Answer:
(a) The probability that the individual likes both vehicle #1 and vehicle #2 is 0.40.
(b) The value of P (A₂ | A₃) is 0.7143.
(c) The events A₂ and A₃ are not independent.
(d) The probability that an individual likes at least one of A₂ and A₃ given they did not like A₁ is 0.7333.
Step-by-step explanation:
The events are defined as follows:
<em>A</em>₁ = an individual like vehicle #1
<em>A</em>₂ = an individual like vehicle #2
<em>A</em>₃ = an individual like vehicle #3
The information provided is:

(a)
Compute the probability that the individual likes both vehicle #1 and vehicle #2 as follows:

Thus, the probability that the individual likes both vehicle #1 and vehicle #2 is 0.40.
(b)
Compute the value of P (A₂ | A₃) as follows:

Thus, the value of P (A₂ | A₃) is 0.7143.
(c)
If two events <em>X</em> and <em>Y</em> are independent then,

The value of P (A₂ ∩ A₃) is 0.50.
The product of the probabilities, P (A₂) and P (A₃) is:

Thus, P (A₂ ∩ A₃) ≠ P (A₂) × P (A₃)
The value of P (A₂ | A₃) is 0.7143.
The value of P (A₂) is 0.65.
Thus, P (A₂ | A₃) ≠ P (A₂).
The events A₂ and A₃ are not independent.
(d)
Compute that probability that an individual likes at least one of A₂ and A₃ given they did not like A₁ as follows:

Thus, the probability that an individual likes at least one of A₂ and A₃ given they did not like A₁ is 0.7333.