SECTION 13.3 Probability 927 Constructing a Probability Model An experiment consists of rolling a fair die once. A die is a cube with each face having 1, 2, 3, 4, 5, or 6 dots on it. See Figure 5. Construct a probability model for this experiment. EXAMPLE 3 Solution A sample space S consists of all the possibilities that can occur. Because rolling the die will result in one of six faces showing, the sample space S consists of { } = S 1, 2, 3, 4, 5, 6 Because the die is fair, one face is no more likely to occur than another. As a result, our assignment of probabilities is ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = P P P P P P 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Figure 5 A six-sided die Now suppose that a die is loaded (weighted) so that the probability assignments are ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = P P P P P P 1 0 2 0 3 1 3 4 2 3 5 0 6 0 This assignment would be made if the die were loaded so that only a 3 or 4 could occur and the 4 was twice as likely as the 3 to occur.This assignment is consistent with the definition, since each assignment is nonnegative, and the sum of all the probability assignments equals 1. Now Work PROBLEM 23 Constructing a Probability Model An experiment consists of tossing a coin. The coin is weighted so that heads (H) is three times as likely to occur as tails (T). Construct a probability model for this experiment. EXAMPLE 4 Solution The sample space S is { } = S H, T . If x denotes the probability that a tail occurs, ( ) ( ) = = P T x P H x and 3 The sum of the probabilities of the possible outcomes must equal 1, so P T P H x x x x 3 1 4 1 1 4 ( ) ( ) + = + = = = Assign the probabilities ( ) ( ) = = P T P H 1 4 3 4 Now Work PROBLEM 27 In working with probability models, the term event is used to describe a set of possible outcomes of the experiment.An event E is some subset of the sample space S. The probability of an event ≠ ∅ E E , , denoted by ( ) P E , is defined as the sum of the probabilities of the outcomes in E. We can also think of the probability of an event E as the likelihood that the event E occurs. If = ∅ E , then ( ) = P E 0; if = E S, then ( ) ( ) = = P E P S 1. In Words P S 1 ( ) = means that one of the outcomes in the sample space must occur in an experiment.
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