WEBVTT

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Silence.

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Silence.

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Silence.

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Silence.

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Okay.

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Me oh, okay. Good afternoon class.

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And this is profitability class 6.

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11021 just a quick check.

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And you hear me.

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Professor great, thank you. And I've got a chat window open, so if there's questions.

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The reason I'm asking, if you can hear me is, I'm not I.

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I can't get audio feedback in any reasonable way without causing some problem. Like, it's delayed by a 2nd, which means, of course, I can't listen to myself or something.

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Key just new tools I like.

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You know, I like new tools, so current tool I'm playing with in this class is I am mirroring my iPad to a window on my laptop. So, the theory is that.

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I can draw notes on that and make it big enough that.

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They actually work said with any luck, it won't crash more than 2 or 3 times.

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During the class, so.

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You can just see.

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See, what is happening here so 6.

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Okay, cool. And then the theory is that I can upload PDFs and this is easier. I had a laptop with a.

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Touch screen before, but that didn't work very well. The iPad works nicely.

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And if you're curious, that's the software I'm using. 2nd.

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This is not the course it's applied electrical engineering.

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You know, I like playing with stuff outside the course. So my current.

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Expensive toy is my house has tests as Tesla power walls in the garage, and it's 2007 kilowatt hour of battery.

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And solar panels on the roof 8 kilowatts peak.

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And it's finally Tuesday got working. I only started the process in August. So, August of February took a bit of time.

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But it's actually working and earlier today, when it was frightened, Sonny was generating at a point 2, kilowatts of power. So, and this is still the winter with snow on the things.

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So, cool stuff. Okay, so we're continuing on in Leon Garcia, chapter 2 and big topic conditional probability.

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So, it's the probability that informal examples last time.

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Now, today, I'm going to give a formal examples and actually throw numbers at you. 1st, some more examples and a precise definition. In point you hear what conditional probability actually means.

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Here's a real world example.

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Darker the defense advance research projects agency.

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Some years ago proposed a futures market applied to real world events.

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So you could, for example, make book on the probability of somebody being assassinated.

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Next year, and the idea darkness stated idea was that they would look at the real world odds.

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And if there was some fairly.

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You know, there were a lot of bets on the president of Iraq being assassinated next year, then they would assume that might actually perhaps start happening and they would learn things. So, the wisdom of the crowds Ha.

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So, there was a bit of a political push back on this.

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And they actually never did do it, but.

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In any case, so this would be probability. So if you looked at somebody.

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Things on the 1st, real king.

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You know, if you'd already been 55 assassination attempts on him, and he survived them what's the probability of him?

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Surviving a 56th attempt.

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Okay, in any case I want to, Here's a real example.

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Okay.

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These are very rough numbers about students that are Pi. Okay.

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So we have any equals 1 plus 3 plus 2 plus 1 that would be 7000.

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Students so a number of engineers.

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And before 1000 number of undergrads.

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Would be, um, let's see, now, I'm reading my own diagram.

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Wouldn't be, um, let's say 5000 or something.

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Now, we can start talking about things such as the probability of being an engineering student.

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That's going to be 4 sevens. You can do stuff like that. Okay. We could talk about the probability of being an undergrad.

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If I said, there's 5000 undergrads and 7000 total.

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5, sevens, we could talk about the probability of being an engineering student and an undergrad.

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That's there's 3000 of them.

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Would be 3 sevens and can show just a 2nd here.

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Okay, so now we can talk about well, if this student is.

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An engineering student, what's the probability of him being an undergrad him or her?

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Sorry.

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Engineering, so given that an undergrad, what this means in words is that.

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Say student.

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Is an undergrad.

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What's the probability as as an engineer.

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Okay, well we can look at the diagram.

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And we can look and it's there's 5000 undergrads 3000 of them are engineering students. That's 3 fifths.

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So, I'm looking at the diagram it's 3 fifths.

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Professor yes wouldn't statement the engineering bar. Undergrad wouldn't that.

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A student that given the student as an engineer.

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What's the priority there? Undergrad just a 2nd. I had my volume turned down a little more. Can you ask the question again? Please where you written? Key energy bar.

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And then you bring up the definition, wouldn't it be if we know that the student is an engineering student.

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What's the problem here and undergrad? Because there's an engineering then undergrad.

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No, the vertical bar means given.

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That so okay, um.

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Hello.

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Are.

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Given an undergrad, so.

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And the probability.

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And the end we're using.

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And from the definition, so the probability.

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Well, there's a couple of things I have it point 4 down here at the bottom of the page.

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And engineering, well, engineer on undergrad were writing that as.

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An undergrad.

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So that's going to be, um.

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Probability engineering and.

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Say give an undergrad time times the probability.

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So so the problem to say engineering.

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Given, he's an undergrad is going to be the probability of being both.

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And engineering an undergrad I had was 3, 7, and the probability undergrad 5, 7.

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You close this, and that looks reasonable. If you look at the diagram here, there's 5000 undergrads 3000 of them.

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Our engineers, so the probability. So, if you know, somebody is an undergrad is 3 fifths the chance the person is an engineer.

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But there is this combo formula that I've got down.

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Like that and so on.

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And we can do all sorts of things we could go the other way to probability. As soon as an undergrad given that he's an engineer.

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That's the probability that they're both, which is.

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Divided by the probability that.

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He's an engineer, which is 4 sevens.

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Which makes sense, there's 4000 engineers. 3000 of them are undergrad. So the probabilities an undergrad given he's an engineer is.

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Is 3 quarters and you can do.

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This this expression this line for here is going to be used.

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It's going to be used a lot so.

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You know, we could go both ways, um.

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We also talk about say, the probability of an undergrad given not an engineer may do that or something.

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Well, how how to find that.

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Probably not being an engineer well, probability of being an engineer is for 7th. So probably, if not being an engineer is going to be 3 7.

