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?

Answer:

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

Final answer:

Using Bayes' theorem, the probability that a defective O-ring came from Galshus and Sons is approximately 0.802 or 80.2%

Explanation:

To find the answer to your question, we need to use Bayes' theorem. This theorem refers to the probability of an event, based on prior knowledge of conditions that might be related to the event. First, let us identify the following:
Probability of choosing an O-ring from Little Rock Plastics (L), P(L) = 0.29
Probability of choosing an O-ring from Galshus and Sons (G), P(G) = 1 - P(L) = 0.71
Probability that an O-ring from Little Rock is defective, P(D|L) = 0.04
Probability that an O-ring from Galshus and Sons is defective, P(D|G) = 0.10

By Bayes' theorem, the probability that a defective O-ring came from Galshus and Sons is given by: P(G|D) = [P(G) * P(D|G)] / [P(L) * P(D|L) + P(G) * P(D|G)]

Upon substitution, P(G|D) = [0.71 * 0.10] / [0.29 * 0.04 + 0.71 * 0.10]. This equates to approximately 0.802, or 80.2%, meaning there is a 80.2% chance that the defective O-ring came from Galshus and Sons.

Learn more about Bayes' theorem here:

brainly.com/question/34293532

#SPJ3