The Watts New Lightbulb Corporation ships large consignments...

The Watts New Lightbulb Corporation ships large consignments of lightbulbs to big industrial users. When the production process is functioning correctly, which is $90 \%$ of the time, $10 \%$ of all bulbs produced are defective. However, the process is susceptible to an occasional malfunction, leading to a defective rate of $50 \%$. If a defective bulb is found, what is the probability that the process is functioning correctly? If a nondefective bulb is found, what is the probability that the process is operating correctly?

Key Concepts

Conditional Probability

Conditional probability measures the probability of an event occurring, given that another event has already occurred. It forms the basis for more advanced probability techniques, helping to analyze how new evidence can affect the likelihood of various outcomes in a system.

Law of Total Probability

The Law of Total Probability is used to compute the overall probability of an event by summing the probabilities of that event across all possible mutually exclusive scenarios, each weighted by their respective probabilities. This concept is crucial when dealing with composite systems or processes with multiple states.

Bayes' Theorem

Bayes' Theorem is an essential principle in probability theory that provides a way to update the probability estimates for a hypothesis as more evidence or information becomes available. It relates the conditional probability of an event given new evidence to the likelihood of obtaining that evidence under different hypotheses.

Transcript

00:01 So we know that this process is functioning correctly 90 % of the time, and so 10 % of the time it is malfunctioning.

00:14 And we know that for the defect rate versus not having defects, that this occurs 10 % of the time and 90 % of the time here.

00:26 And when it is malfunctioning, it actually still can produce something that is good.

00:32 But it has a defect rate of 50 % though, way higher, and a 50 % rate that it will not be defective.

00:39 And we're asked to find a couple of questions.

00:43 What's the probability that given that they choose a part and it is defective, what's the likelihood that we really have this is functioning correctly? and then we want to find what is the probability that if they get a non -defective part when they check it, that it actually was functioning correctly because we know regardless of whether the machine is functioning correctly or malfunctioning can produce good or bad parts.

01:14 So we need to find the probability of being defective and the defective is 0 .9 times 0 .1 plus 0 .1 times 0 .5 and then we want to find the probability of what part of that comes from the functioning correctly and that's that point nine over point one and so we have the point nine times point one divided by and then we have that point nine times point one plus point one times point five in the denominator and this comes out to be i'll put it over here point six four two and we'll round it to nine so about a 64 % chance of that happening.

02:07 Now, this denominator, by the way, the likelihood of it, not of it being defective is going to be the complement of this...

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