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Chapter 7: Problem 24

Let \(S=\\{1,2,3\\}\) be a sample space associated with an experiment. a. List all events of this experiment. b. How many subsets of \(S\) contain the number 3 ? c. How many subsets of \(S\) contain either the number 2 or the number 3 ?

Short Answer

Expert verified
a. The events of this experiment are the subsets of the sample space S: {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}. b. There are 4 subsets containing the number 3: {3}, {1, 3}, {2, 3}, {1, 2, 3}. c. There are 6 subsets containing either the number 2 or the number 3: {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.

Step by step solution

01

List all events of the experiment

The events of this experiment are the subsets of the sample space S. To list all possible subsets, consider each element and whether it is included or not in the subset. Since there are 3 elements, each having 2 possibilities (in or out), there are a total of \(2^3 = 8\) subsets (including the empty set and the set S itself). The subsets are: 1. {} 2. {1} 3. {2} 4. {3} 5. {1, 2} 6. {1, 3} 7. {2, 3} 8. {1, 2, 3} These 8 subsets are the events of this experiment.
02

Count the subsets containing the number 3

To count the subsets that contain the number 3, observe that now we only need to consider the elements 1 and 2 and whether they are included in the subset or not. For example, the subsets containing the number 3 could have only 3, or 3 with 1, or 3 with 2, or 3 with both 1 and 2. Since there are 2 elements remaining, each with 2 possibilities (in or out), there are \(2^2 = 4\) subsets containing the number 3: 1. {3} 2. {1, 3} 3. {2, 3} 4. {1, 2, 3}
03

Count the subsets containing either the number 2 or the number 3

To count the subsets containing either the number 2 or the number 3, we can simply list all subsets from Step 1 and remove those that contain neither 2 nor 3. The remaining subsets are: 1. {2} 2. {3} 3. {1, 2} 4. {1, 3} 5. {2, 3} 6. {1, 2, 3} There are 6 subsets containing either the number 2 or the number 3.

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