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So, the probability of being an undergrad, not being an engineer is a probability.

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Of both, which is 3 sevens divided by.

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What did I do wrong here? Sorry I need to have the probability of undergrad and not an engineer.

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There you go to the, we're going to go see either.

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Well, undergrads, who are not engineers there's 2 fifths so.

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So, up here is going to be like.

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Because.

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3 7 C calls.

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Just a minute, wouldn't it be 5 7 over.

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You probably.

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Silence.

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So probably of undergrad and not engineer.

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Which would be.

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To fifths divided by the probability of.

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Not being an engineer.

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Okay.

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Yes.

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Can you explain to us and not 1 sentence.

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Good question. Let's see. Undergrad and not engineer.

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We've 7000 students and.

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5000 are the white thing I grabbed since 5000 of them are undergrad.

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Not engineer.

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Is to FIS.

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And probability.

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Okay, I got myself all or is it is it to.

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Is it 2, 7, 2, 7 over too fast so.

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Yeah, if you're not an engineer or is that.

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Probably would be an undergrad if you're not an engineer.

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Be 2 thirds.

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Okay, everyone think about that. This is a good chance to think. Okay, so.

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And I may come back.

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Everyone we'll come back to that in a minute. A K.

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Let's move on.

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Okay, the binary communication is a big example.

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So, we're transmitting.

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A 0T or a 1 and we're transmitting the 1 was probability. P and the 0T was probability 1 minus P.

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And they answer probably comes back correct. Gets transmitted correctly, but maybe not. So.

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And it goes bad with the probability. So E.

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And then so basically that.

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Silence.

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Then we want to get the probabilities of.

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The various events, so I'm going to need some, some notation here.

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So, what I'm going to say is.

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We got 1, 1 0.

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Heroin.

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Or 1 receive 1 so you've got things like the probability of.

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Transmit 1 equals P. for example.

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Probability of receive 0.

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Given that we transmitted a 0.

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Is 1 minus EE probability we received a 0.

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Equals E, and so on.

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Okay, because we've got those areas things there and then you could say, well.

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The probability that we transmitted as 0T and received a 0.

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Well, we transmit it to 0T. That's probability.

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1 minus P and then given that, that we received a zeros 1 minus. See, let's say so.

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Transmitted 0T and received a 1.

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1 minus B we translated a 0T times. E.

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And probability.

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Transmitted 1 and received 0.

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Month there we go.

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201 is a p and received a 0.

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He, and probably translated 1 and received 1 is P and 1 minus D.

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So we get to 4 different 4 different possibilities.

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Um, but you see, now we can start talking about.

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Say the probability that received a 1 given, we transmit it on.

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And I'll be a probability transmit 1 and received 1.

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If I invited probability transmitted and 1.

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And that's going to be 1 minute to see.

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Divided by probably the translating 1, which was.

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P, which is, which is correct. Okay. Is which is what we defined it as.

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So, that that works out okay now.

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What we're actually interested in.

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And say probability of.

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I E, what's the probability that if we saw a 0T or 1, that it really was 0T? Well, we can do that to.

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So you have to get the probability of the received arrow, which I hadn't yet.

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Calculated that yet, so.

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I'm anticipating a little here, but the probability that I received is, 0T is a probability that we transmitted to 0T and received a 0T. Plus the probability that we transmitted a 1 and received to 0.

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And I'm just assuming it's intuitive that we can just add them like that.

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So, transmitted 0T and received 0.

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Is 1 minus P1 minus C.

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Transmitted 1, and received a 0.

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This P. E.

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Which we could simplify as.

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Doesn't really simplify, but we could try 1 minus P minus E. plus 2. P. E. didn't really simplify, but.

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Ok, so, now, going up here, what we wanted was.

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We wanted to profitability that we.

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We wanted that to probability that if we saw a 0T that it was a 0T that was transmitted.

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Well, that if I can.

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Silence.

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Well, let me throw some numbers at you, let's say.

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Let's suppose equals 1 half.

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And he equals point 1 or something. So now that probability.

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A quote.

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Silence.

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This case was really simple so if we receive 0T, the probability of 0T was really transmitted was.

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90%, which is a Congress, the probability of error, but that's.

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That's because we were transmitting zeroes and ones equally.

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Let's do another example, let's assume zeroes and ones are not equal and this teaches you something.

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Let's assume the probability of translating a 0T is also point 1 and the chance of errors point 1.

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So, the probability of transferring that, if we received a 0T, the probability to the 0T was transmitted.

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Was Scott is going to be, um.

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Times point 9 over.

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That's going to be roughly point 9 8.

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So, if we received 0T, then it was 98% chance that a 0T was transmitted.

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Um, cause the thing is this that.

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It was a 90 you see if we saw nothing at all, it's a 90% chance that a 0T was transmitted.

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If we before we see anything to 90% chance, 0T was transmitted.

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Because 90% of the transmission to zeros, if we after we see the 0T that raises it to 98.

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So, and this is a realistic thing, because a lot of the time you transmitted symbols are not equally problem.

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And we do it the, if we receive a 1, what's the probability that a 1 was transmitted?

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And 1 was transmitted, given or received a 1 and this will be the probability that.

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Divided by the probability.

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Receiving a 1 now transmitting a 1 and receiving a 1.

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Is probably transmitting the 1 was.

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That's P, and they're probably receiving a 1.

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Is we don't have to work that out here?

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Quickest way is.

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See, up here, probability of receiving a 1 was.

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Right okay translating 0T and receiving a 1 plus the probability.

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Of translating a 1 and receiving a 1.

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Graduating a 0T and receiving a 1 would be 1 minus P times E translating a 1 and receiving a 1 would be P times 1 minus.

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So, we put that back in here.

