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Monday, May 26, 2008

Introduction to Probability

Charles M. Grinstead

Swarthmore College

J. Laurie Snell

Dartmouth College

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments.

This text is designed for an introductory probability course taken by sophomores, juniors, and seniors in mathematics, the physical and social sciences, engineering, and computer science. It presents a thorough treatment of probability ideas and techniques necessary for a firm understanding of the subject. The text can be used in a variety of course lengths, levels, and areas of emphasis.

For use in a standard one-term course, in which both discrete and continuous probability is covered, students should have taken as a prerequisite two terms of calculus, including an introduction to multiple integrals. In order to cover Chapter 11, which contains material on Markov chains, some knowledge of matrix theory is necessary.

The text can also be used in a discrete probability course. The material has been organized in such a way that the discrete and continuous probability discussions are presented in a separate, but parallel, manner. This organization dispels an overly rigorous or formal view of probability and offers some strong pedagogical value in that the discrete discussions can sometimes serve to motivate the more abstract continuous probability discussions. For use in a discrete probability course, students should have taken one term of calculus as a prerequisite.

Very little computing background is assumed or necessary in order to obtain full benefits from the use of the computing material and examples in the text. All of the programs that are used in the text have been written in each of the languages TrueBASIC, Maple, and Mathematica.

This book is on the Web at http://www.dartmouth.edu/˜chance, and is part of the Chance project, which is devoted to providing materials for beginning courses in probability and statistics. The computer programs, solutions to the odd-numbered exercises, and current errata are also available at this site. Instructors may obtain all of the solutions by writing to either of the authors, at jlsnell@dartmouth.edu and cgrinst1@swarthmore.edu

For detail, please look at : http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/amsbook.mac.pdf

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Some Example in Probability Theory

Problem:	A spinner has 4 equal sectors colored yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner? What are the chances of landing on red?	Top of Form Bottom of Form
Solution:	The chances of landing on blue are 1 in 4, or one fourth.
	The chances of landing on red are 1 in 4, or one fourth.

This problem asked us to find some probabilities involving a spinner. Let's look at some definitions and examples from the problem above.

Definition	Example
An experiment is a situation involving chance or probability that leads to results called outcomes.	In the problem above, the experiment is spinning the spinner.
An outcome is the result of a single trial of an experiment.	The possible outcomes are landing on yellow, blue, green or red.
An event is one or more outcomes of an experiment.	One event of this experiment is landing on blue.
Probability is the measure of how likely an event is.	The probability of landing on blue is one fourth.

In order to measure probabilities, mathematicians have devised the following formula for finding the probability of an event.

Probability Of An Event

P(A) =	The Number Of Ways Event A Can Occur
	The Total Number Of Possible Outcomes

The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. Let's take a look at a slight modification of the problem from the top of the page.

Experiment 1:

A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?

Outcomes:

The possible outcomes of this experiment are yellow, blue, green, and red.

Probabilities:

P(yellow)	=	number of ways to land on yellow	=	1
		total number of colors		4

P(blue)	=	number of ways to land on blue	=	1
		total number of colors		4

P(green)	=	number of ways to land on green	=	1
		total number of colors		4

P(red)	=	number of ways to land on red	=	1
		total number of colors		4

Experiment 2:

A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?

Top of Form

Bottom of Form

Outcomes:

The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6.

Probabilities:

P(1)	=	number of ways to roll a 1	=	1
		total number of sides		6

P(2)	=	number of ways to roll a 2	=	1
		total number of sides		6

P(3)	=	number of ways to roll a 3	=	1
		total number of sides		6

P(4)	=	number of ways to roll a 4	=	1
		total number of sides		6

P(5)	=	number of ways to roll a 5	=	1
		total number of sides		6

P(6)	=	number of ways to roll a 6	=	1
		total number of sides		6

P(even)	=	# ways to roll an even number	=	3	=	1
		total number of sides		6		2

P(odd)	=	# ways to roll an odd number	=	3	=	1
		total number of sides		6		2

Experiment 2 illustrates the difference between an outcome and an event. A single outcome of this experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. Rolling an even number (2, 4 or 6) is an event, and rolling an odd number (1, 3 or 5) is also an event.

In Experiment 1 the probability of each outcome is always the same. The probability of landing on each color of the spinner is always one fourth. In Experiment 2, the probability of rolling each number on the die is always one sixth. In both of these experiments, the outcomes are equally likely to occur. Let's look at an experiment in which the outcomes are not equally likely.

Experiment 3:

A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

Outcomes:

The possible outcomes of this experiment are red, green, blue and yellow.

Probabilities:

P(red)	=	number of ways to choose red	=	6	=	3
		total number of marbles		22		11

P(green)	=	number of ways to choose green	=	5
		total number of marbles		22

P(blue)	=	number of ways to choose blue	=	8	=	4
		total number of marbles		22		11

P(yellow)	=	number of ways to choose yellow	=	3
		total number of marbles		22

The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

Experiment 4:

Choose a number at random from 1 to 5. What is the probability of each outcome? What is the probability that the number chosen is even? What is the probability that the number chosen is odd?

Outcomes:

The possible outcomes of this experiment are 1, 2, 3, 4 and 5.

Probabilities:

P(1)	=	number of ways to choose a 1	=	1
		total number of numbers		5

P(2)	=	number of ways to choose a 2	=	1
		total number of numbers		5

P(3)	=	number of ways to choose a 3	=	1
		total number of numbers		5

P(4)	=	number of ways to choose a 4	=	1
		total number of numbers		5

P(5)	=	number of ways to choose a 5	=	1
		total number of numbers		5

P(even)	=	number of ways to choose an even number	=	2
		total number of numbers		5

P(odd)	=	number of ways to choose an odd number	=	3
		total number of numbers		5

The outcomes 1, 2, 3, 4 and 5 are equally likely to occur as a result of this experiment. However, the events even and odd are not equally likely to occur, since there are 3 odd numbers and only 2 even numbers from 1 to 5.

Summary:

The probability of an event is the measure of the chance that the event will occur as a result of an experiment. The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:

If P(A) > P(B) then event A is more likely to occur than event B.
If P(A) = P(B) then events A and B are equally likely to occur.

Source: http://www.mathgoodies.com/lessons/vol6/intro_probability.html

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Probability

Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics.

There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution.

A properly normalized function that assigns a probability "density" to each possible outcome within some interval is called a probability function (or probability distribution function), and its cumulative value (integral for a continuous distribution or sum for a discrete distribution) is called a distribution function (or cumulative distribution function).

A variate is defined as the set of all random variables that obey a given probabilistic law. It is common practice to denote a variate with a capital letter (most commonly ). The set of all values that can take is then called the range, denoted (Evans et al. 2000, p. 5). Specific elements in the range of are called quantiles and denoted , and the probability that a variate assumes the element is denoted .

Probabilities are defined to obey certain assumptions, called the probability axioms. Let a sample space contain the union () of all possible events , so