A corporation regularly takes deliveries of a particular...

A corporation regularly takes deliveries of a particular sensitive part from three subcontractors. It found that the proportion of parts that are good or defective from the total received were as shown in the following table: $$ \begin{array}{lccc} \hline &{}{}{\text { Subcontractor }} \\ \text { Part } & \text { A } & \text { B } & \text { C } \\ \hline \text { Good } & 0.27 & 0.30 & 0.33 \\ \text { Defective } & 0.02 & 0.05 & 0.03 \\ \hline \end{array} $$ a. If a part is chosen randomly from all those received, what is the probability that it is defective? b. If a part is chosen randomly from all those received, what is the probability it is from subcontractor $\mathrm{B}$ ? c. What is the probability that a part from subcontractor $B$ is defective? d. What is the probability that a randomly chosen defective part is from subcontractor $\mathrm{B}$ ? e. Is the quality of a part independent of the source of supply? f. In terms of quality, which of the three subcontractors is most reliable?

A corporation regularly takes deliveries of a particular sensitive part from three subcontractors. It found that the proportion of parts that are good or defective from the total received were as shown in the following table:
$$
\begin{array}{lccc}
\hline &{}{}{\text { Subcontractor }} \\
\text { Part } & \text { A } & \text { B } & \text { C } \\
\hline \text { Good } & 0.27 & 0.30 & 0.33 \\
\text { Defective } & 0.02 & 0.05 & 0.03 \\
\hline
\end{array}
$$
a. If a part is chosen randomly from all those received, what is the probability that it is defective?
b. If a part is chosen randomly from all those received, what is the probability it is from subcontractor $\mathrm{B}$ ?
c. What is the probability that a part from subcontractor $B$ is defective?
d. What is the probability that a randomly chosen defective part is from subcontractor $\mathrm{B}$ ?
e. Is the quality of a part independent of the source of supply?
f. In terms of quality, which of the three subcontractors is most reliable?

Key Concepts

Total Probability Rule

This concept involves calculating the overall probability of an event by summing the probabilities of that event occurring under several mutually exclusive scenarios, each weighted by the probability of each scenario. In problems involving multiple sources or subgroups, the total probability rule helps combine the contribution of each subgroup to the overall event likelihood.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another related event has already occurred. It is a crucial concept when the probability of one outcome depends on a condition or a particular subgroup, allowing for more precise calculations in context-specific scenarios.

Bayes' Theorem

Bayes' Theorem provides a way to update our probability estimates based on new information. It relates the conditional probability of an event given a condition with the reverse probability, meaning it allows us to find the probability of a source or cause given an observed outcome, which is particularly useful in diagnostic or reverse inferencing problems.

Independence of Events

Independence of events refers to situations where the occurrence of one event does not influence the probability of another event. Establishing whether events are independent is crucial to simplify probability computations and to correctly model the relationship between different factors in a process.

Reliability Analysis

Reliability analysis focuses on evaluating the quality and performance of different components or sources by assessing their failure rates or defect frequencies. In a broader context, it involves comparing probabilities of successful outcomes to determine which source or system is likely to perform best in terms of quality and consistency.

Transcript

00:01 So we have a chart given to us, and we have three suppliers, a, b, and c, and we have good parts and defective parts.

00:09 And we know what the distribution is.

00:11 We have 0 .27 .02.

00:13 These do add to 0 .29.

00:15 Not that we necessarily need that, but a helps to have the row totals and column totals.

00:21 And this is 0 .33 and 0 .03, and this is 0 .36.

00:27 And let's see if we add this way.

00:29 This is going to end up being 10%.

00:32 So this should add up to 90.

00:34 These two add up to 60 plus 30, 90, and then that adds down to 1, and that adds across to 1.

00:40 So we're good to go.

00:42 And our first question is, if you randomly pick a part, what's the probability that it's defective? and that is going to be this row total.

00:49 It's going to be that 10%.

00:52 Then we want to know what's the probability that the part, if it's randomly chosen, comes from b.

00:57 And that is 0 .35.

01:03 Then in part c, it says, given that you know, it doesn't say it exactly in those wording, but knowing that the part came from b, what's the likelihood that it's defective? and so we only care about column b, and we know the likelihood of it coming from column b or from supplier b is 0 .35, and what fraction of those are defective, that is 0 .05.

01:29 So that is 5 out of 35, which is a one -seventh chance.

01:34 Now, letter d asks, what's the likelihood that it came from b if you knew that it had been defective? so now we know it's defective, which is only this row.

01:48 And so we know we only are worried about those 10%.

01:51 And what fraction of those would come from b? well, that looks like we're going to add half of them.

01:57 So if you know it's defective, it's most likely coming from b.

02:03 So that would be one -half a chance.

02:06 And then e asks us, do we have independence between the source and being defective or being good? and the, we'll just say, quality.

02:23 So do we have independence? and a couple things.

02:28 First of all, this answer right here, and this answer would be the same if we had independence.

02:36 And likewise, these two answers would be the same...