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And let me put in the case also, let's suppose.

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Example again, let's suppose that there is a point 1% chance of transmitting of 1.

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00:29:31.679 --> 00:29:35.909
And a point 1 chance of error. So now that probabilty up here.

232
00:29:38.669 --> 00:29:42.298
Graduating a 1 given we received a 1 is going to be.

233
00:29:42.298 --> 00:29:46.169
P times 1 minus E that's point 0 9.

234
00:29:48.778 --> 00:29:52.019
So, that makes this thing up here to be point 0, 9 plus.

235
00:29:52.019 --> 00:29:58.259
9 equals point 1, 8 so down here point 1, 8.

236
00:29:58.259 --> 00:30:02.969
Equals point 5.

237
00:30:02.969 --> 00:30:07.108
So, um.

238
00:30:07.108 --> 00:30:10.169
Just a 2nd, I wanted to.

239
00:30:20.578 --> 00:30:26.878
So, if we received a 1, it's only a 50, 50 chance that a 1 was transmitted.

240
00:30:26.878 --> 00:30:30.808
But before we say anything was only 10% chance.

241
00:31:07.949 --> 00:31:12.358
So this is worth.

242
00:31:12.358 --> 00:31:15.598
This is, we're thinking about.

243
00:31:16.648 --> 00:31:21.898
So we receive 1, it makes it more likely that a 1 was transmitted, but it's still not.

244
00:31:21.898 --> 00:31:29.338
Certain, and the reason it's not certain is it transmitted? Ones are fairly unlikely. They're only 10% of the chat of the time.

245
00:31:29.338 --> 00:31:34.949
And also there's things that translated errors. So.

246
00:31:36.118 --> 00:31:42.538
This may be counterintuitive here. Perhaps the surprising.

247
00:31:45.659 --> 00:31:49.288
Yeah, but that's.

248
00:31:50.489 --> 00:31:58.949
So, weird things, binary communication, some total probability from.

249
00:32:00.179 --> 00:32:04.409
That says if you add up all the different cases, and you get to a 1.

250
00:32:05.578 --> 00:32:14.338
So, if we've got grads and undergrads here, then the.

251
00:32:15.838 --> 00:32:18.959
Suppose a is undergrads then.

252
00:32:20.159 --> 00:32:24.209
The probability of being an undergrad given that.

253
00:32:24.209 --> 00:32:32.159
You're an engineering, plus probably being an undergrad given times of probability that you are in engineering and everything added up.

254
00:32:32.159 --> 00:32:35.729
It has to sum up to 1.

255
00:32:35.729 --> 00:32:40.048
So.

256
00:32:40.048 --> 00:32:43.499
Let me work through something like that.

257
00:32:51.749 --> 00:32:54.989
So, in this case here there are.

258
00:32:54.989 --> 00:33:01.229
How many students? This is a total probability.

259
00:33:06.689 --> 00:33:10.199
Silence.

260
00:33:12.568 --> 00:33:18.388
Okay, how many students are there? 912.

261
00:33:18.388 --> 00:33:21.868
1320 underground.

262
00:33:24.358 --> 00:33:27.358
And grads.

263
00:33:29.159 --> 00:33:36.929
An undergrad equals 9000 to 100.

264
00:33:38.999 --> 00:33:44.219
So, the probability of.

265
00:33:50.699 --> 00:33:54.568
And engineering.

266
00:33:54.568 --> 00:33:57.749
Say, 2500.

267
00:33:57.749 --> 00:34:02.098
He's our fictitious numbers and science.

268
00:34:02.098 --> 00:34:06.179
Equals 1400.

269
00:34:06.179 --> 00:34:09.268
And the manager and.

270
00:34:09.268 --> 00:34:14.759
800 arc.

271
00:34:14.759 --> 00:34:23.278
Because 600 Haas, plus say 120 or something.

272
00:34:23.278 --> 00:34:26.369
Then we can say.

273
00:34:32.489 --> 00:34:36.809
So, the probability of being an engineering.

274
00:34:36.809 --> 00:34:43.079
That's the on the total number, total, big and.

275
00:34:44.278 --> 00:34:47.969
A close 5420.

276
00:34:47.969 --> 00:34:51.898
So, probably being in engineering is going to be things like.

277
00:34:56.759 --> 00:35:01.619
5420 and so on, so we can get conditional. Now.

278
00:35:10.498 --> 00:35:16.559
Engineering and undergrad. Well, that is 500.

279
00:35:17.759 --> 00:35:24.449
Divided by 5420 and we can get things like the probability of being in.

280
00:35:25.889 --> 00:35:29.849
Engineering given you're an undergrad.

281
00:35:29.849 --> 00:35:34.438
Is a problem of being an engineering and undergrad.

282
00:35:36.389 --> 00:35:40.918
Engineering, an undergrad is.

283
00:35:41.938 --> 00:35:45.478
Bye.

284
00:35:45.478 --> 00:35:50.128
And probably being an undergrad is 4100 over 5480.

285
00:35:53.969 --> 00:36:00.568
Close probably being an engineer for an undergrad and there are.

286
00:36:02.369 --> 00:36:07.378
4100 undergrad, 500 of them are engineers and so on.

287
00:36:07.378 --> 00:36:10.708
And the total probability theorem is that.

288
00:36:10.708 --> 00:36:15.119
If we add up all these various cases, and it gets out.

289
00:36:15.119 --> 00:36:20.429
Things add up so what I mean is.

290
00:36:23.789 --> 00:36:27.358
And undergrad is.

291
00:36:33.329 --> 00:36:39.898
Probability of being in science, given undergrad.

292
00:36:39.898 --> 00:36:45.898
Is, um.

293
00:36:45.898 --> 00:36:52.289
Right before 22 no undergrad is 4100 by mistake.

