Problem 31 - Algebra and Trigonometry

Question

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

Two events are mutually exclusive if they cannot both occur at the same time. If the occurrence of one event precludes the occurrence of the other, they are mutually exclusive.

Step 2: Analyze Event Pair (a)

For part (a), consider the events: - Event E: The person is male. - Event F: The person is female. A person cannot be both male and female at the same time, so these events are mutually exclusive.

Step 3: Analyze Event Pair (b)

For part (b), consider the events: - Event E: The person is tall. - Event F: The person is blond. A person can be both tall and blond, so these events are not mutually exclusive.

Key Concepts

Understanding Probability and Mutually Exclusive Events

Probability is a fundamental concept in statistics that helps us quantify the likelihood of an event occurring. When we say we are determining probability, we are essentially measuring how likely it is that a particular outcome will happen when a random event occurs. In this context, for any event occurring in a probability space, its value lies between 0 and 1, where 0 indicates impossibility, and 1 denotes certainty.
When dealing with mutually exclusive events, these are events that cannot both occur at the same time. In probability terms, if two events, A and B, are mutually exclusive, then the probability of both events happening simultaneously is zero:

• Mathematically, this is expressed as: \(P(A \cap B) = 0\)

For example, consider tossing a coin: the outcomes "heads" and "tails" are mutually exclusive because the coin cannot land on both sides at the same time.

Event Analysis: Using Real-World Scenarios

Event analysis involves evaluating circumstances to understand whether certain events can or cannot occur simultaneously. This step is crucial in determining if events are mutually exclusive. Let us apply this analysis to everyday scenarios to make it clearer. In our original exercise scenario:

• **(a) Event E:** The person is male.
  **Event F:** The person is female.
  In this case, a person cannot be both male and female simultaneously, reinforcing that these events are mutually exclusive.
• **(b) Event E:** The person is tall.
  **Event F:** The person is blond.
  Here, a person can indeed be both tall and blond, so these events are not mutually exclusive.

By examining these situations, one can clearly see which events cannot co-occur, making the concept of mutually exclusive events easier to grasp.

Statistics and Probability Applications

Statistics often employs the principles of probability to solve real-world problems and analyze data sets. Understanding mutually exclusive events in statistics helps with simplifying computations of probabilities and analyzing outcomes more effectively. When analyzing data:

• Calculating probabilities for events helps in decision making, such as predicting outcomes from past data trends.
• Recognizing mutually exclusive events ensures that misinterpretations do not occur when planning or analyzing complex situations.
• Proper event categorization using the mutually exclusive concept aids in creating more accurate models and forecasts.

These applications show how probability, event analysis, and mutually exclusive events serve as vital tools within the field of statistics, leading to precise and meaningful data interpretations.