Specialty Manufacturing gets 29% of its O-rings from Little Rock Plastics and the rest of its O-rings from Galshus and Sons. Historically 4% of the O-rings it gets from Little Rock Plastics are defective and 10% of the O-rings it gets from Galshus and Sons are defective. An O-ring is found to be defective, what is the probability the O-ring came from Galshus and Sons?

The probability that the O-ring came from Galshus and Sons given that it is defective is 0.359.

Step-by-step explanation:

Probability of getting O-ring from Little Rock Plastics = 0.29

Probability of getting O-ring from Galshus and Sons = 0.71

Probability of getting Defective Rings from Little Rock Plastics = 0.04

Probability of getting Defective Rings from Galshus and Sons = 0.10

Denoting Little Rock Plastics as LRP, Galshus and Sons as GS and Defective as D, we can write:

P (LRP) = 0.29

P (GS) = 0.71

P (D ∩ LRP) = 0.04

P (D ∩ GS) = 0.10

We are given that an O-ring is found to be defective and we need to find the probability that it came from Galshus and Sons so we will use the conditional probability formula for calculating the probability that the O-ring came from Galshus and Sons given that it is defective.

P (GS|D) = P (D ∩ GS) / P (D)

We need to compute P (D) first. So,

P (D) = P (D|GS) + P (D|LRP)

= P (D∩GS) / P (GS) + P (D∩LRP) / P (LRP)

= 0.10 / 0.71 + 0.04/0.29

= 0.1408 + 0.1379

P (D) = 0.2787

P (GS|D) = P (D ∩ GS) / P (D)

= 0.10/0.2787

= 0.3587

P (GS|D) = 0.359