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lecture4.2

# lecture4.2 - Stat 302 Introduction to Probability Jiahua...

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Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 4.2 January-April 2011 1 / 36

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Conditional Probability Motivation. Updating the probability of an event E to take into account the information that another event F occurred. Intuition : Assume that all the outcomes are equally likely. Once F has occured, F contains all possible outcomes so it is the new sample space. Hence Number of Possibles = # F . For E to occur, it is necessary that the outcome is in E F . Hence Number of Favorable = # ( E F ) . Therefore the updated probability of E should be # ( E F ) # ( F ) = ( # ( E F )) / ( # S ) ( # F ) / ( # S ) = P ( E F ) P ( F ) . Jiahua Chen () Lecture 4.2 January-April 2011 2 / 36
Conditional Probability Definition of Conditional Probability . Consider a random experiment with sample space S and probability measure P ( · ) . Let E and F be two events such that P ( F ) > 0. The conditional probability of E given F is defined as P ( E | F ) = P ( E F ) P ( F ) . Jiahua Chen () Lecture 4.2 January-April 2011 3 / 36

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Conditional Probability Definition of Conditional Probability . Consider a random experiment with sample space S and probability measure P ( · ) . Let E and F be two events such that P ( F ) > 0. The conditional probability of E given F is defined as P ( E | F ) = P ( E F ) P ( F ) . The symbol P ( E | F ) is read as follows: conditional probability of E given F or for brevity probability of E given F . Jiahua Chen () Lecture 4.2 January-April 2011 3 / 36
Example (A family with 2 kids) A family has 2 children. Suppose the 4 possible configurations { ( B , B ) , ( B , G ) , ( G , B ) , ( G , G ) } are equally likely. What is the probability that both are girls if 1) no additional information is given, 2) the elder child is a girl, 3) one of the two kids is a girl. Jiahua Chen () Lecture 4.2 January-April 2011 4 / 36

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Answer to 1) 1) probability that both are girls when no additional information is given. Answer: Since P ( { G , G } ) = P ( { G , B } ) = P ( { B , G } ) = P ( { B , B } ) = 1 4 P ( { G , G } ) = 1 / 4. Jiahua Chen () Lecture 4.2 January-April 2011 5 / 36
Answer to 2) 2) probability that both are girls when it is known that the elder child is a girl. Answer: It is known that F = ( { G , G } , { G , B } ) , P ( F ) = 1 / 2 has occured. Now, P ( { G , G }| F ) = P ( F ∩ { G , G } ) P ( F ) = P ( { G , G } ) P ( F ) = 1 4 2 4 = 1 2 . Knowing the elder is a girl makes “both girl” more likely. Jiahua Chen () Lecture 4.2 January-April 2011 6 / 36

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Answer to 3) 3) probability that both are girls when one of the two kids is a girl. Answer: It is known that F = ( { G , G } , { G , B } , { B , G } ) P ( F ) = 3 / 4 has occured. Hence, P ( { G , G }| F ) = P ( F ∩ { G , G } ) P ( F ) = P ( { G , G } ) P ( F ) = 1 4 3 4 = 1 3 . Jiahua Chen () Lecture 4.2 January-April 2011 7 / 36
Answer to 3) 3) probability that both are girls when one of the two kids is a girl. Answer: It is known that F = ( { G , G } , { G , B } , { B , G } ) P ( F ) = 3 / 4 has occured. Hence, P ( { G , G }| F ) = P ( F ∩ { G , G } ) P ( F ) = P ( { G , G } ) P ( F ) = 1 4 3 4 = 1 3 . What would be the outcome if F is the event that one of two kids is known to be a boy? Jiahua Chen () Lecture 4.2 January-April 2011 7 / 36

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Example (Passing the Second Test) Example : A probability class has two midterms. The prob of acing the first is 42% and the prob of acing both is 25%.
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