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notes09c3 - 3 Probability Counting Techniques Some...

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3. Probability – Counting Techniques Some probability problems can be attacked by specifying a sample space S = { a 1 , a 2 , . . . , a n } in which each simple event has probability 1 n (i.e. is “equally likely"). Thus, if a compound event A consists of r simple events, then P ( A ) = r n . To use this approach we need to be able to count the number of events in S and in A , and this can be tricky. We review here some basic ways to count outcomes from “experiments". These approaches should be familiar from high school mathematics. 3.1 General Counting Rules Most of us think we know how to count, but it might help to review two basic rules for counting which can deal with most problems involving large sample spaces. We phrase the rules in terms of “jobs" which are to be done. 1. The Addition Rule: Suppose we can do job 1 in p ways and job 2 in q ways. Then we can do either job 1 or job 2, but not both, in p + q ways. For example, suppose a class has 30 men and 25 women. There are 30 + 25 = 55 ways the prof. can pick one student to answer a question. 2. The Multiplication Rule: Suppose we can do job 1 in p ways and an unrelated job 2 in q ways. Then we can do both job 1 and job 2 in p × q ways. For example, to ride a bike, you must have the chain on both a front sprocket and a rear sprocket. For a 21 speed bike there are 3 ways to select the front sprocket and 7 ways to select the rear sprocket. This linkage of OR with addition and AND with multiplication will occur throughout the course, so it is helpful to make this association in your mind. The only problem with applying it is that questions do not always have an AND or an OR in them. You often have to play around with re-wording the question for yourself to discover implied AND’s or OR’s. 15
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