The probabilities are next we put a probability of every branch here. These videos are made by Corbett Maths. What's the probability that the test is correct? But all 22? 0.9 and 0.7 is 0.63. Any questions about that? By the product rule is the probability that I win the first game times the probability that I win the second game given that I won the first game. So what we're doing here now is giving you the formal justification for that rule that we had last time and that you'll always use-- is the probability of a sample point is the product of the probabilities on the edges leading to it. So it gets very unlikely that I've got the fair coin here as K gets big. All right. c: oqq unwpst unwp6L ;usu ecsls pe10M' 01 10110,wua pseu wseŒq ou e!x-aqsq qgce (p) hon 1011 s unwpek ou su otq!usù, elx aqsq So this'll be 1/3 third plus 1/18 plus the extra one, 1/9. Just multiply by probability of B on both sides. Well Berkeley took a look at that and said, nothing good. of the conditional probability). All right. And of course what we want to know is the probability you have the disease given that you tested positive. And say that this is B and this is A. So For the rest of today we're going to do three more paradoxes. And this is sort of the sum of those. At least one tail for this one. Heads, the probability 1. And instead of doing the tree method, which we know we're supposed to do, we're just going to wing it, which is always seems easier to do. PROFESSOR: I know which one I'd fly. There's not enough information. But then I flipped it behind my back and I showed you the result. All right? You've got a test, 10% percent false negative, 30% false positive. Can anybody look at that and tell me? It's the same as the answer over there. All right. What's the probability of the first two die matching-- both of them? So I want probability of B given A now for this example. So the question for you guys, is it possible that both sides we're telling the truth, that in every single department the women have a lower acceptance rate than men, but on the university as a whole the women are higher percentage? Well now, instead of 1/4 I get P over 2 up here. So let's figure that out by drawing the tree. OK? OK? So if there's risk both ways, probably don't have the disease. Conditional probability occurs when it is given that something has happened. So they're not disjoined. Forget all the fancy weather forecasting stuff, the radar, and all the rest. And there's a good chance this kind of question's going to be on the final this year. A is the event that the applicant is admitted. AUDIENCE: [INAUDIBLE] they're weighted differently? Yeah. What's the probability of playing three games? Now the key here is we knew the probability of picking the fair coin in the first place. Somebody talked about the series lasting three games. So what's the probability of A and B? This probability shows the chanc… Similarly, if you lost you're sort of down and the other guy has a better chance of beating you. Every single city, American Airlines is better. So we have event A, event B, and A intersect B. They're weighted very differently. Now here you're on the path where you lost the first game. 0% is less than 50. Here's America West. What's the probability the test is right in general? B will be the event that we win the first game. And they're not disjoined because there's a sample point were two or more of the die could come up the same being a winner, which means the same sample point, namely all die are N, comes up in each of these three. I didn't technically assume that. It's hard. Here's the fair coin-- heads and tails. All right. Now you could understand some of the departments accepting more male PhDs than female PhDs. And now I need to figure out for the sample points what's their probability. All right? Any questions about that? What's the probability I'm holding the fair coin? Video by Art of Problem Solving's Richard Rusczyk, a MATHCOUNTS alum. Might be I've drawn it as 1/3 third if it was uniform. » But like most medical tests, they're not perfect. It's 10% false negative right, 30% false positive right. So how could there be any probability there? Of course, if I'd have shown you tails, You would have known for sure it was the fair coin because that's the only one with the tails. There's a disease out there. Won the first, lost the second. 1/3 because what you would do is add up these three sample points. So you can't-- Carroll didn't apply. All right? And every time it comes up heads. OK. Any questions? Flash and JavaScript are required for this feature. » And just for fun, let's make A be here-- include all of them. For now, normal dice-- nothing fishy. We know that now. Let's take-- say C and D are disjoined. Instructor's Note: The actual details of the Berkeley sex discrimination case may have been different than what was stated in the lecture, so it is best to consider the description given in lecture as fictional but illustrative of the mathematical point being made. Probability of Single Events: Videos. So instead of a single heads I get K straight heads and no tails. 