3. A service station owner sells Firm-on-the-road tyres which are ordered from a local tyre distributor. The distributor receives tyres from two plants, A and B. When the owner of the service station receives an order from the distributor, there is a 0.5 probability that the order consists of tyres from plant A or plant B. However, the distributor will not tell the owner which plant the tyres come from. The owner knows that 20% of all tyres produced at plant A are defective, whereas only 10% of the tyres produced at plant B are defective. When an order arrives at the station, the owner is allowed to inspect it briefly. The owner takes this opportunity to inspect one tyre to see if it is defective. If the owner believes the tyre came from plant A, the order will be sent back. Using Baye’s rule, determine the posterior probability that a tyre is from plant A, given that the owner finds that it is defective.

To determine the posterior probability that a tyre is from plant A, given that the owner finds it defective, we can use Bayes' Rule.

Let's define the following events:
A: The tyre is from plant A.
B: The tyre is from plant B.
D: The tyre is defective.

We are given the following probabilities:
P(A) = 0.5 (probability that the order consists of tyres from plant A or plant B)
P(B) = 0.5 (probability that the order consists of tyres from plant A or plant B)
P(D|A) = 0.2 (probability that a tyre is defective given that it is from plant A)
P(D|B) = 0.1 (probability that a tyre is defective given that it is from plant B)

We need to calculate P(A|D), the posterior probability that a tyre is from plant A given that it is defective, using Bayes' Rule:

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

Substituting the given values:
P(A|D) = (0.2 * 0.5) / (0.2 * 0.5 + 0.1 * 0.5)

Simplifying:
P(A|D) = 0.1 / (0.1 + 0.05)

Calculating:
P(A|D) = 0.1 / 0.15

Simplifying further:
P(A|D) = 2/3

Therefore, the posterior probability that a tyre is from plant A, given that the owner finds it defective, is 2/3 or approximately 0.67.

To find the posterior probability that a tire is from plant A, given that the owner finds it defective, we can use Bayes' rule.

Let's define the events:
A: Tire is from plant A
B: Tire is from plant B
D: Tire is defective

Now, we can use Bayes' rule:

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

P(D|A) is the probability that the tire is defective given that it is from plant A. According to the information provided, 20% of the tires produced at plant A are defective, so P(D|A) = 0.20.

P(A) is the prior probability that a tire is from plant A, without considering any additional information. The distributor receives tires from two plants, A and B, so the probability that an order consists of tires from plant A is 0.5. Therefore, P(A) = 0.5.

P(D) is the probability that a tire is defective, regardless of plant origin. This can be calculated by considering the probabilities of the tire being defective from both plants:

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

P(D|B) is the probability that the tire is defective given that it is from plant B. According to the information provided, 10% of the tires produced at plant B are defective, so P(D|B) = 0.10.

P(B) is the probability that an order consists of tires from plant B. Since there are only two plants (A and B) and the probability that an order consists of tires from plant A is 0.5, P(B) = 0.5.

Therefore, we can calculate P(D):

P(D) = P(D|A) * P(A) + P(D|B) * P(B)
= 0.20 * 0.5 + 0.10 * 0.5
= 0.10 + 0.05
= 0.15

Now, we can substitute the values back into Bayes' rule to find P(A|D):

P(A|D) = P(D|A) * P(A) / P(D)
= 0.20 * 0.5 / 0.15
= 0.10 / 0.15
= 0.67

Therefore, the posterior probability that a tire is from plant A, given that the owner finds it defective, is 0.67.