225-Notes02

225-Notes02 - Stat 225 Lecture Notes "Counting Techniques"...

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Stat 225 Lecture Notes “Counting Techniques” Ryan Martin Spring 2008 1 Basic Counting Rule Recall that in the classical probability model, the probability of an event E is P ( E ) = N ( E ) N (Ω) , (1.1) the number of outcomes in E divided by the total number of outcomes. Counting the number of outcomes that make up a given event can be very diFcult and impractical since, even for relatively simple situations, there can be a tremendous number of such outcomes. ±or example: There are 2,598,960 ways to deal a poker hand. A sequence of 10 tosses of a coin has 1024 possible outcomes. There are 13,983,816 to select 6 out of 49 lottery numbers. Let’s consider an easier example ²rst. Example 1.1. Suppose we ³ip a coin and then roll a 6-sided die. Let E be the event that we get a “Tail” on the coin and an even number on the die. To use formula (1.1) to ²nd P ( E ), we need to know the total number of outcomes. ±rom a tree diagram (see ±igure 1) it is easy to see that there are 12 possible outcomes. Three of the outcomes result in a “Tail” and an even roll, therefore, P ( E ) = 3 / 12 = 0 . 25. Exercise 1.2. I own 5 shirts, 2 pair of pants and two pair of shoes. How many di´erent out²ts could I wear? Draw a tree diagram. Mathematicians are lazy and did not want to work this hard. They developed a set of counting rules, which are included in what is called combinatorics . The ²rst is the Basic Counting Rule (BCR), often called the multiplication rule for obvious reasons. Proposition 1.3 (BCR) . Suppose that r actions are to be performed in a deFnite order. Suppose further that there are m k possibilities for action k , for k = 1 , . . ., r . Then there are m 1 m 2 ··· m r possibilities altogether for the r actions. 1

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H , 1 H , 2 H , 3 H , 4 H , 5 H , 6 T , 1 T , 2 T , 3 T , 4 T , 5 T , 6 Fig 1: A tree diagram for Exercise 1.1. The ±rst set of branches denote the outcome of the coin toss (H or T) and the second set denote the outcome of the die roll (1–6). Exercise 1.4. If you rolled four fair six-sided dice, what is the probability that you get at least one 3? Exercise 1.5. Prove that a set containing n elements has 2 n possible subsets. How many di²erent pizzas can be made with 10 distinct toppings? Exercise 1.6. Let A and B be ±nite sets, each with n elements. How many one-to-one functions are there from A to B ? Relate your answer to the number of tries it would take to successfully decode a simple substitution cipher by guessing. (Note: a substitution cipher is the coding scheme used in a standard cryptogram.) 2 Sampling With and Without Replacement Defnition 2.1. Consider a population with N members. Sampling with replacement is where n members of the population are selected, one at a time, and after the observation is made the member is returned to the population for possible re-selection.
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This note was uploaded on 06/15/2008 for the course STAT 225 taught by Professor Martin during the Spring '08 term at Purdue University.

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225-Notes02 - Stat 225 Lecture Notes "Counting Techniques"...

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