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Unformatted text preview: Chapter 7 Elementary Probability In this chapter we will introduce the basics of probability needed to under- stand statistical concepts and ideas. The literature identifies three defini- tions of probability; 1) the classical definition, 2) the empirical or frequentist definition, and 3) the subjective definition. The classical definition is based on the assumption that the outcomes of an experiment are equally likely, e.g probability of even numbers when rolling a die. This is basically the ratio of the number of favorable outcomes over the total number of possible outcomes, which for our example would be 3 out of 6 or 1 / 2. the empirical definition is based on relative frequencies, e.g one could talk of the probability of Manchester United beating Chelsea, and this based on the number of games won by the former team over the latter. Lastly, the subjective definition of probability is probability that is based on an individual’s evaluation of available opinions and other information, e.g a student’s belief or subjective probability that (s)he will get an A in a statistics course like this one. We mention the three types of probabilities, but what follows, what can be called the mathematics of probability, does not depend on the definition. We will treat the general mathematics of probability applicative to whatever the definition of probability assumed. 1 2 CHAPTER 7. ELEMENTARY PROBABILITY 7.1 The probability space, experiment and out- come Tossing a coin, for example, can produce a heads or a tails. In the jargon of probability, the outcome of an experiment is uncertain and is thus handled with a probability model. An experiment, in our example tossing a coin, will produce exactly one out outcome (either heads or tails) out of a number of possible outcomes (heads and tails). The set of all possible outcomes is called the sample space of the ex- periment, and is denoted by S . A subset of the sample space, that is, a collection of possible outcomes, is called an event . There is no restriction in what constitutes an experiment. For example, it could be a single toss of a coin, or two tosses, or three tosses, or even infinite number of tosses. What is important to realize is that each is only one experiment, that is, three tosses of a coin constitute a single experiment and not three separate experiments. To concretize things, let’s continue with the experiment of tossing three coins. The sample space S is HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. It is reasonable to assume that each possible outcome has the same chance of occurring, that is, 1 / 8. Note that, probabilities always lie between 0 and 1 (or 0 % and 100 %.) From here, it follows that P ( A ) = 1- P ( ∼ A ) (7.1) That is, the probability of something happening equals 100 % minus the probability of the opposite thing, and vice versa, that is, the probability of something not happening equals 100 % minus the probability of that thing happening, i.e P ( ∼ A ) = 1- P ( A )....
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