91Ó°ÊÓ

Conditional probability helps us determine the likelihood of an event occurring, given that another event has already happened. In the exercise, we are given the conditional probability that a head of household earns more than \$100,000 per year, given they have a bachelor's degree. This is represented as \( P(A|B) = 0.412 \). The formula for conditional probability is:

\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]

This tells us the probability of event A happening provided that event B has already occurred. In the example, the fact that \( P(A|B) = 0.412 \) means there is a 41.2% chance of earning more than \$100,000 per year if the head of the household has a bachelor's degree. This concept is crucial in probability theory, as it shows how related events influence each other.

Probability theory is a branch of mathematics that deals with calculating the likelihood of various outcomes. It provides a framework for understanding random events and their occurrences. In the context of the exercise, probability theory is applied to determine how likely it is for a head of household to earn more than \$100,000 per year both in general and under specific conditions. The formulas and concepts used, such as independent events and conditional probability, are part of this broader field.

Key elements in probability theory include:

• Probability of single events: This is the chance of a specific outcome occurring out of all possible outcomes, noted as \( P(A) \).
• Joint probability: The probability of two events occurring together, represented as \( P(A \text{ and } B) \).
• The addition rule: Used to find the probability of either one of two events happening, noted as \( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \).

Understanding these fundamentals allows us to analyze more complex situations, such as those given in the exercise.

Event independence in probability is when the occurrence of one event does not affect the probability of another event occurring. For events A and B to be considered independent, the probability of A given B must be the same as the probability of A alone:

\( P(A|B) = P(A) \)

In the exercise, we compare \( P(A|B) \) and \( P(A) \) to test for independence. We discovered:

• \( P(A) = 0.202 \)
• \( P(A|B) = 0.412 \)
• \( P(A|B) eq P(A) \)

Therefore, earning more than \$100,000 per year is not independent of having a bachelor's degree because the probabilities are different. This means the occurrence of having a bachelor's degree impacts the likelihood of earning more than \$100,000 per year, thus showing event dependence rather than independence.