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Now that we understand the addition rule and the multiplication rule, and we started to talk about

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conditional probability, we want to talk about Bayes Theorem, which is a really important concept

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in probability.

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And just walking through the theorem itself is a little confusing, but when we use an example, it's

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actually not too tough.

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So the theorem itself says the probability of event a happening given that event B has already occurred,

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is equal to the probability of B given A multiplied by the probability of a divided by the probability

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of B.

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And remember, with all these probability formulas, we can let event A and event B be anything we want

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to.

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We're just using A and B as placeholders.

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So as an example so that we can understand what this theorem is actually doing, let's say that we have

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two factories or assembly lines, so we have assembly line one and assembly line two.

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And let's say that 3% of the parts produced on line one are defective, while 1% of the parts produced

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on line two are defective.

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In other words, we've been studying these assembly lines over a long period of time.

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We've taken lots of samples and we already know that the defective rate for line one is 3%.

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The defective rate for line two is 1%.

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That's kind of historical information that we have.

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OC So we have defective rates for each of these lines.

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What Bayes Theorem allows us to do is answer a question like this one.

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So given that we got a defective part, what is the probability that it came from line one or we could

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say came from line two?

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In other words, we're getting defective parts from line one, we're getting defective parts from line

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two, but we're getting them at different rates.

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Now.

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We have a defective part in our hand, and we want to know the probability that it came from one line

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or the other.

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It's almost like calculating the probability of something in the past or going back in time.

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I have a defective part at this point.

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I don't know which assembly line it came from.

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I want to know the likelihood that it came from line one versus line two.

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What's the probability that the defective part came from one assembly line versus the other?

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So to use Bayes Theorem to answer a question like this one, the first thing we always need to do is

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identify event A and event B, right?

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Our Bayes Theorem formula is given to us in terms of events A and B, we need to now assign those things

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real names so that we don't get confused.

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So this is saying, given a defective part, what is the probability that it came from line one?

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And Bayes theorem allows us to calculate the probability of A given B, So if we match up that language

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to the question that we're trying to answer, the probability of event A given event B given that event

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B has already occurred.

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Well, in our question, we're saying given that we have a defective part, so we're going to say that

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event B.

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Is the event that we get a defective part.

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So given that we pull a defective part.

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What is the probability that it came from line one?

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What is the probability of a a must be coming from?

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Line one.

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So we're saying event A is the part coming from line one.

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Event B is the part being defective.

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Now we need to calculate these three probabilities on the right hand side of the theorem.

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So let's start with the probability of event A So the probability of event A is the probability of the

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part coming from line one.

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Well, let's just say that there's an equal likelihood that we choose apart from line one or line two.

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So we'll say that this probability is one half.

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Now, this could change, for instance, if line one produces 90% of all of the parts and line two produces

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only 10% of all of the parts, then the probability that the part comes from line one would be 90% or

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nine over ten.

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But if we say that 50% of all the parts come from line one and 50% come from line two, then there's

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an equivalent chance of choosing line one or line two, which means the probability of event a choosing

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line one is one out of two, one and two one half 50%.

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Now the probability of event B, given that A has already happened is simple as well.

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In fact, we've already been given it in the information in the question.

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This probability is the probability of a defective part event.

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B Given that the part comes from line one event.

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A Well, we already know that that probability is 3% or three over 100, because if the part is coming

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from line one, we know that line one has a 3% defective rate.

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So the probability of the part is defective, given that it comes from line one is three over 100.

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So these things are easy to calculate.

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The toughest probability to figure out is the probability of event B, the probability that the part

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is defective, because we have to consider both line one and line two.

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Let's first think about the probability that the part comes from line one and is also defective.

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Well, remember our multiplication rule.

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If we say the probability that the part comes from line one and is defective, or we could write this

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as line one.

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And.

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Defective is equal to.

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This is our multiplication rule.

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We have to multiply the probability that the part comes from line one, which we know is.

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One half by the probability that the part is defective, which we know is three over 100.

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So three over 100 or three over 200.

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Now the probability that the part comes from.

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Line two and is.

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Defective again is our multiplication rule for independent events.

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We get the probability that the part comes from line two multiplied by the probability that it's defective.

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For line two, that's one over 100.

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So we're going to get one over 200, which means then that the probability.

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Of a defective part.

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We get using the addition rule.