294
00:37:01.318 --> 00:37:10.318
Probability of being in management if an undergrad is.

295
00:37:12.748 --> 00:37:18.358
Ability of being an arc given undergrad.

296
00:37:18.358 --> 00:37:21.659
Yes.

297
00:37:21.659 --> 00:37:27.028
Really being in hos, if an undergrad is.

298
00:37:28.438 --> 00:37:33.659
Okay, and they add up to.

299
00:37:34.768 --> 00:37:38.668
They add to.

300
00:37:52.438 --> 00:37:56.579
By source 9121320.

301
00:38:15.840 --> 00:38:19.320
Sorry, I know.

302
00:38:21.389 --> 00:38:27.809
I was saying I was using numbers for grads here, not undergrads all along the way here.

303
00:38:27.809 --> 00:38:33.210
We correct this, um.

304
00:38:49.050 --> 00:38:53.309
Every time I said.

305
00:38:53.309 --> 00:38:58.050
Undergrad or grad.

306
00:39:00.239 --> 00:39:05.369
Crack given grad grad, given grad.

307
00:39:09.210 --> 00:39:19.590
Um, just a 2nd, here.

308
00:39:19.590 --> 00:39:25.110
I'm going to start this 1 again. I was mixing up grads and undergrads. So.

309
00:39:41.099 --> 00:39:45.389
Problem of engineering given grad is 500.

310
00:39:45.389 --> 00:39:52.619
I just want to again.

311
00:39:55.739 --> 00:40:00.840
Okay, sorry about that. Okay.

312
00:40:01.949 --> 00:40:06.030
Total probability continue.

313
00:40:07.260 --> 00:40:10.590
Probability of.

314
00:40:13.860 --> 00:40:17.489
Under the simple case, the probability of being an engineering.

315
00:40:19.380 --> 00:40:24.719
Equals the probability of engineering and undergrad.

316
00:40:24.719 --> 00:40:28.139
Plus the probability of engineering.

317
00:40:28.139 --> 00:40:33.599
And grad probability of engineering and.

318
00:40:33.599 --> 00:40:37.889
An undergrad that is.

319
00:40:37.889 --> 00:40:41.369
500 over the 5420.

320
00:40:44.159 --> 00:40:52.619
2000 is 2000 over 5420 plus 500 over 5420.

321
00:40:54.510 --> 00:41:01.829
And that's correct is 2500 engineering students.

322
00:41:04.739 --> 00:41:08.880
Yeah, you're correct, Nicholas. Yeah, that's what I did. Wrong sent me.

323
00:41:08.880 --> 00:41:12.599
Send me an email. That's okay. Um.

324
00:41:12.599 --> 00:41:19.019
You're right, I swapped undergrad and grad, so that's a total there. And then.

325
00:41:21.389 --> 00:41:24.510
And then we can take it to the next level of detail.

326
00:41:24.510 --> 00:41:27.690
Because the probability of, um.

327
00:41:30.420 --> 00:41:39.300
Of course of engineering an undergrad equals the probability of engineering.

328
00:41:39.300 --> 00:41:44.159
Given an undergrad as a probably of undergrad.

329
00:41:44.159 --> 00:41:48.059
So, we can stretch the whole thing out and so on.

330
00:41:48.059 --> 00:41:51.510
And you can run this both ways. It's a conditional thing.

331
00:41:53.610 --> 00:41:59.880
Yes, just a 2nd, here.

332
00:42:03.659 --> 00:42:08.610
Right.

333
00:42:18.329 --> 00:42:23.849
Okay, good. A laptop became on plugged. Okay.

334
00:42:23.849 --> 00:42:26.849
Um.

335
00:42:26.849 --> 00:42:29.849
So, quality control.

336
00:42:29.849 --> 00:42:37.650
Is an important topic will be getting lots of examples like that like, around here.

337
00:42:37.650 --> 00:42:50.039
Okay, and then we might say that if we manufacture a good chip, it's at a time.

338
00:42:50.039 --> 00:43:01.050
Probably so it's still alive at time teas. He does a minus alpha tea and if it's bad, it might still live a long time but it is more likely to be dead.

339
00:43:01.050 --> 00:43:04.980
So, he might want to talk about the probability.

340
00:43:04.980 --> 00:43:09.510
That some random chip is still alive at time tea.

341
00:43:09.510 --> 00:43:15.329
And we could do that and start a new page here.

342
00:43:17.400 --> 00:43:26.940
So, chip quality control.

343
00:43:26.940 --> 00:43:37.860
Okay, um.

344
00:43:37.860 --> 00:43:44.099
T, Kevin good.

345
00:43:45.480 --> 00:43:52.139
Times the probability of good, plus the probability.

346
00:43:52.139 --> 00:43:57.329
Live at T, even bad.

347
00:43:57.329 --> 00:44:02.159
Probability of bad for example.

348
00:44:04.409 --> 00:44:09.960
Because that's probably the live at T and good. Plus the probability if alive at T and benches it.

349
00:44:12.510 --> 00:44:17.969
T and good so your whole life.

350
00:44:19.079 --> 00:44:22.230
At T and bad. Okay.

351
00:44:22.230 --> 00:44:27.900
A total probability thing. So now what we can do so alive at T.

352
00:44:29.940 --> 00:44:33.989
So, probably a live at T. Kevin is good.

353
00:44:35.969 --> 00:44:39.599
Has the probability of being good.

354
00:44:41.250 --> 00:44:46.230
Which is 1 minus P plus the problem it is alive at keeping bad.

355
00:44:47.340 --> 00:44:54.030
I am so probability of it being that. Okay.

356
00:45:00.059 --> 00:45:06.539
So, you can reverse it and what is a probability.

357
00:45:06.539 --> 00:45:10.949
Of being good given.

358
00:45:10.949 --> 00:45:16.230
So either T, okay.

