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Chapter 7: Problem 29

A die is rolled and the number that falls uppermost is observed. Let \(E\) denote the event that the number shown is a 2 , and let \(F\) denote the event that the number shown is an even number. a. Are the events \(E\) and \(F\) mutually exclusive? b. Are the events \(E\) and \(F\) complementary?

Short Answer

Expert verified
a. No, events \(E\) and \(F\) are not mutually exclusive, as the occurrence of Event \(E\) (rolling a 2) is part of the outcome of Event \(F\) (rolling an even number). b. No, events \(E\) and \(F\) are not complementary, since their occurrence does not imply the non-occurrence of the other event and they do not cover all possible outcomes together.

Step by step solution

01

Define Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time. The occurrence of one event excludes the occurrence of the other event.
02

Define Complementary Events

Complementary events are such that one of these events must occur, and the occurrence of one event implies the non-occurrence of the other event.
03

Evaluate If Events \(E\) and \(F\) are Mutually Exclusive

To determine if events \(E\) and \(F\) are mutually exclusive, we must check if their occurrences are mutually exclusive (meaning the occurrence of one event excludes the occurrence of the other event). Event \(E\) - Number shown is 2 Event \(F\) - Number shown is even number (2, 4, or 6) Since the occurrence of Event \(E\) (rolling a 2) is part of the outcome of Event \(F\) (rolling an even number), these events are not mutually exclusive.
04

Answer Part (a)

Since the events can occur together, they are not mutually exclusive.
05

Evaluate If Events \(E\) and \(F\) are Complementary

To determine if events \(E\) and \(F\) are complementary, we must check if the occurrence of one event implies the non-occurrence of the other event and that they cover all possible outcomes. Event \(E\) - Number shown is 2 Event \(F\) - Number shown is even number (2, 4, or 6) For the events to be complementary, one of the events should cover the outcomes that the other event does not. However, we can observe that event \(F\) also includes the outcome for event \(E\). Thus, events \(E\) and \(F\) cannot be complementary since their occurrence does not imply the non-occurrence of the other event.
06

Answer Part (b)

Since the occurrence of one of the events does not imply the non-occurrence of the other event, they are not complementary events.

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