Question

What is the probability that a randomly selected car is electric?

Answer 1

Probability of an Electric Car

Using the Total Probability Theorem, we define:

• A: The event that the car is manufactured by Amber.
• B: The event that the car is manufactured by Bonzi.
• C: The event that the car is manufactured by Comet.
• E: The event that the car is electric.

Applying the law of total probability:

\[ P(E) = P(E|A) P(A) + P(E|B) P(B) + P(E|C) P(C) \]

Substituting values:

\[ P(E) = (0.20 \times 0.60) + (0.10 \times 0.30) + (0.05 \times 0.10) \] \[ P(E) = 0.12 + 0.03 + 0.005 = 0.155 \]

Final Answer:

P(Electric Car) = 0.155

Answer 2

Probability of a Petrol Car

Since a car is either electric or petrol, we use:

\[ P(\text{Petrol Car}) = 1 - P(E) \]

Substituting the value of \( P(E) \):

\[ P(\text{Petrol Car}) = 1 - 0.155 = 0.845 \]

Final Answer:

P(Petrol Car) = 0.845

Answer 3

Bayes' Theorem: Probability of Comet Manufacturing an Electric Car

Using Bayes' Theorem, we compute:

\[ P(C|E) = \frac{P(E|C) P(C)}{P(E)} \]

Substituting values:

\[ P(C|E) = \frac{(0.05 \times 0.10)}{0.155} \] \[ P(C|E) = \frac{0.005}{0.155} \approx 0.03226 \]

Final Answer:

P(Comet — Electric Car) ≈ 0.032

Answer 4

We need to compute: \[ P(A \cup B | E) = P(A | E) + P(B | E). \] Using Bayes’ Theorem: \[ P(A|E) = \frac{P(E|A) P(A)}{P(E)}. \] \[ = \frac{(0.20 \times 0.60)}{0.155} = \frac{0.12}{0.155} \approx 0.774. \] Similarly, \[ P(B|E) = \frac{P(E|B) P(B)}{P(E)}. \] \[ = \frac{(0.10 \times 0.30)}{0.155} = \frac{0.03}{0.155} \approx 0.193. \] Step 3: Compute final probability. \[ P(A \cup B | E) = 0.774 + 0.193 = 0.967. \] Final Answer: \[ P(\text{Amber or Bonzi | Electric Car}) \approx 0.967. \]