Probability

Probability - A Note About Probability There are many files...

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A Note About Probability There are many files and videos. You need to read and view all of them! Practically every homework problem assigned to you is developed at least in part somewhere in these readings. Some of your homework problems require you to extend or combine concepts beyond the reading. Use your textbook to expand your example base. You will find similar examples in any standard textbook about probability!
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Probability, An Introduction Probability is as intimately related to counting processes as you are to your skeleton! Without counting we cannot cover the concept except in fuzzy generalities. By the end of this section you should believe this. However, we do need to recognize some differences between our counting vocabulary and the probability vocabulary. Let’s begin with a definition of probability. Probability is the study of the likelihood or chance that an outcome or event can occur. This sentence is replete with words that need explanation! An outcome is some basic result that we can record in a process. Example: Say we flip a coin. The outcomes are to see heads or tails . We discount the on-edge possibility as a nonevent. Example: We spin a dial with areas marked a, b, c . Then the outcomes are a, b, or c . A sample space , usually denoted by S, is the collection of all possible outcomes from some action or process. Example: Say we flip a coin. The outcomes are a heads or a tails . These are the only two possible outcomes. We discount “edgies” and require a re-toss just as in most sporting events. S = {heads, tails} Example: Say we flip a coin two times. The outcomes are of each spin are heads or tails . However, the sample space is composed of the results of both spins. We list them as ordered pairs. S = {(heads, heads), (heads, tails), (tails, heads) (tails, tails)} The number of elements in the sample space is simply an application of previous counting methods. There are two choices at each flip. So there are outcomes. 2 22 2 4  Example: We spin a dial with areas marked a, b, c . S = {a, b, c} . Example: We spin a dial with areas marked a, b, c three times. Similarly to the coin flip the outcomes are not a, b, or c . They are the ordered triples listed as ( first spin, second spin, third spin ). The sample space is listed in the table below. aaa aab aac aba abb abc aca acb acc baa bab bac bba bbb bbc bca bcb bcb caa cab cab cba cbb cbc cca ccb ccc Listing the 27 outcomes is tedious. Fortunately, we are more interested than the count of the set than the list itself in many cases. Again counting the sample space is an application of the multiplication principle. There are three choices at each spin. The sample space has outcomes. 3 3333 2 7  
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Probability, An Introduction Page 2 of 5 A word of caution about sample spaces: These are not universal sets, strictly speaking. Sets are unordered , complete listing of elements with each one listed exactly once . As we will see later, sometimes it is convenient to allow repetition in a sample space to better understand the probabilities.
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Probability - A Note About Probability There are many files...

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