KUNTAL GHOSH

KUNTAL GHOSH

Answered question

2022-05-08

Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A_1, A_2, and A_3 by A_1=likes vehicle #1, A_2= likes vehicle #2, A_3=likes vehicle #3.

Suppose that

a. What is the probability that the individual likes both vehicle #1 and vehicle #2? Determine and interpret.

b. What is the probability that the individual likes either vehicle #2 or vehicle #3? Determine and interpret.

c. Are A_1 and A__2 independent events? Answer in two different ways.

d. If you learn that the individual did not like vehicle #2, what now is the probability that he/she liked at least one of the other two vehicles?

Answer & Explanation

Jazz Frenia

Jazz Frenia

Skilled2023-05-05Added 106 answers

We are given the following events:
A1: The individual likes vehicle #1.
A2: The individual likes vehicle #2.
A3: The individual likes vehicle #3.
We are asked to find the probability that the individual likes both vehicle #1 and vehicle #2, which can be expressed as the intersection of events A1 and A2: P(A1A2).
Since we are assuming that the individual is randomly selected, each event Ai has the same probability of occurring, which we denote by P(Ai). Thus, P(A1)=P(A2)=P(A3)=13.
To find P(A1A2), we need to determine the probability that the individual likes both vehicle #1 and vehicle #2. Since the individual is testing three vehicles, there are three possible outcomes for the test drive:
1. The individual likes vehicle #1 only. This corresponds to the event A1A2cA3c, where A2c and A3c denote the complements of events A2 and A3, respectively.
2. The individual likes vehicle #2 only. This corresponds to the event A1cA2A3c.
3. The individual likes both vehicle #1 and vehicle #2. This corresponds to the event A1A2A3c.
Note that the event A1cA2cA3c corresponds to the individual not liking any of the vehicles, which we assume has probability zero since the individual must like at least one of the vehicles.
Therefore, we have:
P(A1A2)=P(A1A2A3c)=13×13×23=227
Interpretation: The probability that the individual likes both vehicle #1 and vehicle #2 is relatively low at 227, indicating that the individual may have different preferences or priorities when it comes to choosing a vehicle.

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