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Lesson8S1.pptx

Lesson8S1.pptx - TAIBAH UNIVERSITY Faculty of...

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TAIBAH UNIVERSITY Faculty of Science Department of Math . ةبيط ةعماج مولعلا ةيلك تايضايرلا مسق Probability and Statistics for Engineers STAT 305 Teacher : Dr. Moustapha Abdellahi First Semester 1438/1439
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Probability of an Event, Additive Rule Lesson 8
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Probability of an Event: · To every point (outcome) in the sample space of an experiment S , we assign a weight (or probability), ranging from 0 to 1 , such that the sum of all weights (probabilities) equals 1 . · The weight (or probability) of an outcome measures its likelihood (chance) of occurrence. · To find the probability of an event A , we sum all probabilities of the sample points in A . This sum is called the probability of the event A and is denoted by P( A ) .
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Definition : The probability of an event A is the sum of the weights (probabilities) of all sample points in A . Therefore, 1) 2) 3) 1 0 A P 1 S P 0 P
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Example : A balanced coin is tossed twice. What is the probability that at least one head occurs? Solution: S = {HH, HT, TH, TT} A = {at least one head occurs}= {HH, HT, TH} Since the coin is balanced, the outcomes are equally likely; i.e., all outcomes have the same weight or probability.
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Outcome Weight (Probability ) 4w =1 w =1/4 = 0.25 P(HH)=P(HT)=P(TH)=P(TT)=0.25 HH HT TH TT P(HH) = w P(HT) = w P(TH) = w P(TT) = w Sum 4w=1
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The probability that at least one head occurs is: P( A ) = P({at least one head occurs})=P({HH, HT, TH}) = P(HH) + P(HT) + P(TH) = 0.25+0.25+0.25 = 0.75
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Theorem 1: If an experiment has n ( S )= N equally likely different outcomes, then the probability of the event A is: S in outcomes of no A in outcomes of no N A n S n A n A P .
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