Problem 32 - Algebra and Trigonometry

Question

Determine whether the events \(E\) and \(F\) in the given experiment are mutually exclusive. The experiment consists of choosing at random a student from your class. (a) \(E :\) The student is female \(F :\) The student wears glasses (b) \(E :\) The student has long hair \(F :\) The student is male

Step-by-Step Solution

Answer

Events in (a) are not mutually exclusive; events in (b) are also not mutually exclusive.

Step 1: Understanding Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occur at the same time. For example, in a single coin toss, the events "getting a head" and "getting a tail" are mutually exclusive because you cannot get both results simultaneously.

Step 2: Analyzing Event Pair (a)

For event (a), we have: - Event E: The student is female - Event F: The student wears glasses These events are not mutually exclusive because a student can be both female and wearing glasses at the same time.

Step 3: Analyzing Event Pair (b)

For event (b), we have: - Event E: The student has long hair - Event F: The student is male These events are not inherently mutually exclusive since a male student can have long hair. Therefore, a student can be male and have long hair at the same time.

Key Concepts

Probability

Probability is the measure of the likelihood that a particular event will occur. Imagine you're rolling a dice; the probability of getting a 3 is one out of six because there are six possible outcomes, and only one of them is a 3.

In mathematics, probability is defined as \[ \text{Probability of an event} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] In our original exercise, when discussing students from the class, probability helps us determine how likely it is to randomly pick a student who is female or wears glasses. This involves recognizing all favorable outcomes, such as all female students, over the total number of students.

Event Analysis

Event Analysis involves examining different outcomes in an experiment. In probability, an event is a specific outcome or a group of related outcomes.

In the context of the original exercise, events are centered around the characteristics of students. For example:

• Event E: The student is female
• Event F: The student wears glasses

When analyzing events, it is essential to recognize whether these events can happen simultaneously or not. If they cannot occur at the same time, they are referred to as mutually exclusive events. However, in our example, a student can indeed be both female and wear glasses at the same time, leading us to conclude these events are not mutually exclusive.

Sample Space

The sample space in probability is the set of all possible outcomes in an experiment. Consider it as a complete list of every potential outcome that can occur.

In our classroom example, the sample space could include combinations of different attributes, such as gender and whether the student wears glasses or not. Visualizing the sample space can help in understanding which events can happen together and which cannot.

• The student is female and wears glasses.
• The student is male and has long hair.
• The student is female and does not wear glasses.
• And so on ...

This complete list helps us better analyze the probability of each event occurring when a student is chosen randomly.

Gender and Characteristics

Gender and characteristics of individuals are often used in probability to define different events. By categorizing groups based on these attributes, we can better analyze complex scenarios and understand how likely specific combinations are.

For instance, in our original problem, different characteristics such as the student being female or male, or wearing glasses or having long hair, define the events we analyze. Exploring such traits allows for a deeper investigation into how these characteristics interact.

In many educational settings, understanding the interaction between these traits is crucial. It helps in determining patterns, probabilities, and the likelihood of encountering certain combinations. These categorizations make the abstract concepts of probability more tangible and relatable through everyday scenarios involving students and their characteristics.