Question

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

Answer 1

To solve the problem, we need to find the probability that a randomly selected car is electric given the market shares and the percentages of electric cars produced by each manufacturer.

1. Understanding the Given Data:
Each manufacturer's contribution to the total market and the percentage of electric cars they produce is as follows:

Amber: 60% of total cars, of which 20% are electric.
Bonzi: 30% of total cars, of which 10% are electric.
Comet: 10% of total cars, of which 5% are electric.

2. Using the Law of Total Probability:
We use the law of total probability to calculate the overall probability that a car is electric:

$ P(E) = P(E|A)P(A) + P(E|B)P(B) + P(E|C)P(C) $
where:
- $P(E|A) = 0.20$, $P(A) = 0.60$
- $P(E|B) = 0.10$, $P(B) = 0.30$
- $P(E|C) = 0.05$, $P(C) = 0.10$

3. Calculating the Probability:
$ P(E) = (0.20)(0.60) + (0.10)(0.30) + (0.05)(0.10) $
$ P(E) = 0.12 + 0.03 + 0.005 = 0.155 $

Final Answer:
The probability that a randomly selected car is electric is $ \boxed{0.155} $ or $ \boxed{15.5\%} $.

Answer 2

To solve the problem, we need to find the probability that a randomly selected car is a petrol car, based on the production shares and the percentage of electric cars produced by each manufacturer.

1. Understanding the Complementary Event:
We know that each car can either be electric or petrol. So, the probability that a car is petrol is the complement of the probability that it is electric:

$ P(\text{Petrol}) = 1 - P(\text{Electric}) $

2. Using the Previously Calculated Probability:
From the previous solution, we know that:
$ P(\text{Electric}) = 0.155 $

3. Calculating the Probability:
$ P(\text{Petrol}) = 1 - 0.155 = 0.845 $

Final Answer:
The probability that a randomly selected car is a petrol car is $ \boxed{0.845} $ or $ \boxed{84.5\%} $.

Answer 3

To solve the problem, we need to find the conditional probability that a car was manufactured by Comet, given that it is electric.

1. Using Bayes’ Theorem:
We want to calculate $P(C | E)$, which is the probability that the car is from Comet given that it is electric. According to Bayes' theorem:

$ P(C | E) = \frac{P(E | C) \cdot P(C)}{P(E)} $

2. Substituting the Values:
From the data given:
- $P(E | C) = 0.05$ (5% of Comet’s cars are electric)
- $P(C) = 0.10$ (Comet makes 10% of all cars)
- $P(E) = 0.155$ (Probability that a car is electric, previously calculated)

3. Calculating the Conditional Probability:
$ P(C | E) = \frac{0.05 \times 0.10}{0.155} = \frac{0.005}{0.155} \approx 0.03226 $

Final Answer:
Given that a car is electric, the probability that it was manufactured by Comet is $ \boxed{0.0323} $ or $ \boxed{3.23\%} $.

Answer 4

To solve the problem, we need to find the conditional probability that a car was manufactured by Amber or Bonzi, given that it is electric.

1. Using the Law of Total Probability with Bayes’ Theorem:
We are required to calculate $P(A \cup B | E)$, the probability that the electric car is from either Amber or Bonzi.
This can be written as:

$ P(A \cup B | E) = P(A | E) + P(B | E) $

2. Apply Bayes’ Theorem Individually:
We already know that:
- $P(E) = 0.155$

Now compute individually:
- $P(A | E) = \frac{P(E | A) \cdot P(A)}{P(E)} = \frac{0.20 \cdot 0.60}{0.155} = \frac{0.12}{0.155} \approx 0.7742$
- $P(B | E) = \frac{P(E | B) \cdot P(B)}{P(E)} = \frac{0.10 \cdot 0.30}{0.155} = \frac{0.03}{0.155} \approx 0.1935$

3. Adding the Probabilities:
$ P(A \cup B | E) = 0.7742 + 0.1935 = 0.9677 $

Final Answer:
Given that a car is electric, the probability that it was manufactured by Amber or Bonzi is $ \boxed{0.9677} $ or $ \boxed{96.77\%} $.