Questions: 10. Concept 5: Subjective Probabilities Script (Instructor): "Not all probabilities are purely data-driven. Sometimes, especially in business forecasting or economics, we rely on expert judgments-known as subjective or Bayesian probabilities. These reflect personal degrees of belief rather than just observed frequencies." Explanation: - Subjective probabilities incorporate expert opinion, past experience, or incomplete data. - Commonly applied in economic outlooks (e.g., probability of recession next year). 11. Concept 6: Independence Script (Instructor): "Two events A and B are independent if knowing that A has occurred does not affect the probability of B, and vice versa. Mathematically, P(A ∩ B)=P(A)P(B)." Explanation: - If independence holds, the conditional probability P(A B) equals P(A). - This concept is critical for simplifying joint probability calculations in modeling and risk assessment.

Solution Steps

Subjective probabilities are based on expert judgment, personal experience, or incomplete data rather than purely on observed frequencies. They are often used in fields like business forecasting and economics, where precise data may not be available. For example, an economist might assign a subjective probability to the likelihood of a recession occurring in the next year based on their analysis of current economic conditions.

Two events \( A \) and \( B \) are independent if the occurrence of one does not affect the probability of the other. Mathematically, this is expressed as: \[ P(A \cap B) = P(A) \cdot P(B) \] If independence holds, the conditional probability \( P(A \mid B) \) is equal to \( P(A) \). This concept is crucial for simplifying joint probability calculations in various applications, such as modeling and risk assessment.

To apply the concept of independence, consider two events \( A \) and \( B \) with known probabilities. If \( P(A) = 0.3 \) and \( P(B) = 0.4 \), and the events are independent, then the probability of both events occurring together is: \[ P(A \cap B) = P(A) \cdot P(B) = 0.3 \times 0.4 = 0.12 \]

Final Answer