Lecture3-4.pdf - Lecture 3 Conditional Probability Definition If P(F)>0 then P E | F P E F is the conditional probability that E occurs P F given that F

Lecture3-4.pdf - Lecture 3 Conditional Probability...

This preview shows page 1 - 4 out of 10 pages.

1 Lecture 3 Conditional Probability Definition If P(F)>0 , then ( ) ( | ) ( ) P E F P E F P F is the conditional probability that E occurs given that F has occurred. Example: Toss 2 dice Given that the 1 st die is a 3, what is the probability that the sum of 2 dices equals 8? Without conditions : 5/36 With condition: 1/6 (all possible outcomes that satisfies the condition are (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)) Example: Diseases distribution for a cohort is given below: Disease B Disease A Yes No Yes 8 2 No 10 80 Prob(Disease A)= 10/100 and Prob (Disease A|Disease B)=8/10 Example: A study contains subjects from three different sites: 100 from site A, 50 from site B and 100 from site C. A subject is chosen at random from the study subjects, and it is noted that the chosen subject is not from site C. What’s the probability that it is from site B? Solution: 50 ( ) 1 250 ( | ) 150 ( ) 3 250 C C C P B C P B C P C . Question : Is ( | ) P F a probability measure? Does it satisfy three axioms of probability? (a) For any E, 0 ( | ) P E F
Image of page 1

Subscribe to view the full document.

2 (b) P(S|F)=1 (c) If , 1,2,3, ....... i E i are mutually exclusive events, then 1 1 ( | ) ( | ) i i i i P E F P E F Proof: (a) ( ) ( | ) 0, ( ) ( ) ( | ) 1 ( ) P E F P E F P E F P F P E F P F (b) P(S F) P(F) P(S|F)= 1 P(F) P(F) (c) 1 1 1 1 1 [ ( )] ( ) ( ) ( | ) ( ) ( ) ( ) i i i i i i i i i i P E F P E F P E F P E F P E F P F P F P F ( i E are mutually exclusive and hence i E F are also mutually exclusive) Law of Total Probability Let 1 2 , , , , k D D D be mutually exclusive and exhaustive events and B be any other event. Then 1 ( ) ( ) i i P B P B D . Example: (P64, K) Schools in London are classified into Government (G), Church (C), and Grant Maintained (GM). The noise level around were rated as L (low), M (moderate), and H (high). 60% of the low-noise schools, 70% of moderate-noise schools, and 83% of high-noise schools are G owned. Among all schools 47.2% were ranked low, 38.2% were ranked moderate, and 14.6% were ranked high. What is the probability that a school randomly chosen in London is Government owned? Solution: Given P(G|L)=0.6, P(G|M)=0.7, P(G|H)=0.83, we have ( ) ( ) ( ) ( ) ( | ) ( ) ( ) ( | ) ( ) ( | ) 0.672 P G P G L P G M P G H P G L P L P M P G M P H P G H
Image of page 2
3 Bayes Theorem 1 1 ( ) | ( ) ( ) ( ) ( | ) ( ) ( | ) ( ) j j j k k j j k k k P E F P F E P E P E F P E F P E F P F P E F P F Where 1 k k F S and , l m F F l m   Example: At a psychiatric clinic the social workers are so busy that, on the average, only 60% of potential new patients that are able to talk immediately with a social worker when they telephone. The other 40% are asked to leave their phone numbers. About 75% of the time a social worker is able to return the call in the same day, and the other 25% of the time the caller is contacted on the following day.
Image of page 3

Subscribe to view the full document.

Image of page 4
  • Fall '18
  • Staff
  • Probability theory, warden,  P, UTSPH

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

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes