Problem 29 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: Answer Part (a)

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

Step 5: 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.

Step 6: 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.

Key Concepts

Mutually Exclusive Events

In probability theory, understanding mutually exclusive events is crucial. These are events that cannot occur simultaneously. If one event happens, the other cannot. Think of it like flipping a switch on or off; you can't do both at the same time. For mutually exclusive events, the probability of both events occurring together is zero. When considering two events, say Event A and Event B, they are mutually exclusive if

• The occurrence of Event A means Event B cannot happen,
• Mathematically defined as: \( P(A \cap B) = 0 \).

In the exercise, Event E (rolling a 2) and Event F (rolling an even number) were examined. Since rolling a 2 is part of rolling an even number (which also includes 4 and 6), both events can happen together. Therefore, they're not mutually exclusive events.

Complementary Events

Complementary events cover all possible scenarios within a defined sample space. If one event occurs, the complementary event does not, and together they encompass every outcome. They are defined such that: if Event A happens, the complementary Event A' does not. This is expressed as:

• \( P(A') = 1 - P(A) \)
• The sum of their probabilities is always equal to 1.

Taking the exercise, for two events to be complementary, like our events E and F:

• Event E: rolling a 2.
• Event F: rolling an even number (2, 4, 6).

They cannot be complementary because Event F includes the scenario of Event E. Complementary events should have no overlap, ensuring one covers all outcomes the other does not. Therefore, these two events do not fill the requirement of being complementary.

Dice Rolling Probabilities

Dice rolling exercises offer a practical way to understand probability concepts. A standard die has six faces, each showing a different number from 1 to 6. The probability of any specific outcome is calculated by dividing the number of favorable outcomes by the total number of possible outcomes, giving: \( P( ext{Specific Outcome}) = \frac{1}{6} \).
When analyzing probabilities on dice:

• Every number 1 through 6 has the same chance, \( P(x) = \frac{1}{6} \).
• Rolling an even number (2, 4, 6) constitutes three favorable outcomes, thus: \( P(F) = \frac{3}{6} = \frac{1}{2} \).
• For a specific number like 2, \( P(E) = \frac{1}{6} \).

Using dice rolls to explore mutually exclusive or complementary probabilities is effective as each roll is independent and consistently offers equal opportunity for each outcome. Thus, a practical application in understanding these basic probability principles.