359
00:45:18.869 --> 00:45:27.059
So, you could work that 1 now that's phase rule and lets you invert these conditional probability. And that's a question you want to ask all the time that.

360
00:45:27.059 --> 00:45:34.650
The transmission case, you receive a 1 what's the probability that a 1 was transmitted given that you received a 1?

361
00:45:35.760 --> 00:45:41.909
Um, if the chip is still alive, what's the probability was a high quality chip instead of a low quality chip.

362
00:45:41.909 --> 00:45:46.619
It could be a bad chip that just happened to be living a long time. That happens.

363
00:45:46.619 --> 00:45:53.429
So, with this reverse concept, this, this is getting into base role here. So.

364
00:45:56.880 --> 00:46:09.030
Silence.

365
00:46:11.130 --> 00:46:19.349
Very important. And so we have the posterior probabilities.

366
00:46:19.349 --> 00:46:22.980
And for then we want to calculate the prior probabilities.

367
00:46:24.780 --> 00:46:28.590
So.

368
00:46:28.590 --> 00:46:31.769
My, um.

369
00:46:31.769 --> 00:46:37.980
My communication example is that was the numbers I gave you in the 2nd version of it.

370
00:46:37.980 --> 00:46:42.510
If you receive a 1, it's still only 50, 50 that a 1 was transmitted.

371
00:46:42.510 --> 00:46:48.420
And I could have picked out the numbers, which would have made it less than a 50 per cent chance.

372
00:46:50.010 --> 00:46:55.079
Or giving an example, which is almost related to the news if.

373
00:46:55.079 --> 00:46:58.679
If you have tests very unlikely on.

374
00:46:59.789 --> 00:47:05.909
Very unlikely occurrences I'll give us with numbers later, but just give an example of.

375
00:47:05.909 --> 00:47:16.559
Where things it'd be surprising I mean, let's suppose that there's a really unusual condition out in the population looking trophy. Okay.

376
00:47:16.559 --> 00:47:20.190
Being aware, and let's suppose that.

377
00:47:20.190 --> 00:47:29.639
1 in a 1M people are where waltz. Okay. So the probability of being aware wealth is 10 of the minus. 6 is going to be 3 in the United States. Let's say.

378
00:47:30.929 --> 00:47:38.400
Well, let's suppose we have a test to see if you're aware well for not. Okay and the test is pretty good.

379
00:47:39.630 --> 00:47:43.889
But it gives a false positive 1 time in a 1000.

380
00:47:43.889 --> 00:47:49.860
Is a pretty good test. Most medical tests are not accurate. 99.9% of the time.

381
00:47:51.239 --> 00:48:00.599
So, she'd take a random person, give them the werewolf test trophy test and the world test reports positive.

382
00:48:01.920 --> 00:48:06.300
So now, what's the probability that the person really is aware.

383
00:48:06.300 --> 00:48:09.719
Before the test, it was 1 in a 1000000.

384
00:48:09.719 --> 00:48:16.469
We gave the test, which is the 100000 chance of error and the test reports positive.

385
00:48:17.699 --> 00:48:21.719
The person is still probably not aware.

386
00:48:21.719 --> 00:48:30.179
So, oh, by doing for the test, it could be wrong. 2 ways. It could get false positives and false negatives.

387
00:48:30.179 --> 00:48:33.300
But, let me work this out actually.

388
00:48:40.110 --> 00:48:55.530
Silence.

389
00:48:57.449 --> 00:49:00.960
Let's say.

390
00:49:00.960 --> 00:49:04.860
Is okay.

391
00:49:14.730 --> 00:49:18.090
There is a test T.

392
00:49:18.090 --> 00:49:21.900
Or this.

393
00:49:23.579 --> 00:49:29.909
It's pretty good.

394
00:49:29.909 --> 00:49:36.059
If you really are a winner wall and you have the test.

395
00:49:36.059 --> 00:49:39.869
He means this means a test comes out positive.

396
00:49:43.619 --> 00:49:48.449
And the probability that you're not aware wealth, given a test was false.

397
00:49:48.449 --> 00:49:51.510
Is still 9, 9 9. okay.

398
00:49:52.530 --> 00:49:59.159
T, means was test was okay, Eva.

399
00:49:59.159 --> 00:50:10.289
Was negative. Okay. So what so, what we want is a probability.

400
00:50:14.070 --> 00:50:18.150
What did I do here? Way? Backwards again.

401
00:50:22.380 --> 00:50:26.070
What I meant is if you are and where.

402
00:50:26.070 --> 00:50:35.849
The problem with the test is positive if you're not aware. Well, it's probably a chest is negative. So, what I want is if the test was positive, what's the chance here aware? Well.

403
00:50:37.469 --> 00:50:42.059
Okay, how are we going to work this out?

404
00:50:45.150 --> 00:50:49.500
My well, the 1st question is what's the probability of the test was positive.

405
00:50:49.500 --> 00:50:55.619
I'm on okay. That's the probability that the.

406
00:50:58.679 --> 00:51:02.219
Well, we're going to work on all the possibilities here.

407
00:51:05.579 --> 00:51:08.940
I'll start back up rerun everything here.

408
00:51:11.550 --> 00:51:18.989
Oh, so the problem probably where wealth.

409
00:51:18.989 --> 00:51:24.119
Was tentative minus 6, the probability of.

410
00:51:25.530 --> 00:51:28.949
Yes, was positive given your where wealth is this?

411
00:51:28.949 --> 00:51:32.610
There'll be a chest as negative given. You're not aware.

412
00:51:32.610 --> 00:51:36.000
What is this?

413
00:51:37.980 --> 00:51:40.980
So, we're going to get is a probability.

414
00:51:42.329 --> 00:51:45.900
Of that to test is positive and you're aware.

