928 CHAPTER 13 Counting and Probability 2 Compute Probabilities of Equally Likely Outcomes When the same probability is assigned to each outcome of the sample space, the experiment is said to have equally likely outcomes . THEOREM Probability for Equally Likely Outcomes If an experiment has n equally likely outcomes, and if the number of ways in which an event E can occur is m, then the probability of E is ( ) = = P E E m n Number of ways that can occur Number of possible outcomes (3) If S is the sample space of this experiment, ( ) ( ) ( ) = P E n E n S (4) Calculating Probabilities of Events Involving Equally Likely Outcomes Calculate the probability that in a 3-child family there are 2 boys and 1 girl. Assume equally likely outcomes. Solution EXAMPLE 5 Begin by constructing a tree diagram to help in listing the possible outcomes of the experiment. See Figure 6, where B stands for “boy” and G for “girl.”The sample space of this experiment is { } = S BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG so ( ) = n S 8. Note from the Multiplication Principle discussed in Section 13.1 that there are ⋅ ⋅ = 2 2 2 8 outcomes. We wish to know the probability of the event E: “having two boys and one girl.” From Figure 6, we conclude that { } = E BBG, BGB, GBB , so ( ) = n E 3. Since the outcomes are equally likely, the probability of E is ( ) ( ) ( ) = = P E n E n S 3 8 Figure 6 B B B B 3rd child 2nd child 1st child B B B G G G G GGG GGB GBG GBB BGG BGB BBG BBB G G G Now Work PROBLEM 37 So far, we have calculated probabilities of single events. Now we compute probabilities of multiple events, which are called compound probabilities . Computing Compound Probabilities Consider the experiment of rolling a single fair die. Let E represent the event “roll an odd number,” and let F represent the event “roll a 1 or 2.” (a) Write the event E and F. What is ( ) ∩ n E F ? (b) Write the event E or F. What is ( ) ∪ n E F ? (c) Compute ( ) P E . Compute ( ) P F . (d) Compute ( ) ∩ P E F . (e) Compute ( ) ∪ P E F . EXAMPLE 6 Solution The sample space S of the experiment is { } 1, 2, 3, 4, 5, 6 , so ( ) = n S 6. Since the die is fair, the outcomes are equally likely.The event E: “roll an odd number” is { } 1, 3, 5 , and the event F: “roll a 1 or 2” is { } 1, 2 , so ( ) = n E 3 and ( ) = n F 2. (a) In probability, the word and means the intersection of two events. The event E and F is { } { } { } ( ) ∩ = ∩ = ∩ = E F n E F 1, 3, 5 1, 2 1 1
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