Three plants manufacture hard drives and ship them to a...

Three plants manufacture hard drives and ship them to a warehouse for distribution. Plant I produces 54$\%$ of the warehouse's inventory with a 4$\%$ defect rate. Plant II produces 35$\%$ of the warehouse's inventory with an 8$\%$ defect rate. Plant III produces the remainder of the warehouse's inventory with a 12$\%$ defect rate. (a) Draw a tree diagram to represent this information. (b) A warehouse inspector selects one hard drive at random. What is the probability that it is a defective hard drive and from Plant II? (c) What is the probability that a randomly selected hard drive is defective? (d) Suppose a hard drive is defective. What is the probability that it came from Plant II?

Three plants manufacture hard drives and ship them to a warehouse for distribution. Plant I produces 54$\%$ of the warehouse's inventory with a 4$\%$ defect rate. Plant II produces 35$\%$ of the warehouse's inventory with an 8$\%$ defect rate. Plant III produces the remainder of the warehouse's inventory with a 12$\%$ defect rate.
(a) Draw a tree diagram to represent this information.
(b) A warehouse inspector selects one hard drive at random. What is the probability that it is a defective hard drive and from Plant II?
(c) What is the probability that a randomly selected hard drive is defective?
(d) Suppose a hard drive is defective. What is the probability that it came from Plant II?

Key Concepts

Bayes' Theorem

Bayes' Theorem provides a way to update the probability of a hypothesis in light of new evidence. It reverses conditional probabilities, allowing one to determine the probability that a specific cause was responsible for an observed effect, such as calculating the likelihood that a defective product originated from a particular plant.

Law of Total Probability

The law of total probability is a method used to compute the probability of an event by considering all possible ways that event can occur, partitioned over mutually exclusive cases. In multi-source scenarios, this law aggregates individual probabilities to arrive at an overall outcome, which is essential for combining data from different production plants in probability calculations.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is crucial when dealing with scenarios where one event influences the probability of another, such as determining probabilities based on production sources or defect status.

Tree Diagram

A tree diagram is a visual representation used to map out events and associated probabilities in a sequential or branching process. It helps break down complex probability problems by clearly showing each decision point and the corresponding outcomes, making it easier to apply techniques like the law of total probability or conditional probability.

Transcript

00:01 So for this problem, we have three plants manufacturing hard drives and shipping them to a warehouse for distribution.

00:10 Plant 1 produces 54 % of inventory, i for inventory, and has 4 % defect rate.

00:26 Plant 2, not pant, plant 2, or produces 35 % of the inventory.

00:39 And has an 8 % defect rate.

00:46 And plant 3 produces the remainder.

00:52 So that's 100 % minus 54 % minus 35 % is equal to 11 % of inventory.

01:06 And it has a 12 % defect rate.

01:12 So the first thing that we want is we want to draw a tree.

01:16 Diagram that represents this information so we start off going to make sure i give myself a lot of space here okay so we start off then we could go plant one i'm going to call that event a one and that has a probability of 0 .54 then once we have determined that we were from plant one then we have the possibility that it is defective.

02:05 I'll say that defective is event b.

02:09 So we have probability of defective given it came from plant 1.

02:15 So that is defective.

02:19 That is going to be 4 % or 0 .4.

02:26 I'm going to zoom in a little bit and i'm going to make my pen a little bit smaller here.

02:32 So that is 0 .4.

02:38 Then we have not defective.

02:42 I'm going to say not defective is event b prime, and i'm just going to say good, puts us there.

02:54 Alternatively, it could come from plant 1, so we have p -a -2, which is equal to the a -2 was, where is it, 35%, or 0 .35, that is plant 2.

03:14 And similarly, we have probability of b given a2, which is equal to...

03:29 Oh, i just realized i made a mistake that was 0 .4 there.

03:33 That should be 0 .04.

03:36 And that probability of not defective up there should be 0 .96.

03:42 And we have probability of defective given it came from plant 2 is 0 .08, which means then...

03:52 That the probability that it's not defective given it came from plant two equal to 0 .92.

04:06 Then we have probability came from plant three is equal to 0 .11.

04:26 And the probability that that is defective, the probability of b given a3 is equal to 0 .12.

04:37 And the probability it is not defective, given it came from plant 3, is equal to 0 .88.

04:59 So that catalogues all the information that we were given...

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