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Are They Independent? For each of the following pair of events \(A\) and \(B\), do you think the events are independent? Explain your reasoning. a. You select an adult U.S. citizen at random. Event \(A\) is "the person is a registered democrat," and event \(B\) is "the person is opposed to the death penalty." b. You select an adult U.S. citizen at random. Event \(A\) is "the person is a baby boomer" (baby boomers are people born 1946-1964), and event \(B\) is "the person favors legalization of marijuana." c. You draw a card at random from a deck of playing cards. Event \(A\) is "the card is the king of hearts," and event \(B\) is "the card is the queen of diamonds." d. You select a student at your college at random. Event \(A\) is "the student is taking a Spanish class," and event \(B\) is "the student has visited Mexico."

Step by step solution

01

Understanding Independence

Two events, \(A\) and \(B\), are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, \(P(A \cap B) = P(A)\times P(B)\). If \(P(A \cap B) = P(A)\times P(B)\), then they are independent.

02

Analyzing Events A and B: Part (a)

For a random U.S. citizen, event \(A\) is being a registered Democrat and event \(B\) is opposing the death penalty. Political affiliation and stance on specific political issues are often correlated, meaning these events are likely dependent rather than independent.

03

Analyzing Events A and B: Part (b)

For a random U.S. citizen, event \(A\) is being a baby boomer and event \(B\) is favoring marijuana legalization. Demographic trends suggest different age groups might have varying opinions, indicating that these events could be dependent.

04

Analyzing Events A and B: Part (c)

When drawing a card from a deck, event \(A\) is drawing the king of hearts and event \(B\) is drawing the queen of diamonds. There is only one king of hearts and one queen of diamonds. Since a single card cannot be both, these events are mutually exclusive and hence independent.

05

Analyzing Events A and B: Part (d)

For a student, event \(A\) is taking a Spanish class and event \(B\) is having visited Mexico. There might be a correlation if students interested in Spanish language and culture are more likely to visit Mexico, suggesting dependency between \(A\) and \(B\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability

Probability is a measure of how likely an event is to occur. It is represented as a number between 0 and 1. Zero means the event will certainly not happen, while one means it is certain to happen.
Probability provides valuable insights not just in games or random experiments, but in real-world scenarios where predicting outcomes can influence decisions.
For example, in the context of the original exercise, understanding the probability of a card being drawn from a deck helps to determine if two events (drawing a king of hearts and drawing a queen of diamonds) are independent. To calculate probability, you use the formula:

• Probability of an event (P) = Number of favorable outcomes / Total number of possible outcomes.

This formula helps us gauge how frequent an event can happen by evaluating past data or experimenting within a controlled setup.

Conditional Probability Explained

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept helps to determine relationships between events and to revise probabilities based on new information.
The formula for conditional probability is:

• \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

This formula is used when you need to know the probability of event A happening given that event B has already happened.
For instance, in the original exercise, if a person is a baby boomer, the probability that they favor marijuana legalization can be analyzed using conditional probability if influences or historical data suggests a correlation.

Understanding Mutual Exclusivity

Mutual exclusivity refers to events that cannot happen at the same time. If one occurs, the other cannot. Think of it as a clear 'one or the other' scenario.
When two events are mutually exclusive, their probabilities cannot be assumed to influence each other, making them independent in nature.
The exercise example where a card drawn can either be a king of hearts or a queen of diamonds but never both at the same time is a perfect depiction of mutual exclusivity. Mutually exclusive events have a key relationship:

• Your combined probability is simply the sum of the two individual probabilities.

This characteristic simplifies understanding and computing probabilities, especially when evaluating complex scenarios where distilling concurrent possibilities is needed.

Decoding Demographic Correlation

Demographic correlation examines the relationship between population characteristics and various phenomena. It's a powerful concept that helps to analyze and predict trends.
By understanding the demographics of a population, it's possible to predict certain behaviors or beliefs shared among that demographic. In the context of the original exercise, being a baby boomer and favoring marijuana legalization may show demographic correlation if evidence suggests that views on issues are often shared across generations.
Demographic correlations often involve large datasets, like surveys or census data, that reveal deeper insights into human behavior and social systems.

• They are vital in market research, public policy making, and social sciences.
• Correlations don't imply causation; they simply suggest a statistical relationship.

It's important to remember these distinctions, as misinterpretation can lead to incorrect conclusions.