1. We can say if a person drives well there is less chance of him to meet with an accident. FCS is the event that the applicant is female and applying to CS. And the beauty here is it's really easy to do. But the medicine for it you take for six months. Let's see. blsce B suq C couscc poxse. Now nobody would do this. And they sample thousands of voters from 1% of the population. Of course, you could imagine debating this for awhile, arguing with somebody. Right? So the test is pretty good. Welcome; Videos and Worksheets; Primary; 5-a-day. Now suppose I pick one of these at random, 50-50, I pick one of these things, and I flip it, which I'm doing behind my back, and lo and behold, it comes out and, you see a heads. So that starts to look a little more complicated. So the probability of A given B is the probability of A and B is p over 2. I need the Ai to be disjoined for this to be true-- the events to be disjoined. What's going on here that make it so that it's not a less than when you look at the union of the departments? Menu Skip to content. And the proof is not hard. If she wins the first set, the probability that she wins the second set is \(\frac{9}{10}\). 1/3, because I just lost the last game. How many people think 1/2? We'll start with an easy one. And it will look a little different each time maybe. So one answer is, I picked the fair coin with 50% probability. That's good. So it's not likely to go three games. And I want to compute the probability of A given B. So for example, in the Monty Hall problem, what's the probability that Carol places the prize in box one and that's the box the contestant chooses? You're going to be right almost all the time. What's the probability of A given D? But I don't get to count them separately. I'll take that down. Much better than the test you paid a lot of money for. What makes them equal? Actually, the woman won the lawsuit. What's the probability of A given C? So if you want to compute the probability of A1 and A2 and all the way up to An, that's simply the probability of a 1 happening all by itself times the probability of A2 given A1 times-- well, I'll do the next one-- times the probability of A3 given A1 and A2, dot, dot dot, times, finally, the probability of An given all the others. What if I do the same game? All right. It doesn't matter which happened first in time. The only outcomes left are in B. Then we have the flip. Why does it come out that way? So you either win or you lose the game depending on if your lucky number came up at least once. Yeah? I flipped it 100 straight times and it's heads every time. Or, using the formula, that's just probability of B and B divided by probability of B. Now we're conditioning on the fact that B happened. But I've got huge weightings on these guys, which changes the numbers quite dramatically. All right. In fact, they give a special name these kinds of things. The following content is provided under a Creative Commons license. Mathematics for Computer Science That's these guys. But it tested positive. So she had to get some mat to back her up. It's 2/3. Sometimes it seems like this in Boston. The probability of having the disease, what is that for a random person? Now this sample point qualifies, that one does, and this one. And in fact, if you try to do fancy stuff, you're probably going to be wrong more of the time. Textbook Solutions. What's the probability this is the fair coin given that you saw a heads after the flip? So this is not true. And she was denied tenure. Videos, worksheets, 5-a-day and much more. And you write that down as a probability of A given B. Is this true? Probability I win the first game is 1/2 times the probability I win the second given that I won the first is 2/3. So we're going to look at the following events. So when are those equal? The probability I win the first is a 1/2. Can I swap things around? Here is the unfair coin-- heads and heads. There's two branches. And this extends to more events. The only difference is that in a probability space, we have weights on the elements. they're based in Phoenix. So you get a higher on-time rate when in a good-weather city. Venn diagrams and probability. 0.09 plus 0.27, which is 0.36. OK. What would you say? And that is then 1/2. 1/3). There's A, B, A and B. And that's the generic case is they're different. All right. OK. All right. Learn conditional probability at your own pace. Maybe I'll leave-- no. And 400 over 450 for 89%, 320, over 450, 71%, 200 over 260 for 77%. Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. A few. But of course, no matter where you're going you're better off with American Airlines. Well, it's just the probability of B and A over the probability of A. So first, do you have the disease? After all, I picked it with probability 1/2. Then we have lose-win-win, lose-win-lose, and then lose-lose. Yeah. And now, game three-- well, it doesn't even take place here. Now if, you don't have the disease, you could test either way. Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; September 4, 2019 corbettmaths. How many people think it's less than 1/2? You got huge waves here. You have to be a little careful though because you got to remember which side you're doing, which what you're putting on either side of the bar here. Conditional probability concepts are presented in this interactive lesson from Math Goodies. So what she did is she looked into Berkeley's practices and she found that in all 22 departments, every single department, the percentage of male PhD applicants that were accepted was higher than the percentage of female PhD applicants that were accepted. And you can see why this makes sense when the picture-- say this is our sample space. So you have a best 2 out of 3 series. Now we define the events of interest. So how could we be discriminating against women? So you have next week for recitation. Yeah. Actually, who do you think had a better-- Yeah. When probability A equals probability B. And then lastly, what's the probability I have the win on the third game here? Probability: tree diagrams Video 252 Probability: conditional Video 247 Maths Genie keyboard_arrow_up. Say that-- let's do it over here. This is from a newspaper article on which airlines are best to fly because they have the best on-time rates. All right. OK. Somebody even said 1/3. Back to Top. OK. Now we're back to the election and there's a pole that says they sampled 1% of the population at random and 60% said they're going to vote Republican. This means that in every 50 matches, she may win the second set 37 times (ie 37 becomes the denominator of the conditional probability). That would be P being 0. Plus, the probably of all of them matching. But it does here. You don't know. And where else? Good. It's just that sample point. The probability of winning today's game given that you won yesterday's game, the probability of winning the series given you already won the first game. I mean, now if I look at exactly this data I might side-- I might side with Berkeley looking at these numbers. Let's figure out the answer. And B is the event that the person tests positive. It's just drawing out the tree and computing the conditional probability. All right. So we normalize. I'll get P over 2 minus P. So the probability with which I picked the coin to start with impacts the answer here. It's really, really easy. Back to Top. Our tips from experts and exam survivors will help you through. What's the probability of A and B? 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. So if I look at the average of the percentages here, well it's 35% for the women versus 75% for the men. This video explores Conditional Probability, a key concept in IB Maths HL Topic 5: Statistics and Probability. OK. Now in addition, all those set laws you did also work for conditional probabilities. And then same thing here. Then I only win once in that case. Use these websites to find topic revision: video examples, practice exam questions and worked solutions: Maths Genie ¦ Corbett Maths ¦ Mr Barton Maths Takeaway ¦ Mr Barton Maths Topic Search ¦ Just Maths 10.1 Combined Events - Use the product rule for finding the number of outcomes for two or more events - List all the possible outcomes of two events in a sample … It sounds like it's-- and just to avoid any confusion here, people only apply to one department and they're only one sex. Roulette, much better. Maths Student videos Venn diagrams; Sets; Unions; Intersections; tutor2u Maths The tutor2u Maths … It's the same formula. I mean, so the immediate conclusion is, well, that's clearly there's sex discrimination going on at Berkeley. If you don't have the disease there's a 30% chance you test positive. No. Home Actually, they make these things look like quarters sometimes. So it's-- actually, we'll call it 0.1. And it's the opposite of these two. That's 1/3 plus 1/18. It's easy to do. She tested positive and it's only got a 30% false positive rate. 1/2, 9/18. So say this was the story. It should because you got to plug something in there for the 1/2 for this to work. The probability of A given B is the probability of both happening, the probability of A and B divided by the probability of B. We're going to compute a conditional probability there. I got these three points. All right? And both A and B are events. OK. All right. Yeah. So let's figure it out. Now what's the principal you used two weeks ago when you did cardinality of a set-- cardinality of a union of sets? Say I flip it 100 times. So the newspaper concluded and literally said that American West is the better airline to fly because they're on-time rate is much better. So we're just going to look at two departments here and try to figure out if it can happen that in both departments the women are worse off but if you take the union they're better off. So for each event that I care about I figure out which sample points are in that event. Be sure to visit his website for top-quality GCSE and A Level revision resources. A bunch. That's what they want us to figure out. Well, that's the probability that the contestant chooses it is 1/3 times the probability that Carol put it there, given the contestant chose it, or actually, vice versa, Is 1/9. And it's easy to work with using the formulas. OK. 0.1 times 1 is 0.01. And then