7 - 18.05 Lecture 7 Bayes Formula Partition B1 Bk k i=1 Bi...

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18.05 Lecture 7 February 18, 2005 Bayes’ Formula. Partition B 1 , ..., B k k B i = S, B i B j = for i = j i =1 P ( A ) = k P ( AB i ) = k P ( A | B i ) P ( B i ) - total probability. i =1 i =1 Example: In box 1, there are 60 short bolts and 40 long bolts. In box 2, there are 10 short bolts and 20 long bolts. Take a box at random, and pick a bolt. What is the probability that you chose a short bolt? B 1 = choose Box 1. B 2 = choose Box 2. P ( short ) = P ( short | B 1 ) P ( B 1 ) + P ( short B 2 ) P ( B 2 ) = 60 2 ) + 10 2 ) | 100 ( 1 30 ( 1 Example: Partitions: B 1 , B 2 , ...B k and you know the distribution. Events: A, A, ..., A and you know the P (A) for each B i If you know that A happened, what is the probability that it came from a particular B i ? P ( B i A ) P ( A B i ) P ( B i ) P ( B i | A ) = = | : Bayes’s Formula P ( A ) P ( A B 1 ) P ( B 1 ) + ... + P ( A B k ) P ( B k ) | | Example: Medical detection test, 90% accurate. Partition - you have the disease ( B 1 ), you don’t have the disease ( B 2 ) The accuracy means, in terms of probability: P (positive B 1 ) = 0 . 9 , P (positive B 2 ) = 0 . 1 | | In the general public, the chance of getting the disease is 1 in 10,000.

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