Problem 41 - Algebra and Trigonometry

Question

\(41-42\) . Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of selecting a person at random. (a) \(E :\) The person is male. F: The person is female. (b) \(E :\) The person is tall. \(F :\) The person is blond.

Step-by-Step Solution

Answer

(a) Mutually exclusive; (b) Not mutually exclusive.

Step 1: Understand the Definition of Mutually Exclusive Events

Two events are said to be mutually exclusive if they cannot occur at the same time. In other words, if one event occurs, the other cannot occur. For example, if you are tossing a coin, getting heads and getting tails are mutually exclusive events.

Step 2: Analyze Case (a)

In part (a), event \(E\) is that the person selected is male, and event \(F\) is that the person is female. These two events cannot happen at the same time because a person cannot be both male and female simultaneously. Therefore, events \(E\) and \(F\) are mutually exclusive.

Step 3: Analyze Case (b)

In part (b), event \(E\) is that the person is tall, and event \(F\) is that the person is blond. These two characteristics can occur simultaneously; a person can be both tall and blond at the same time. Hence, events \(E\) and \(F\) are not mutually exclusive.

Key Concepts

Probability Theory

Probability theory is a fascinating branch of mathematics focused on analyzing the likelihood of different events occurring. At the core of this theory is the concept of a probability, which can be understood as a measure, often expressed as a number between 0 and 1, of the chance that a certain event will happen. A probability of 0 suggests the event will not occur at all, while a probability of 1 indicates that it is certain to occur. Understanding probability involves several key components:

• Experiments: Any action or process that results in well-defined outcomes. For instance, selecting a person at random is an experiment.
• Sample Space: This is the set of all possible outcomes from an experiment. In our example, the sample space would include every possible individual you could select.
• Events: These are specific outcomes or sets of outcomes from the sample space, like selecting a person who is male or tall.

Probability theory helps in understanding how likely it is for events to occur and assists in making informed predictions in both simple and complex situations.

Event Analysis

Event analysis involves breaking down and examining the components and potential outcomes of an event. This process often requires defining the events clearly and analyzing their relationships. In our example, we have two cases of event analysis: For (a), we define event \(E\) as selecting a male and event \(F\) as selecting a female. It is essential to note that these outcomes are straightforward due to the clear distinction between male and female in this context. For (b), event \(E\) is selecting someone tall, while event \(F\) is selecting someone blond. This case requires careful consideration of whether these characteristics can coexist, rather than being inherently separate attributes. Key steps in event analysis include:

• Defining the events precisely.
• Considering possible sample spaces and outcomes for each event.
• Evaluating whether the events are overlapping or distinct.

Event analysis is crucial in probability theory because it lays the groundwork for determining how events relate, such as whether they are mutually exclusive or not.

Mutual Exclusivity Determination

Determining whether events are mutually exclusive is an important aspect of probability theory. Events are mutually exclusive when they cannot both occur at the same time. In simple terms, the occurrence of one event means the other cannot happen. In the original exercise, this happens in case (a), where selecting a person as male excludes the possibility of selecting the same person as female. The occurrence of one clearly denies the occurrence of the other, thus making them mutually exclusive. In case (b), the determination changes because being tall and being blond are not mutually exclusive; a person can possess both characteristics at the same time. Therefore, these events can overlap, and if you are trying to categorize events based on mutual exclusivity, it's important to look at such overlaps. To effectively determine mutual exclusivity, consider:

• Analyzing whether there is any overlap between the events.
• Checking if the occurrence of one event completely excludes the other.
• Thinking critically about the nature of the events and their contexts.

This determination helps in correctly evaluating probabilities and better understanding the structure of different event scenarios.