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Allan G. Bluman - Probability Demystified

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Book:
Probability Demystified
Author:
Allan G Bluman
Publisher:
McGraw-Hill Companies
Genre:
Books / Children
Year:
2012
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Stack the odds in your favor for mastering probability Dont leave your knowledge of probability to chance. Instead, turn to Probability Demystified, Second Edition, for learning fundamental concepts and theories step-by-step.This practical guide eases you into the subject of probability using familiar items such as coins, cards, and dice. As you progress, you will master concepts such as addition and multiplication rules, odds and expectation, probability distributions, and more. Youll learn the relationship between probability and normal distribution, as well as how to use the recently developed Monte Carlo method of simulation. Detailed examples make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce key ideas.Its a no-brainer! Youll learn about: Classical probability Game theory Actuarial science Addition rules Bayes theorem Odds and expectation Binomial distribution Simple enough for a beginner, but challenging enough for an advanced student, Probability Demystified, Second Edition, helps you master this essential subject.

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Acknowledgments

I would like to thank my wife, Betty Claire, for helping me with the preparation of this book and my editor, Judy Bass, for her assistance in its publication. I would also like to thank Eugene Mastroianni for his error checking and helpful suggestions.

Answers to Quizzes and Final Exam

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appendix

Bayes Theorem

A somewhat more difficult topic in probability is called Bayes theorem.

Given two dependent events, A and B, the previous formulas allowed you to find P( A and B) or P( B | A). Related to these formulas is a principle developed by an English Presbyterian minister, Thomas Bayes (17021761). It is called Bayes theorem.

Knowing the outcome of a particular situation, Bayes theorem enables you to find the probability that the outcome occurred as a result of a particular previous event. For example, suppose you have two boxes containing red balls and blue balls. Now if it is known that you selected a blue ball, you can find the probability that it came from Box 1 or Box 2. A simplified version of Bayes theorem is given next.

For 2 mutually exclusive events, A and B, where event B follows event A,

EXAMPLE

__________________________________

Box 1 contains two red balls and one blue ball. Box 2 contains one red ball and three blue balls. A coin is tossed; if it is heads, Box 1 is chosen, and a ball is selected at random. If the toss yields tails, Box 2 is chosen. If the ball is red, find the probability it came from Box 1.

SOLUTION

_________________________

Let A = selecting Box 1 and = selecting Box 2. Since the selection of a box is based on a coin toss, the probability of selecting Box 1 is and the probability of selecting Box 2 is ; hence, P( A) and P( ) . Let B = selecting a red ball and = selecting a blue ball. From Box 1, the probability of selecting a red ball is , and the probability of selecting a blue ball is since there are two red balls and one blue ball. Hence, P( B| A) and P( | A) . Since there is one red ball in Box 2, P( B| ) is , and since there are 3 blue balls in Box 2, P( | ) . The probabilities are shown in .

FIGURE A-1

Hence,

In summary, if a red ball is selected, the probability that it came from Box 1 is .

EXAMPLE

__________________________________

Two video products distributors supply video tape boxes to a video production company. Company A sold 100 boxes of which five were defective. Company B sold 300 boxes of which 21 were defective. If a box was defective, find the probability that it came from Company B.

SOLUTION

_________________________

Let D = the event that the box is defective and = the event that the box is not defective.

Let P( A) = probability that a box selected at random is from Company A; also, let B = . Then P( A) ; P( B) = P( ) . Since there are 5 defective boxes from Company A, P( D| A) and there are 21 defective types from Company B or , P( D| ) . The probabilities are shown in .

FIGURE A-2

PRACTICE

__________________________________

1. Box 1 contains six green marbles and four yellow marbles. Box 2 contains five yellow marbles and five green marbles. A box is selected at random and a marble is selected from the box. If the marble is green, find the probability that it came from Box 1.

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