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...

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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

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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
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.

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• Fall '18
• Staff
• Probability theory, warden,  P, UTSPH

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