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# Lecture4standard - Statistics 511: Statistical Methods Dr....

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 Lecture 4: Counting Techniques, Conditional Probability and Independence Devore: Section 2.3-2.5 Jan 2012 Page 1

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 Why are counting tecniques needed? Sample space is too large The outcomes are not equally likely Jan 2012 Page 2
Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 The product rule Ordered pairs: if the ﬁrst element can be selected in n 1 ways and the second in n 2 ways we have n 1 n 2 possible ordered pairs. Note that this is sampling with replacement . An ordered k-tuple is an ordered collection of k objects. Example: { 1 , 2 , 3 , 4 , 5 , 6 } is an ordered 6 -tuple. If each of i elements can be selected in n i ways, the total number of ordered k -tuples possible is n 1 n 2 × ... × n k Jan 2012 Page 3

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 Examples Each of the 2 clinics has 3 specialists in internal medicine, 2 general surgeons. You require the services of both and beneﬁt most by choosing both from the same clinic. How many opportunities do you have to do it? Answer: 4 × 3 = 12 What if you also have 2 pediatricians for each clinic and need to choose a pediatrician as well? Answer: 4 × 3 × 2 = 24 . Jan 2012 Page 4
Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 Example Consider a system of ﬁve identical components connected in a series. Let F be a failure and S be a success for any single component. What is the probability of an event A = { system fails } ? Answer: A 0 = { S,S,S,S,S } , thus, P ( A ) = 1 - P ( A 0 ) = 1 - P ( S ) 5 . E.g. if P ( S ) = 0 . 9 , P ( A ) = 1 - 0 . 9 5 = 0 . 41 . Jan 2012 Page 5

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2012 Sampling without replacement An ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k . The notation is P k,n . The number of possible permutations is P k,n = n ( n - 1)( n - 2) × ... × ( n - ( k - 1)) For any positive integer m , m ! = m ( m - 1) × ... × 1 . Using factorial notation, we have P k,n = n ! ( n - k )! . Jan 2012 Page 6
Dr. Levine Purdue University Spring 2012 Example A boy has 4 beads red, white, blue, and yellow. In how many different ways can three of the beads be strung together in a row? This is a permutation since the beads will be in a row (order). Thus, the answer

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## This note was uploaded on 02/20/2012 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue University-West Lafayette.

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Lecture4standard - Statistics 511: Statistical Methods Dr....

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