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It's the probability that the part came from line one and was defective or came from line two and was

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defective because either of those scenarios would result in a defective part.

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And so we have to then add these two probabilities together.

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So three over 200 plus one over 200 is for over 200 or.

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One over 50.

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So the probability that the part is defective is one over 50.

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Now, with all three of these values, we can actually calculate the probability we're looking for using

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Bayes Theorem.

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So let's actually write this all out.

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We'll say the probability that the part came from line one, given that it was.

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Defective is equal to the probability that it was defective, given that it came from line one three

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over 100 multiplied by the probability of event A, the probability that it came from line one.

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One half.

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All divided by probability of event B, the probability that it was defective, which in this case is

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one over 50.

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And when we do that math, we get in the numerator here three times one is three, 100 times two is

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200.

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So we get three over 200 divided by one over 50.

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When we divide by a fraction, that's the same as multiplying by the reciprocal.

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So we say three over 200.

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And then instead of dividing by one over 50, we multiply by 50 over one.

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We flip that fraction upside down and we get 150 over 200, which simplifies to three over four or.

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75%.

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So there's a three in four chance or a 75% chance that the part came from assembly line number one,

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given that the part was defective.

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So if we're holding a defective part in our hand and we don't know whether it came from line one or

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came from line two, we know that the probability tells us that there's a 75% chance, there's a three

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out of four chance that the defective part came from line one compared to line two.

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And of course, given what we know about complimentary probability, if there's a 75% chance, the defective

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part came from line one, that means by definition that there's a 25% chance that the defective part

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came from line two.

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Now there is another tool we can use to answer questions using Bayes Theorem, and it's this idea of

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a tree diagram.

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And this is something that helps a lot of people because this formula itself can be confusing.

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It takes some practice.

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Sometimes it helps to see a visual.

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So the idea here with this same question is that we have an equally likely chance of choosing line one

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or line two, because these assembly lines each produce 50% of all of our parts.

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So we have a 50% chance of choosing line one and a 50% chance of choosing line two.

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We could also write this as one half and one half.

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And so we build out this first decision in our tree diagram.

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And then from here off of assembly line one, we can get defective parts or we can get what we call

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compliant parts or non defective parts.

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Well, we know that line one produces defective parts at a 3% rate.

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So we could say that this is 3% or three in 100, which means compliant parts are produced at a 97%

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rate or 97 in 100.

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And then off of assembly line two, we can get defective parts.

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Line two produces defective parts at a 1% rate.

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So we can say 1% here or one in 100, which means that line two produces compliant parts at a 99% rate

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or 99 in 100.

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Now, going back to our same question, given that we have a defective part, what is the probability

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that it came from line one?

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Once we've built out this tree diagram, this decision tree, what we want to do is sort of trim off

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all the branches or ignore all of the paths in the tree that don't lead to the question we're trying

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to answer.

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In other words, we're looking at defective parts only, not compliant parts.

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So we can sort of ignore this whole part.

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This whole part will cancel out that part of the tree, cancel out this part of the tree.

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We're ignoring all of the compliant sides of our tree and just looking at the defective sides.

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So what we can say then is that the probability that we chose, apart from line one, given that the

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part was defective.

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So we'll say the probability of choosing a part from line one, it's going to be equal to.

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We look then at the number of branches that lead to defective from line one, this number right here.

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So three and we divide that by the total number of branches that lead to defective anywhere in our tree,

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which is this value three, but also this value of one.

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We have one branch leading to defective here, three branches leading to defective here.

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And so we say three plus one or three over four, which is that same three out of four number.

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We calculated earlier that same 75% figure as the chance that our part came from line one.

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Given that it was defective, we could obviously use this same tree diagram to see the probability that

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the part came from line two, given that it was defective.

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So we could say the probability that it came from line two is equal to we would take this value one,

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divide it by the total possibilities, three and one, so three and one, and we would get one over

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four or 25%, which of course we already know to be the case because 100% -75% gives us that 25%.

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These are complementary probabilities.

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So that's the fundamental idea behind Bayes Theorem, given that I'm holding a defective part in my

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hand and I know that assembly line number one produces defective parts, and assembly line number two

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produces defective parts.

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What's the probability what's the likelihood that the defective part I'm holding was built on assembly

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line number one or was built on assembly line number two?

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It's kind of a way for us to look back in time to figure out the likelihood that some event occurred

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in the past, given the fact that we know that the part we have is defective.