### Probability Analysis of Vehicle PreferencesConsider a scenario where an individual tests three different vehicles. Define the events \( A_1, A_2, A_3 \) as follows:- \( A_1 \): Likes vehicle #1- \( A_2 \): Likes vehicle #2- \( A_3 \): Likes vehicle #3Given probabilities:- \( P(A_1) = 0.45 \)- \( P(A_2) = 0.55 \)- \( P(A_3) = 0.70 \)- \( P(A_1 \cup A_2) = 0.80 \)- \( P(A_2 \cap A_3) = 0.50 \)- \( P(A_1 \cup A_2 \cup A_3) = 0.88 \)#### (a) Probability of Liking Both Vehicle #1 and Vehicle #2What is the probability that the individual likes both vehicle #1 and vehicle #2?- Answer: \( 0.20 \)#### (b) Determine \( P(A_2 \mid A_3) \)Calculate the conditional probability \( P(A_2 \mid A_3) \):- Answer: \( 0.7143 \)**Interpret \( P(A_2 \mid A_3) \):**This probability indicates that if a person likes vehicle #3, there is a 71.43% chance that they will also like vehicle #2.#### (c) Independence of Events \( A_2 \) and \( A_3 \)Evaluate whether \( A_2 \) and \( A_3 \) are independent events:- No, \( P(A_2 \mid A_3) \neq P(A_2) \), indicating that \( A_2 \) and \( A_3 \) are not independent.#### (d) Probability Given Not Liking Vehicle #1If it is known that the individual did not like vehicle #1, calculate the probability that they liked at least one of the other two vehicles:- Correct calculation pending: Miscalculation shown as \( 0.56 \) indicates an error.This material provides a foundational overview of probability assessment in the context of preferences for multiple choices, focusing on event independence and conditional probability.

The provided information is as follows:The is the event that like vehicle #1.The is the event that…

Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A₁, A₂, and A3 by A₁ A₂ = likes vehicle #2 A3 = likes vehicle #3. = likes vehicle #1 Suppose that P(A₁) = 0.45, P(A₂) = 0.55, P(A3) = 0.70, P(A₁ U A₂) = 0.80, P(A₂ A3) = 0.50, and P(A₁ U A₂ U A3) = 0.88. (a) What is the probability that the individual likes both vehicle #1 and vehicle #2? 0.2 (b) Determine P(A₂ | A3). (Round your answer to four decimal places.) P(A₂ | A3) Interpret P(A₂ | A3). If a person likes vehicle #2, this is the probability he or she will also like vehicle #3. = 0.7143 This is the probability a person does not like both vehicle #2 and vehicle #3. If a person likes vehicle #3, this is the probability he or she will also like vehicle #2. This is the probability a person likes both vehicle #2 and vehicle #3. (c) Are A₂ and A3 independent events? Answer in two different ways. (Select all that apply.) Yes. P(A₂ | A3) # P(A₂). Therefore, A₂ and A3 are independent. No. P(A₂ | A3) # P(A₂). Therefore A₂ and A3 are not independent. Yes. P(A₂ A3) = P(A₂)P(A3). Therefore, A₂ and A3 are independent. No. P(A₂ | A3) = P(A₂). Therefore, A₂ and A3 are not independent. Yes. P(A₂ | A3) = P(A₂). Therefore, A₂ and A3 are independent. No. P(A₂ A3) # P(A₂)P(A3). Therefore, A₂ and A3 are not independent. (d) If you learn that the individual did not like vehicle #1, what now is the probability that he/she liked at least one of the other two vehicles? (Round your answer to four decimal places.) 0.56 X

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...