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.

Question

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.