415
00:51:45.900 --> 00:51:55.380
Is 10 and a minus 6 the probability the test is negative and given you're not aware of.

416
00:51:55.380 --> 00:52:02.460
Is is the same sorry is I'm.

417
00:52:16.019 --> 00:52:29.880
Okay.

418
00:52:33.269 --> 00:52:38.190
So this is.

419
00:52:40.949 --> 00:52:46.199
This here is very roughly point 9, 9, 9, that's approximately.

420
00:52:48.150 --> 00:52:48.960
Okay,

421
00:52:49.644 --> 00:53:04.434
so that the test is positive and you're not aware well,

422
00:53:05.065 --> 00:53:06.295
that's about point 0.

423
00:53:06.324 --> 00:53:07.105
0 1.

424
00:53:10.050 --> 00:53:16.199
The problem is the test being positive is point 0 0 1 plus point.

425
00:53:18.420 --> 00:53:26.039
And that's roughly point 0. 0 1.

426
00:53:27.840 --> 00:53:36.449
So so probably the testing positives roughly point 0. 0 1, we want.

427
00:53:41.309 --> 00:53:47.429
Okay, so the probability.

428
00:53:47.429 --> 00:53:53.699
Of where well, given the task was positive probability where will and the tests being positive.

429
00:53:57.000 --> 00:54:01.800
I have a probability of the test where wealth and test being positive.

430
00:54:01.800 --> 00:54:10.289
Is fine and probably the test being positive.

431
00:54:11.670 --> 00:54:19.260
Something like that.

432
00:54:20.280 --> 00:54:23.309
Approximately, I may have dropped a 0T here.

433
00:54:24.690 --> 00:54:31.320
I probably did I still here I should go back in.

434
00:54:34.230 --> 00:54:38.909
And triphosphate, so, in other words.

435
00:54:41.670 --> 00:54:45.179
Even if the test.

436
00:54:45.179 --> 00:54:52.500
It's positive.

437
00:54:54.239 --> 00:54:58.440
So, where will.

438
00:54:58.440 --> 00:55:03.659
Is only 1 0. 0. 1.

439
00:55:04.829 --> 00:55:11.070
Because what happened here is the.

440
00:55:11.070 --> 00:55:14.699
Looking is such an unusual condition.

441
00:55:14.699 --> 00:55:21.630
That the positive tests are dominated by the fact that the test procedure.

442
00:55:21.630 --> 00:55:35.550
Isn't accurate. Okay. So if you had a positive test for being aware well, before the positive test, you had a 1 in a 1M chance after the positive test still only a 1000 chance. You're aware of.

443
00:55:35.550 --> 00:55:39.539
So this is base role here.

444
00:55:40.769 --> 00:55:46.440
We're inverting the conditional probabilities. So.

445
00:55:46.440 --> 00:55:55.440
You might say we're going left to right probability through. Probably the test is positive given your werewolf and right to left. You might say.

446
00:55:55.440 --> 00:56:02.039
Given probably you're aware, we'll if given the test is positive.

447
00:56:03.150 --> 00:56:08.460
So, and.

448
00:56:11.429 --> 00:56:22.440
The, I'll come back to the engineering diagram next time, maybe and again, the receiver example, which you'll be saying to tell you a sec of it, because it's important and it's easy to work with.

449
00:56:22.440 --> 00:56:25.980
I'm looking at the chip quality control thing.

450
00:56:27.269 --> 00:56:31.139
Um, okay, so you got the good.

451
00:56:31.139 --> 00:56:36.329
On here example, 12 here, that's basically like the where we'll thing to slightly different numbers.

452
00:56:36.329 --> 00:56:40.139
Okay, now the chip quality control thing.

453
00:56:41.760 --> 00:56:48.389
Remember we said we had the good chips and the bad chips and the good chips last longer.

454
00:56:50.159 --> 00:57:00.480
Now, if I scroll back up here, a little just to make people see, 6, slightly here, a chip quality control this here is called an exponential distribution.

455
00:57:00.480 --> 00:57:04.980
The interesting thing about that, is that the previous history.

456
00:57:04.980 --> 00:57:08.940
Of the chip doesn't matter. So.

457
00:57:08.940 --> 00:57:13.199
It doesn't matter where we say time 0T if the good ship is alive.

458
00:57:13.199 --> 00:57:19.349
Now, the probability that it will be alive at tea in the future is either the minus a tea.

459
00:57:19.349 --> 00:57:22.739
It doesn't matter how long it's been living.

460
00:57:23.820 --> 00:57:26.969
This isn't complete the accurate, but, you know.

461
00:57:26.969 --> 00:57:30.449
It's not totally crazy that if.

462
00:57:30.449 --> 00:57:34.139
The chip say lives to let cosmic re, had said.

463
00:57:34.139 --> 00:57:40.650
And that is going to be the exactly the lifetime.

464
00:57:40.650 --> 00:57:44.340
Okay, so you got the good chips and you got the bad chips.

465
00:57:45.570 --> 00:57:49.409
And what you want to do, maybe if you want to burn them both in for a while.

466
00:57:49.409 --> 00:57:55.199
And you hope that what you'll do is, you'll burn the bad chips will die and will be left of the good chips.

467
00:57:55.199 --> 00:57:58.199
And a few of the good ships will die also.

468
00:57:58.199 --> 00:58:06.780
But mostly bad chips will die and look, you'll have done yield, sort of, refined your product mix. Your what? The surviving chips are more likely to be good.

469
00:58:06.780 --> 00:58:11.940
So, the question is.

470
00:58:14.639 --> 00:58:18.360
That how long do we have to bring in the chips? So.

471
00:58:18.360 --> 00:58:25.710
The surviving chips are probably good. So, the cost of that, what the cost of that is.

472
00:58:25.710 --> 00:58:29.010
Is while the time it takes to burn in all the chips.

473
00:58:29.010 --> 00:58:35.550
Plus the fact that we're going to lose a few good chips that we could have sold.

474
00:58:36.570 --> 00:58:39.630
But the benefit of it is that.

475
00:58:40.739 --> 00:58:45.030
We're selling fewer bad chips and bad chips will make our customers.

476
00:58:45.030 --> 00:58:48.480
Angry so they may jump to someone else.

477
00:58:52.829 --> 00:59:04.380
So that's that example, point 11 here example, 2 and 30 it's a little messy for me to do by hand unnecessarily. I may work it out. Next time. You can think about that.

478
00:59:04.380 --> 00:59:07.409
False positives. I mentioned.

479
00:59:08.969 --> 00:59:14.280
I mentioned with the thing, so that can be survive. That can be.

480
00:59:16.769 --> 00:59:22.170
Surprising and counterintuitive. This actually is economic.

481
00:59:22.170 --> 00:59:28.469
Consequences that the people, the accountants working in health care think about.

482
00:59:28.469 --> 00:59:34.260
You, as a consumer might think it's cold blooded, but they do think about questions like this.

483
00:59:34.260 --> 00:59:39.059
Because there are some diseases where are the tests have.

484
00:59:39.059 --> 00:59:43.829
A lot of false positives and the treatments are expensive.

485
00:59:43.829 --> 00:59:50.699
And the treatments you might argue, maybe they don't actually help a lot. That's a separate argument.

486
00:59:51.840 --> 01:00:00.539
But if you give the patient, the test, and the test is positive, then you have to treat the patient for that disease that costs a lot of money.

487
01:00:00.539 --> 01:00:04.380
And if the patient, in fact, was negative.

488
01:00:04.380 --> 01:00:13.710
1st, that treatment that was wasted money and wasted time for the patient. And 2nd, the treatment itself might be dangerous.

489
01:00:13.710 --> 01:00:19.019
Give us say some examples cancer treatment.

490
01:00:19.019 --> 01:00:27.119
Of the radiation and the chemo their guiding principle of them is that you almost kill the patient.

491
01:00:27.119 --> 01:00:41.760
But you totally kill the cancer. I'm sounding rather ruthless about this, but this really is the guiding principle you want to take the patient almost to the edge of dying. But what you did is you completely kill the cancer.

492
01:00:41.760 --> 01:00:45.179
But now, here's the thing if the patient didn't have the cancer.

493
01:00:45.179 --> 01:00:48.840
You nearly killed the patient so you.

494
01:00:48.840 --> 01:00:53.010
And the account may not care about that, but account cares about the money that was spent.

495
01:00:53.010 --> 01:00:57.869
So the false positive so there are some answers out there with.

496
01:00:57.869 --> 01:01:03.090
The medical policy makers, I was saying, do not do these tests so often.

497
01:01:04.289 --> 01:01:11.909
Really? Okay, so this is real world applications of base real, and so on and profitability.

498
01:01:16.110 --> 01:01:20.639
Multi nomial there was a question about that last time.

499
01:01:20.639 --> 01:01:26.340
I talked about it a little last time. Here's the, here's the.

500
01:01:28.949 --> 01:01:32.579
Is the formula again so that.

501
01:01:36.690 --> 01:01:41.460
If I can say, give a dartboard multi nomial example.

502
01:01:56.460 --> 01:02:01.559
And maybe we've got bang, I don't play darts, but.

503
01:02:06.900 --> 01:02:10.739
And maybe we've got something like, I don't know, it's, um.

504
01:02:10.739 --> 01:02:19.860
Like, 1% chance being there called this region, a region B maybe it's point 3 and Regency. It's point 6.

505
01:02:20.605 --> 01:02:21.324
Okay,

506
01:02:36.954 --> 01:02:38.094
and we get,

507
01:02:38.125 --> 01:02:38.934
I don't know.

508
01:02:41.369 --> 01:02:45.960
To a to B and C. that's the probability.

509
01:02:47.280 --> 01:02:52.619
And the formula is going to be.

510
01:02:55.500 --> 01:02:58.889
I have choose to to 1, 1.

511
01:02:58.889 --> 01:03:03.179
Times 1 squared.

512
01:03:03.179 --> 01:03:07.170
1, 3 squared point 6 to the 1.

513
01:03:07.170 --> 01:03:11.130
And 5, 2, 2, 1, that is going to be.

514
01:03:14.610 --> 01:03:18.599
Okay, so we can calculate that.

515
01:03:23.460 --> 01:03:26.760
Random phone numbers I may talk about next time.

516
01:03:30.480 --> 01:03:35.489
I want to hit this thing random number generation here.

517
01:03:35.489 --> 01:03:38.519
You're generating random number to do tests.

518
01:03:38.519 --> 01:03:44.219
And there's a section in the book, I mean, if people ask me to, I'll cover it.

519
01:03:44.219 --> 01:03:51.780
But I'll give you the executive summary. Is it generating good? Random numbers is surprisingly difficult.

520
01:03:51.780 --> 01:03:56.190
Okay, commercial software commercial routines.

521
01:03:56.190 --> 01:04:00.750
And software packages have sometimes gotten it wrong.

522
01:04:00.750 --> 01:04:07.949
Here's a case several decades ago. The Arizona lottery got it wrong.

523
01:04:08.905 --> 01:04:24.264
Eva.

524
01:04:24.269 --> 01:04:32.730
Okay, so what happened was the Arizona laundry they were picking if you pick round numbers, then you.

525
01:04:33.780 --> 01:04:39.480
You know, if you pick the numbers, you when.

526
01:04:39.480 --> 01:04:45.150
And it turned out that the lottery was never generating a 9.

527
01:04:45.150 --> 01:04:48.809
And try and call it here. What the person thought.

