Problem 30 - 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 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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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

Key Concepts

Mutually Exclusive Events

When learning about probability, one fundamental concept is that of mutually exclusive events. These are events that can't happen simultaneously — the occurrence of one event excludes the possibility of the other occurring. For example, if you flip a coin, it can only land on either heads or tails, not both. That makes these outcomes mutually exclusive.

In the context of rolling a die, if event E represents rolling an even number, and event F represents rolling an odd number, these are also mutually exclusive. Rolling a die will always result in a single number, and each number is classified distinctly as either even (2, 4, 6) or odd (1, 3, 5), without any overlap between the two.

Complementary Events

In probability, complementary events are two outcomes of an experiment that are the only possible outcomes and that cover all the potential scenarios. They are like two sides of the same coin, quite literally. In the coin example, heads and tails are complementary — a coin will surely land on one side, and together, heads and tails represent all possible outcomes.

Similarly, when a die is rolled, as in our exercise, even numbers (2, 4, 6) and odd numbers (1, 3, 5) together make up the set of all possible outcomes. This completeness means that the events of rolling an even number and rolling an odd number are perfectly complementary. Understanding this can help students check their work by ensuring that their total probability adds up to 1, the certainty of an event occurring.

Outcomes of Rolling A Die

The roll of a die presents a classic case of discrete probability, which deals with outcomes that are individually distinct and countable. A standard die has six faces, each bearing a number from 1 to 6. The outcomes are naturally 1, 2, 3, 4, 5, and 6. When dealing with dice rolls, it's important to know these are equally likely outcomes if the die is fair. This means the probability of any single number appearing after a roll is \( \frac{1}{6} \).

Understanding the possible outcomes is crucial for solving a variety of probability problems and is often the first step in the process. To visualize the concept, you might imagine a real die or draw a diagram of a die, emphasizing the likelihood of each side landing up after a roll.