{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 4.2 January-April 2011 1 / 36
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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%.
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}