Questions: A number cube with faces labeled 1 to 6 is rolled once. The number rolled will be recorded as the outcome. Consider the following events. Event A : The number rolled is even. Event B : The number rolled is less than 4. Give the outcomes for each of the following events. If there is more than one element in the set, separate them with commas. (a) Event "A or B": (b) Event "A and B": (c) The complement of the event B:

Solution Steps

The sample space \( S \) for a single roll of a number cube with faces labeled 1 to 6 is: \[ S = \{1, 2, 3, 4, 5, 6\} \]

• Event \( A \): The number rolled is even. The even numbers in the sample space are 2, 4, and 6. Thus, \[ A = \{2, 4, 6\} \]

• Event \( B \): The number rolled is less than 4. The numbers less than 4 in the sample space are 1, 2, and 3. Thus, \[ B = \{1, 2, 3\} \]

The event " \( A \) or \( B \) " includes all outcomes that are in either event \( A \) or event \( B \) or in both. This is the union of sets \( A \) and \( B \): \[ A \cup B = \{2, 4, 6\} \cup \{1, 2, 3\} = \{1, 2, 3, 4, 6\} \]

The event " \( A \) and \( B \) " includes all outcomes that are in both event \( A \) and event \( B \). This is the intersection of sets \( A \) and \( B \): \[ A \cap B = \{2, 4, 6\} \cap \{1, 2, 3\} = \{2\} \]

The complement of event \( B \) includes all outcomes in the sample space that are not in event \( B \): \[ B^c = S - B = \{1, 2, 3, 4, 5, 6\} - \{1, 2, 3\} = \{4, 5, 6\} \]

Final Answer