

$\mathit{P}(\{T\})=0.5$ ,

$\mathit{P}(\{H, T\})=1$ .

Measure theory does assume considerable mathematical maturity, so it is usually ignored until one reaches an advanced undergraduate level. Once measure theory is covered, however, probability becomes a lot easier to deal with.

Types of Probability

Part of a comprehensive understanding of basic probability includes an understanding of the differences between different kinds of probability problems.

algebraic probability
combinatorial probability problems involve counting outcomes.
geometric probability

Important subdivisions of probability include

Example Problems

Introductory

Intermediate

Resources

Three Types of Probability

Introduction to Counting and Probability by David Patrick

Intermediate Counting and Probability by David Patrick

@@ Line 1: / Line 1: @@
-'''Probability''' is one of the most difficult areas of [[mathematics]] to define, explain, or understand.  Probability can be loosely defined as the chance that an event will happen.
+'''Probability''' is traditionally considered one of the most difficult areas of [[mathematics]], since probabilistic arguments often come up with apparently paradoxical or counterintuitive results. Examples include the [[Monty Haul paradox]] and the [[birthday problem]].  Probability can be loosely defined as the chance that an event will happen.
@@ Line 9: / Line 8: @@
 == Formal Definition of Probability ==
-Probability is formally defined as a triple <math>(\Omega, \mathfrak{a}, \mathit{P})</math>. Here <math>\Omega</math> is a set called the sample space, <math>\mathfrak{a}</math> is a class of events from the sample space, and <math>\mathit{P}:\mathfrak{a}\to [0,1]</math> is an assignment with certain properties called the probability function.
+The foundations of probability reside in an area of [[analysis]] known as [[measure theory]]. Measure theory in general deals with [[integral|integration]], in particular, how to define and extend the notion of "area" or "volume." Intuitively, therefore, probability could be said to consider how much "volume" an event takes up in a space of outcomes.
+In the language of measure theory, probability is formally defined as a triple known as a [[measure space]], denoted <math>(\Omega, \mathfrak{a}, \mathit{P})</math>. Here <math>\Omega</math> is a set called the sample space, and <math>\mathfrak{a}</math> is a class of events given by certain subsets of <math>\Omega</math>. <math>\mathfrak{a}</math> must satisfy certain properties (it must be a [[<math>\sigma</math>-algebra]]) to qualify as a class of events. Together, <math>\Omega</math> and <math>\mathfrak{a}</math> form what is known as a [[measurable space]], <math>(\Omega, \mathfrak{a}</math>. <math>\mathit{P}:\mathfrak{a}\to [0,1]</math> is an assignment with certain properties (it is a special kind of [[measure]]), called the probability function, or probability measure. It assigns a "volume" to each possible event.
+As a simple example, consider a single coin-toss. In this case, <math>\Omega = \{H, T\}</math>, <math>\mathfrak{a} = \{\emptyset, \{H\}, \{T\}, \{H, T\}\}</math>, and <math>\mathit{P}</math> assigns the following probabilities to events in <math>\mathfrak{a}</math>:
+<math>\mathit{P}(\emptyset)=0</math>,
+<math>\mathit{P}(\{H\})=0.5</math>,
+<math>\mathit{P}(\{T\})=0.5</math>,
+<math>\mathit{P}(\{H, T\})=1</math>.
+Measure theory does assume considerable mathematical maturity, so it is usually ignored until one reaches an advanced undergraduate level. Once measure theory is covered, however, probability becomes a lot easier to deal with.
 === Types of Probability ===
@@ Line 16: / Line 29: @@
 * [[combinatorial probability]] problems involve counting outcomes.
 * [[geometric probability]]
+Important subdivisions of probability include
+* [[stochastic processes]]
+* [[mathematical statistics]]
 === Example Problems ===
@@ Line 33: / Line 50: @@
 * [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=302 Intermediate Counting and Probability] by [[David Patrick]]
+[[Category:Mathematics]]

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