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Chapter 7: Problem 30

A die is rolled and the number that falls uppermost is observed. Let \(E\) denote the event that the number shown is even, and let \(F\) denote the event that the number is an odd number. a. Are the events \(E\) and \(F\) mutually exclusive? b. Are the events \(E\) and \(F\) complementary?

Short Answer

Expert verified
a. Yes, events \(E\) and \(F\) are mutually exclusive because they cannot both occur at the same time. If we roll an even number, it cannot be an odd number and vice versa. b. Yes, events \(E\) and \(F\) are complementary because together they cover all possible outcomes of rolling a die.

Step by step solution

01

Understanding Mutually Exclusive Events

Two events are said to be mutually exclusive if they cannot both occur at the same time. In other words, if one event occurs, the other event cannot occur. In this problem, event \(E\) is getting an even number, and event \(F\) is getting an odd number.
02

Determine if E and F are Mutually Exclusive

Let's list all the possible outcomes of rolling a die: 1, 2, 3, 4, 5, and 6. Even numbers (\(E\)): 2, 4, 6 Odd numbers (\(F\)): 1, 3, 5 As we can see, there is no overlap between these two sets of outcomes. If we roll an even number, it cannot be an odd number and vice versa. So, events \(E\) and \(F\) are mutually exclusive.
03

Understanding Complementary Events

Two events are said to be complementary if their union covers all the possible outcomes of an experiment. In other words, if one event occurs, the other event must not occur, and together they make up all possible outcomes.
04

Determine if E and F are Complementary

Using the list of possible outcomes for events \(E\) and \(F\) we generated earlier, let's check if their union covers all the possible outcomes when rolling a die: Even numbers (\(E\)): 2, 4, 6 Odd numbers (\(F\)): 1, 3, 5 All possible outcomes: 1, 2, 3, 4, 5, 6 Since the union of events \(E\) and \(F\) includes all the possible outcomes, we can say that events \(E\) and \(F\) are complementary events. In conclusion: a. The events \(E\) and \(F\) are mutually exclusive b. The events \(E\) and \(F\) are complementary

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