chapter4 - 2011/09/13 Introduction to Probability and...

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2011/09/13 1 Introduction to Probability and Statistics Thirteenth Edition Chapter 4 Probability and Probability Distributions What is Probability? • In Chapters 2 and 3, we used graphs and numerical measures to describe data sets which were usually samples. • We measured “how often” using Relative frequency = f/n Sample And “How often” = Relative frequency Population Probability • As n gets larger, Basic Concepts • An experiment is the process by which an observation (or measurement) is obtained. Experiment: Record an age Experiment: Toss a die Experiment: Record an opinion (yes, no) Experiment: Toss two coins Basic Concepts • A simple event is the outcome that is observed on a single repetition of the experiment. –The basic element to which probability is applied. –One and only one simple event can occur when the experiment is performed. • A simple event is denoted by E with a subscript.
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2011/09/13 2 Basic Concepts • Each simple event will be assigned a probability, measuring “how often” it occurs. • The set of all simple events of an experiment is called the sample space, S. Example The die toss: • Simple events: Sample space: 1 2 3 4 5 6 E 1 E 2 E 3 E 4 E 5 E 6 S ={E 1 , E 2 , E 3 , E 4 , E 5 , E 6 } S E 1 E 6 E 2 E 3 E 4 E 5 Basic Concepts • An event is a collection of one or more simple events. The die toss: –A: an odd number –B: a number > 2 S A ={E 1 , E 3 , E 5 } B ={E 3 , E 4 , E 5 , E 6 } B A E 1 E 6 E 2 E 3 E 4 E 5 Basic Concepts • Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. Experiment: Toss a die –B: observe a number greater than 2 –C: observe a 6 –D: observe a 3 Not Mutually Exclusive Mutually Exclusive B and C? B and D?
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2011/09/13 3 The Probability of an Event • The probability of an event A measures “how often” we think A will occur. We write P(A). • Suppose that an experiment is performed n times. The relative frequency for an event A is Number of times A occurs f nn n f A P n lim ) (  •If we let n get infinitely large, The Probability of an Event • P(A) must be between 0 and 1. –If event A can never occur, P(A) = 0. If event A always occurs when the experiment is performed, P(A) =1. • The sum of the probabilities for all simple events in S equals 1. •The probability of an event A is found by adding the probabilities of all the simple events contained in A. –10% of the U.S. population has red hair. Select a person at random. Finding Probabilities • Probabilities can be found using –Estimates from empirical studies –Common sense estimates based on equally likely events. P(Head) = 1/2 P(Red hair) = .10 Examples: –Toss a fair coin. Example • Toss a fair coin twice. What is the probability of observing at least one head?
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This note was uploaded on 09/19/2011 for the course MTH 1250C taught by Professor Any during the Fall '08 term at St. Johns Duplicate.

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chapter4 - 2011/09/13 Introduction to Probability and...

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