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Unformatted text preview: QABE Lecture 8 Probability I: Permutations and Combinations School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Why Counting? 2 3 Basic Counting 3 3.1 By Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 By Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Permutations 5 4.1 By the formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 Combinations 6 6 Summary 8 1 Introduction We now take a couple of lectures to think about probability. This is partly because it is another key feature of various economic problems (especially situations to do with un certainty ) but also partly because it helps as a primer for further thinking in probability and its cousin statistics . For this lecture, we will focus on ways of counting things up. This may not seem related to probability at first, but the fact is, if we cannot count the number of possible occurrences, then we cannot determine the likelihood or probability of seeing the one we have before us. More on this application in the following lecture. Before we begin, one more note on ‘probability’. This is a very common word in the popular media and discussions in general, for instance, the weatherman talks about ‘a 25% probability of rain tomorrow’, or our family doctor talks about ‘a onein1,500 chance’ that we might inherit a particular disease trait, or even the government might say ‘there is a very low probability of any further interest rate rise in the next two years.’ Do they all mean the same thing by the word ‘probability’? Sort of, however, sometimes we slip into using the word a bit informally. The first two examples are quite correct – in that the weather man and the doctor are asserting that ‘of all the cases we know about in the past that are like this one, x % of them had y characteristic’. To translate for the weather man, he is saying, ‘of all the 1 ECON 1202/ECON 2291: QABE c School of Economics, UNSW days that are meteorologically similar to what we expect tomorrow to be, 25% of them experienced rainfall.’ For the doctor this is, ‘of all the people we know of who have your heritage, an average of 1 in 1,500 of them had this particular disease.’ Notice that in both cases, each professional is asserting that they know about the population (‘all days like tomorrow’, or ‘all people with the same heritage as you’), and therefore can say something about the instance in front of them. This is what probability is about – knowing information about a whole population gives rise to estimations of likelihoods about a particular event. Statistics , on the other hand, ‘goes the other way’ (sample to population). More on this presently....
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 LorettiIsabellaDobrescu
 Economics, ECON 1202/ECON, QABE, QABE Lecture

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