Lecture Notes on Probability - Chapter 7 1 Section 7.3 2...

Info icon This preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 7 1
Image of page 1

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

View Full Document Right Arrow Icon
Section 7.3 2
Image of page 2
3 Section Summary y Bayes’ Theorem y Generalized Bayes’ Theorem y Bayesian Spam Filters
Image of page 3

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

View Full Document Right Arrow Icon
4 Motivation for Bayes’ Theorem y Bayes’ theorem allows us to use probability to answer questions such as the following: y Given that someone tests positive for having a particular disease, what is the probability that they actually do have the disease? y Given that someone tests negative for the disease, what is the probability, that in fact they do have the disease? y Bayes’ theorem has applications to medicine, law, artificial intelligence, engineering, and many diverse other areas.
Image of page 4
5 Bayes’ Theorem Bayes’ Theorem : Suppose that E and F are events from a sample space S such that p ( E ) ് 0 and p ( F ) ് 0 . Then: Example : We have two boxes. The first box contains two green balls and seven red balls. The second contains four green balls and three red balls. Bob selects one of the boxes at random. Then he selects a ball from that box at random. If he has a red ball, what is the probability that he selected a ball from the first box. y Let E be the event that Bob has chosen a red ball and F be the event that Bob has chosen the first box. y By Bayes’ theorem the probability that Bob has picked the first box is:
Image of page 5

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

View Full Document Right Arrow Icon
6 Derivation of Bayes’ Theorem y Recall the definition of the conditional probability p ( E |F): y From this definition, it follows that: , continued ՜
Image of page 6
7 Derivation of Bayes’ Theorem On the last slide we showed that continued ՜ , , Solving for p ( E | F ) and for p ( F | E ) tells us that Equating the two formulas for p(E|F) shows that
Image of page 7

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

View Full Document Right Arrow Icon
8 Derivation of Bayes’ Theorem On the last slide we showed that: Note that Hence, since because and By the definition of conditional probability,
Image of page 8
9 Applying Bayes’ Theorem Example : Suppose that one person in 100,000 has a particular disease. There is a test for the disease that gives a positive result 99 % of the time when given to someone with the disease. When given to someone without the disease, 99.5 % of the time it gives a negative result.Find a) the probability that a person who test positive has the disease. b) the probability that a person who test negative does not have the disease. y Should someone who tests positive be worried?
Image of page 9

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

View Full Document Right Arrow Icon
10 Applying Bayes’ Theorem Solution : Let D be the event that the person has the disease, and E be the event that this person tests positive. We need to compute p ( D | E ) from p (D), p ( E | D ), p ( E | ) , p ( ).
Image of page 10
11 Applying Bayes’ Theorem y What if the result is negative?
Image of page 11

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

View Full Document Right Arrow Icon
Image of page 12
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