P ab p b a p a 0001 095 00187 p b 0001

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Unformatted text preview: what is the probability that he/she has the disease? Let A ={The person has the disease}, and B ={the test result is positive}. P (A|B ) = P (B |A) P (A) 0.001 = 0.95 = 0.0187. P (B ) 0.001 · 0.95 + 0.999 · 0.05 (80% of the questioned persons in US leading hospital answered the probability that the person has the decease is 0.95!) M. Chen (IE@CUHK) ENGG2430C lecture 4 10 / 19 Counting Based on Discrete Uniform Law Throw two dies. Event A ={sum of two dies gives 7}; P (A) =? Since all outcomes are equally likely, thus P (A) = M. Chen (IE@CUHK) number of outcomes in A total number of outcomes in Ω ENGG2430C lecture 4 11 / 19 Counting Principle r steps ni choices at step i Total umber of choices is: n1 n2 · · · nr (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientific Publishers, 2002) M. Chen (IE@CUHK) ENGG2430C lecture 4 12 / 19 Permutation: Order of Selection Matters The total number of ways to order n elements = n! 128 bits long IPv6 (Internet Protocol version 6) addresses will be used to identify endhosts on the Internet. How many are they? 2128 ≈ 3.4 × 1038 . Number of subsets of {1, 2, . . . , n} = 2n “Mark Six" lottery: six Drawn Numbers and one Extra Number, all between 1 and 49. Number of outcomes in Ω : Event A ={all numbers of a outcome are even} Number of outcomes in A: P (A) =? M. Chen (IE@CUHK) ENGG2430C lecture 4 13 / 19 Combination: Order of Selection Does Not Matter k Number of k -element subsets of an n-element set: Cn = (notations: http://en.wikipedia.org/wiki/Combination) n k n Question: ∑n =0 k =? Answer: 2n . Hint: how to relate it to the total k number of subsets of an n-element set? k How to obtain the closed-form expression of Cn ? Consider the following two ways of constructing an ordered sequence of k distinct items Choose the k items, one at a time: n(n − 1) · · · (n − k + 1) = n! (n − k )! Choose k items at one shot, then order them k Cn · k ! Thus k Cn = M. Chen (IE@CUHK) n! (n − k )!k ! ENGG2430C lecture 4 14 / 19 Partition 52...
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This document was uploaded on 03/31/2014.

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