Chapter_5 - Chapter 5 Probability 5.1 What is probability...

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Chapter 5: Probability 5.1 What is probability anyway? Probability is a branch of mathematics which intrudes on everyday conversation perhaps more than any other (except for just plain arithmetic and counting). A TV weather report might announce that “there is a 70% chance of rain today.” When you hear that statement you automatically infer that there must be a 30% chance that it will not rain, don’t you? When you do that you’re intuitively using one of the basic properties of probability. Other common expressions that implicitly have to do with probability are statements like, “N.C. State is a 2 to 1 favorite to beat Wake Forest today,” or “I think there’s about a 50-50 chance that I’ll pass English this semester.” Often when we talk about probability in everyday conversation we speak in terms of “percentages” or “odds” or “chances.” When speaking mathematically we usually treat probabilities as fractions (or decimal numbers) rather than percentages, and we usually use the word “probability.” The three statements in the paragraph above are equivalent to these: (1) The probability of rain today is .7. (2) The probability that N.C. State will beat Wake Forest today is 2/3. (3) I think the probability that I will pass English this semester is about 1/2. Many of the examples which are most useful in developing our understanding of probabilities involve simple experiments. With each experiment there are certain outcomes for the experiment. For instance, if a coin is tossed then the possible outcomes are HEADS and TAILS. If a class holds an election to elect a class president, then the possible outcomes are the individual members of the class, any of whom could be elected president. When talking about probabilities, it’s always a good idea to have a firm understanding of just what the possibilities are. In fact, this idea is so important that there is a special name for the set of all possibilities. It’s called the sample space. Definition: Sample space The sample space for an experiment is the set of all possible outcomes. Example 5.1. A coin is tossed twice. In this case the sample space could be considered to be the set S = { HH , HT , TH , TT }. The expression TH , for example, would
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Section 5.1: What Is Probability Anyway? 163 represent the outcome in which a “tails” occurs on the first toss and a “heads” occurs on the second toss. Example 5.2. A card is drawn from a standard deck of 52 cards. Here the possible outcomes are the 52 cards in the deck, so the sample space is the set S = {2 , 2 ,2 ,2 ,3 ,..., K , Α♣ , A ,A ,A }. Don’t get the idea that you have to be an artist, however, in order to represent the elements of the sample space. The important concept is that the sample space consists of the 52 cards. Whatever notation you wish to use to represent the 52 cards is your business.
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