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# Week04Notes - Stat 311 Introduction to Mathematical...

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Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 22-24, 2009 Example Suppose we have 6 cards { A,K,Q,J,10,9 } and we select a pair of diﬀerent cards from the list without replacement. Q: P ( A ) = P ( { both letter cards selected } ) Solve: List all possible pairs Problems If the sample space has many items, this kind of work becomes tedious, we will use counting method. Combinatorial principles The science of counting is called combinatorics . Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and combinations. Counting rules Multiplication rule 1: total number of outcomes of a sequence when each has k possi- bilities = k n . Multiplication rule 2: total number of outcomes of a sequence when each has a diﬀerent number of possibilities k 1 · k 2 · k 3 ··· k n . 0

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In set notation S = { ( s 1 ,...,s n ) : s i S i } = S 1 × ··· × S k | S | = | S 1 | × ··· × | S k | . Example Suppose we ﬂip three coins and roll two fair six-sided dice. What is the probability that all three coins come up heads, and both dice come up 6? Example: Traveling salesman problem Start from city C0, will visit n other cities one by one Counting formula 1 A permutation is an ordered arrangement of distinct items. The total number of permutations of n distinct items is n ( n - 1) ··· (2)(1) = n ! Theorem 3.1 The total number of permutations of a set
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Week04Notes - Stat 311 Introduction to Mathematical...

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