University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

252

Pages:

3
Author:

Niklas

Problem Statement: Motor vehicles sold in the US are classified as either cars or light trucks and as either domestic or imported. Given data about the proportion of vehicles sold in the previous year. Calculations: Calculate the probability of a vehicle being domestic or a light truck using the inclusion-exclusion principle. Calculate the probability of a vehicle being imported and a light truck. Calculate the probability of a vehicle being a car. Calculate the probability of a vehicle being domestic and a car. Calculate the probability of a vehicle being imported and a car. Conditional Probability: Defines conditional probability as the probability of one event given that another event has occurred. Provides the formula for conditional probability. Discusses the cases where the two events are mutually exclusive or one event implies the other. Additional Calculations: Calculate the probability of a vehicle being a truck, given that it is imported. Calculate the proportion of imports that are trucks. Calculate the proportion of trucks that are imports. Multiplication Rule: Defines the probability of the intersection of two events A and B using conditional probability. Provides the formula: P(A ∩ B) = P(B) * P(A | B) Statistical Independence: Defines two events A and B as statistically independent if and only if P(A ∩ B) = P(A) * P(B). Derives the properties P(A | B) = P(A) and P(B | A) = P(B) for independent events. Example: Describes a jar containing 3 red balls and 7 black balls. Defines events A and B as drawing a red ball on the first and second draws, respectively. Calculates the probability of B under two scenarios: replacing the first ball and not replacing it. Determines whether A and B are independent in each scenario. Law of Total Probability: States that the probability of an event A can be calculated by considering the conditional probabilities of A given different events B and their respective probabilities. Provides the formula: P(A) = P(A ∩ B) + P(A ∩ B̄) = P(B) * P(A | B) + P(B̄) * P(A | B̄) Multiplication Rule: If A and B are independent, then P(A ∩ B) = P(A) * P(B). Otherwise, P(A ∩ B) = P(A) * P(B | A). Addition Rule: If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B). Otherwise, P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Independence: Independence cannot be depicted in a Venn diagram because it involves the probabilities of the events rather than the outcomes. P(A | B) = 0 indicates that A and B are not independent. Important Concepts: P(Ā) = 1 - P(A) P(A ∪ B) = P(A) + P(B) - P(A ∩ B) P(B | A) = P(A ∩ B) / P(A) Conditional Probability: Defines conditional probability as the probability of one event considering the other event has already occurred. Provides the formula: P(A | B) = P(A ∩ B) / P(B) Law of Total Probability: States that the probability of an event A can be calculated by considering the conditional probabilities of A given different events B and their respective probabilities.