528
01:04:48.809 --> 01:04:53.400
You know, this is.

529
01:04:53.400 --> 01:05:02.010
Totally simple things. No, 1 ever actually tried, they rented our gender, looked at the numbers and saying this is funny. Still know nice coming out.

530
01:05:02.010 --> 01:05:07.139
And also what is happening here.

531
01:05:08.400 --> 01:05:14.130
Is if it's some random number, assuming R and D or on a number of returns of fraction.

532
01:05:14.130 --> 01:05:19.050
And you modify the fraction and say a fraction from 0T to 1.

533
01:05:19.050 --> 01:05:24.989
Evolve applied by 9 and assume and route chunk hates it down to the closest.

534
01:05:24.989 --> 01:05:31.980
Well, the only way that will ever produce a 9 is just around a number generator produced a 1.0.

535
01:05:31.980 --> 01:05:38.460
Probably that is like, 0T in any case. So real world example getting round numbers wrong.

536
01:05:39.539 --> 01:05:45.599
So, in fact, several decades ago, the book was sold.

537
01:05:45.599 --> 01:05:48.929
With familiar with her and the numbers in it so you buy the book.

538
01:05:48.929 --> 01:05:52.320
And you would just, you know.

539
01:05:52.320 --> 01:06:00.179
To ground new numbers, if you're doing a psychology experiment, you'd go into the book, you'd pick random numbers off the book, and you just cross them out as you use them.

540
01:06:00.179 --> 01:06:04.170
There's quite a fun book. Actually it's available for sale.

541
01:06:08.699 --> 01:06:12.030
By the way the story is that the New York.

542
01:06:12.030 --> 01:06:17.190
Public library, but this book in abnormal psychology, because since it talked about deviance.

543
01:06:17.190 --> 01:06:22.920
All you got to see the reviews for this book here hysterical.

544
01:06:24.960 --> 01:06:33.449
Hello.

545
01:06:36.539 --> 01:06:42.809
It's not all random subsequential numbers in the book.

546
01:06:43.829 --> 01:06:45.114
Not a very good plot

547
01:06:48.715 --> 01:07:20.065
site.

548
01:07:32.699 --> 01:07:38.880
Okay, any case so, random number of generators so difficult.

549
01:07:43.949 --> 01:07:51.030
And so the suit of random things, but they, um.

550
01:07:51.030 --> 01:07:54.179
There are ways to do it, but it takes some skill.

551
01:07:54.179 --> 01:07:57.449
Now, this is important, of course, with cryptography.

552
01:07:57.449 --> 01:08:00.449
If you're picking a key.

553
01:08:00.449 --> 01:08:11.760
For your secret cipher, you maybe the key is based on a 1000 bit random number, you would read the secret. You would really like those to be random bits.

554
01:08:11.760 --> 01:08:17.340
And so you really hope that whatever the operating system is doing, they're not predictable.

555
01:08:17.340 --> 01:08:22.050
Otherwise somebody might steal your bed coins. Okay. Um.

556
01:08:24.539 --> 01:08:28.439
And base here, I'm going to review again.

557
01:08:28.439 --> 01:08:33.600
And probably of extra terrestrial life.

558
01:08:35.220 --> 01:08:39.149
And I'll hit it again and I'll hit chip quality control again.

559
01:08:39.149 --> 01:08:45.270
I just want to review some terminology here.

560
01:08:45.270 --> 01:08:48.479
If we toss a coin once.

561
01:08:48.479 --> 01:08:52.680
It's called unfair coin that saved called it.

562
01:08:52.680 --> 01:08:58.739
Try we're nearly there were the name of a family of mathematicians a couple 100 years ago that.

563
01:08:58.739 --> 01:09:05.069
Like, gambling and still work at some probability to do gambling it's a trial 1 toss.

564
01:09:05.069 --> 01:09:13.890
You have lots of tosses of the coin. That's normally a law. They don't realize probably of exactly say 5 heads and 10 tosses.

565
01:09:13.890 --> 01:09:28.529
And the coin might be unfair, but the tosses are not correlated with each other. If, you know, 1 toss it doesn't help you on another task. The multi nomial law is the coin has several phases. Like, it's a dye with 6 spaces. Let's say.

566
01:09:28.529 --> 01:09:39.180
And that, but I might not be fair, each faces different probability. And that would be a multi nomial law wheel or drawing cards or something.

567
01:09:39.180 --> 01:09:46.829
Okay, and I'll come back and hit some of these I guess Monday we get off.

568
01:09:46.829 --> 01:09:51.569
Go skiing or something might be skiing and.

569
01:09:55.109 --> 01:10:06.899
And I'll hit some of the, I'll review some of these things and continue on more base serum. You're going to see lots of examples of base theorem because it's important.

570
01:10:06.899 --> 01:10:10.350
Okay, well Thank you.

571
01:10:12.359 --> 01:10:16.680
I'll rework that student number question also for next time.

572
01:10:16.680 --> 01:10:21.149
So that's for today, we saw some new stuff.

573
01:10:21.149 --> 01:10:32.039
Total probability, the big new thing that we saw. Well, conditional probability we saw the formal definition and the big idea today was base rule.

574
01:10:33.479 --> 01:10:39.000
Okay, well have fun. Relax for a couple of days.

575
01:10:39.000 --> 01:10:45.239
And I'll see you Thursday next week, and I'll stay around for a couple of.

576
01:10:45.239 --> 01:10:48.600
Minutes in case everyone has any questions.

577
01:10:53.220 --> 01:11:08.430
Silence.

578
01:11:11.399 --> 01:11:28.409
Silence.

579
01:11:30.989 --> 01:11:52.439
Silence.

580
01:11:59.279 --> 01:12:02.489
Silence.

581
01:12:11.850 --> 01:12:16.050
Silence.