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Chapter 7: Problem 44

Let \(S\) be a sample space for an experiment. Show that if \(E\) is any event of an experiment, then \(E\) and \(E^{c}\) are mutually exclusive.

Short Answer

Expert verified
To show that if E is any event of an experiment, then E and \(E^c\) are mutually exclusive, we need to prove that there's no overlapping element between both events. By definition, a complementary event \(E^c\) is the subset of S containing all outcomes not in E. Therefore, there are no common elements between E and \(E^c\). Since their intersection is an empty set \((E 鈭� E^c = 鈭�)\), E and \(E^c\) are mutually exclusive.

Step by step solution

01

Understand the Definitions

Before we start proving that E and E^c are mutually exclusive, let's recall the definitions of events and complementary events: 1. An event E is a subset of the sample space S. 2. A complementary event, denoted by E^c, is the subset of S containing all outcomes not in E. 3. Mutually exclusive events are events that cannot occur simultaneously. In set theory terms, two events E and F are mutually exclusive if their intersection is empty (E 鈭� F = 鈭�).
02

Identify the Intersection

To show that E and E^c are mutually exclusive, we need to find the intersection of E and E^c and prove that it's an empty set. So let's examine the intersection: \(E 鈭� E^c\)
03

Apply the Definition of Complementary Events

Now we use the definition of complementary events to substitute what we know in the expression: \(E 鈭� (S - E)\) Here, (S - E) represents the complementary event, which is the subset of S containing all outcomes not in E.
04

Observe the Intersection

Let's look closely at the intersection between E and (S - E). Since E is a subset of S and (S - E) contains all outcomes not in E, there are no common elements between E and (S - E). Think of it this way, E is the set of outcomes that are in E, while (S - E) are outcomes that are specifically not in E. Therefore, their intersection is an empty set: \(E 鈭� (S - E) = 鈭匼)
05

Conclusion

We found that the intersection of E and E^c is an empty set. According to the definition of mutually exclusive events, two events are mutually exclusive if their intersection is empty. Thus, we have shown that if E is any event of an experiment, then E and E^c are mutually exclusive.

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