Probability Problem Homework: Find P(Defective|Test Positive)

[teme92]

Homework Statement

Suppose that upon using a particular testing device, a defective component has a 70% chance of being found as defective, and a non-defective component has just a 10% chance of being found defective. Suppose also that within a quality control batch, the probability of a randomly selected component being defective is 12%.

i)What is the probability that a component selected at random from the batch is found to test positive (i.e. tests as being defective)?

ii)What is the probability that a randomly selected component is defective, given it is found to test positive?

Homework Equations

The Attempt at a Solution

i)I said:

(0.12)(0.7) +(0.88)(.1)=0.172

ii)'m having trouble wrapping my head around this one. Did I use the correct method for the first part? Also any help in the second part would be much appreciated.

 

[Curious3141]

Homework Statement

Suppose that upon using a particular testing device, a defective component has a 70% chance of being found as defective, and a non-defective component has just a 10% chance of being found defective. Suppose also that within a quality control batch, the probability of a randomly selected component being defective is 12%.

i)What is the probability that a component selected at random from the batch is found to test positive (i.e. tests as being defective)?

ii)What is the probability that a randomly selected component is defective, given it is found to test positive?

Homework Equations

The Attempt at a Solution

i)I said:

(0.12)(0.7) +(0.88)(.1)=0.172

ii)'m having trouble wrapping my head around this one. Did I use the correct method for the first part? Also any help in the second part would be much appreciated.

This problem is a simple application of Bayes' Theorem. Are you familiar with it?

The answer to the first part is correct, by the way, although it's better to express the probability as a percentage since you're given percentages to begin with.

Before applying Bayes' Theorem, start defining your events and your conditional probabilities, e.g. $p(D^+)$ is probability of a random article being defective and $p(T^+|D^+)$ is probability of test positive IF article is defective. Surely, you've seen this sort of notation?