there's the test. Why is this happening? And this is called a false negative, because the test comes back negative but it's wrong, because you have the disease. All right. But it really depends on the numbers as to which one you might, if you had to vote, which way you'd vote. Probability of being admitted in EE if you're a female is less than if you're a male. Read about our approach to external linking. So we've got a sample space here. Next Relative Frequency Video. The product rule says that the probability of A and B for two events is equal to the probability of B times the probability of A given B. And I encourage you to pause this video before I work through it. That's what you can conclude because the population really is fixed in this case. And similarly, the probability of winning after a loss is 1/3. Doesn't matter what happened on the first. How many people think that it's a least a 70% chance you got the disease? Now, the example from Monty Hall-- and actually, we had several-- but you might have B being the event that Carol places the prize in box 1. And we use our formula. Is that a theorem? You can go through each individual law but it all comes out to be fine. And they said that if the university as a whole-- look at the University as a whole, actually, the women, the females have a higher acceptance rate for the PhD Program than the men. Therefore, given that she wins the second set, the probability she won the first set is, Home Economics: Food and Nutrition (CCEA). How many people think it's better than 50-50 you have the disease? Back to Top. But we won't go through it. So it would be 0.09 plus 0.63. Balbharati Solutions (Maharashtra) Samacheer Kalvi Solutions (Tamil Nadu) NCERT Solutions; RD Sharma Solutions; RD Sharma Class 10 Solutions; RD Sharma Class 9 Solutions; Lakhmir Singh Solutions; HC Verma … Anybody think it's not a fair game? Corbett Maths; Mathster keyboard_arrow_up. And in this case they were comparing American Airlines and America West, looking at on-time rates. Let's see. These videos are made by Corbett Maths. Absolutely. Last time we began our discussion on probability with the Monty Hall game-- the Monty Hall problem. 1/2. So if I extended this nice proof technique I couldn't have probability of 7/6 of winning with seven die. I see. Explore materials for this course in the pages linked along the left. These videos are made by Corbett Maths. PROFESSOR: That's a great point, great point, because there's additional information conditioning this in the personal example I cited. 10%. And even that's a terrible game to play. Anybody tell me places where it's come up? Corbett Maths with a thorough video including how we can use Venn diagrams to calculate probabilities. Yeah? OK. Any questions about that example? All right. 86 versus 77. Say I picked the fair coin with probability P and the unfair coin with 1 minus P. And this is the same heads and tails, 1/2, 1/2. Menu Skip to content. So that becomes 1/3. Because presumably the one you paid for is better, even though accurate less of the time. What you want to know is the probability they have the disease given that it's a random person. Because if you look at how many people have the disease and test positive, it's 0.09. Let's compute that scenario. And then in EE, 70 out of 100 women were admitted compared to the men, which had 1 out of 1. Well, the probability that I picked the fair coin is 0 over 2, which is 0. And you've got 6,520 over 7,260 for 90%. It's historically been a difficult area. They had a lot of flights. Yeah. That's right. So it looks like an easy game. And they got most of their flights there. All right. The fair coin is equally likely to be heads or tails, each with 1/2. This sample point is win-lose-win. Both independent and conditional probability are covered. This one is 0.1 times 0.9 is 0.09. So we're going to do the same thing over and over again because it just works for every problem. Well, this is just A intersect B because you still have to have A happen, but now you're inside of B. That equals the probability of B and A over a probability of A. So I've got 2/3 and 1/3. May 22, 2020 corbettmaths. All right. This video shows examples of using probability trees to work out the overall probability of a series of events are shown. Let's draw that. That's just the probability of A and B over the probability of B. But they did their own study of PhD applicants. For example, the sample space for rolling a normal dice is {1,2,3,4,5,6} … And the player wins if and only if the number he picked comes up on at least one of the dice. AUDIENCE: [INAUDIBLE] you have to take into account [INAUDIBLE]. So many people don't have the disease that even with a small false positive rate, this number swamps out that number. When it is hard to find a worse game than this it you take for six months where. Over 450, 71 %, 0.1 all ads for each corbett maths conditional probability video that we the... Be different on-time rates, the reason we have, which is 1 and equations. Third plus 1/18 plus the extra one, 1/9 looks something like the.! 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