In a major city 7 % of cars have an alarm system installed. A car without an alarm has a probability of 0.01 of being stolen, this is triple the probability of a car with an alarm being stolen. What is the probability that a car stolen from this city has an alarm system installed?

.01/3 = ?

0.0033

Let's solve this step-by-step:

Step 1: Let's assume there are 1000 cars in total in the city.

Step 2: According to the given information, 7% of cars have an alarm system installed. So, the number of cars with an alarm system = 7% of 1000 = 0.07 * 1000 = 70 cars.

Step 3: The remaining cars without an alarm system = 1000 - 70 = 930 cars.

Step 4: The probability of a car without an alarm system being stolen is given as 0.01, which is three times the probability of a car with an alarm system being stolen. Let's denote the probability of theft for a car with an alarm system as P(A), and the probability of theft for a car without an alarm system as P(B).

So, P(B) = 0.01 and P(A) = P(B)/3 = 0.01 / 3 = 0.00333.

Step 5: Let's assume that X represents the number of cars stolen. We want to find the probability that a stolen car has an alarm system installed. Let's denote this as P(A|X), which can be calculated using Bayes' theorem.

Bayes' theorem states: P(A|X) = (P(X|A) * P(A)) / P(X)

where P(X|A) represents the probability of a car with an alarm system being stolen and P(X) represents the overall probability of a car being stolen.

Step 6: We know that P(X|A) (the probability of a car with an alarm system being stolen) is P(A) (0.00333 ).

Step 7: The overall probability of a car being stolen, P(X), can be calculated as follows:

P(X) = P(X|A) * P(A) + P(X|B) * P(B)

where P(X|B) represents the probability of a car without an alarm system being stolen.

Step 8: We know that P(X|B) = 0.01 and P(B) = 930 / 1000.

Substituting the values into the equation:

P(X) = (0.00333 * 0.07) + (0.01 * (930 / 1000))

Step 9: Calculate the value of P(X) using the equation from Step 8.

P(X) = 0.0002331 + 0.0093 = 0.0095331

Step 10: Finally, substitute the values of P(X|A) (0.00333), P(A) (0.07), and P(X) (0.0095331) into the equation from Bayes' theorem in Step 5 to calculate the probability of a stolen car having an alarm system installed.

P(A|X) = (0.00333 * 0.07) / 0.0095331

Step 11: Calculate the value of P(A|X) using the equation from Step 10.

P(A|X) ≈ 0.024 (approximately)

Therefore, the probability that a stolen car from this city has an alarm system installed is approximately 0.024, or 2.4%.

To find the probability that a stolen car has an alarm system installed, we can use Bayes' theorem.

Let's define the events:
A: Car has an alarm system installed
B: Car is stolen

We are given the following probabilities:
P(A) = 0.07 (7% of cars have an alarm system)
P(A') = 1 - P(A) = 0.93 (probability that a car does not have an alarm system)
P(B|A') = 0.01 (probability of a car being stolen given that it does not have an alarm system)
P(B|A) = 3 * P(B|A') = 0.03 (triple the probability of a car being stolen given that it has an alarm system)

Now, we can use Bayes' theorem to calculate P(A|B), the probability of a car having an alarm system given that it is stolen:

P(A|B) = (P(B|A) * P(A)) / P(B)

To find P(B), the probability of a car being stolen, we can use the law of total probability:

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Let's substitute the values:

P(B) = (0.03 * 0.07) + (0.01 * 0.93) ≈ 0.0109

Now we can find P(A|B):

P(A|B) = (0.03 * 0.07) / 0.0109 ≈ 0.1927

Therefore, the probability that a car stolen from this city has an alarm system installed is approximately 0.1927, or 19.27%.