Lec5ConditProb

Lec5ConditProb - ORIE 270 Engineering Probability and...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page Page 1 of 27 Go Back Full Screen Close Quit ORIE 270 Engineering Probability and Statistics Lecture 5: Conditional Probability Sidney Resnick School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA http://legacy.orie.cornell.edu/ sid [email protected] September 5, 2007
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 Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page Page 2 of 27 Go Back Full Screen Close Quit 1. Conditional Probability Suppose we have a probability model ( S, A , P ) reflecting our estimates of likelihood of outcomes and then we get additional information that some event must occur. What to do? Examples: Example 1: Suppose 270 consists of W women, M males; WS female soccer players MS male soccer players. Each member of the population is characterized by 2 traits: (gender, play soccer or not). Experiment: select a student from this population. Probability model for this experiment (w=female, m=male, s=soccer player, n=not) S (w,s) (w,ns) (m,s) (m,ns) P WS W + M W - WS W + M MS W + M M - MS W + M
Image of page 2
Conditional Probability Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page Page 3 of 27 Go Back Full Screen Close Quit Now restrict attention to the subpopulation of women. Within this subpopulation, we only distinguish by one trait: play soccer or not. The model changes to S (w,s) (w,ns) P WS W W - WS W OR S (w,s) (w,ns) (m,s) (m,ns) P WS W + M / W M + W W - WS W + M / W M + W 0 0 Note In the first table, we calculate WS W = % of women students who are soccer players by restricting attention to the subpopulation of women but as we see in the 2nd table, it is also possible to do the calculation relative to the orignal population of both males and females:
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 Multiplication Rule Law of Total Probability. Bayes Rule. Independence. Random Variables. Title Page Page 4 of 27 Go Back Full Screen Close Quit The ratio WS W + M W W + M = % of class consisting of female soccer players % of class consisting of females effectively does the calculation relative to the bigger popula- tion of the whole class but renormalizes the percentages. Example 2: In the skunky beer example, suppose I am second in line and I get additional information that the first in line received a good (or bad) beer. How do I revise my probability estimates? Definition: Given the model ( S, A , P ), the conditional probability of A given B is P ( A | B ) = P ( AB ) P ( B ) provided P ( B ) > 0. [ Reminder: AB = A B .] Could regard the new sample space as B but usually don’t and simply redistribute the probability onto B , making sure B gets 0 probability. Note P ( B | B ) = P ( B B ) P ( B ) = P ( ) P ( B ) = 0 .
Image of page 4
Conditional Probability Multiplication Rule Law of Total Probability.
Image of page 5

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

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern