So let's do an example. All right. So to this point, we've seen examples of a conditional probability where it's A given B where A follows B, like, we're told B happened. You don't know. 4,900 out of 5,300 for 92%. Probability: tree diagrams Video 252 Probability: conditional Video 247 Maths Genie keyboard_arrow_up. Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; June 18, 2013 August 20, 2019 corbettmaths. I mean you're pretty sure you got the unfair coin because you never saw a tails. So I'm just going to give you the numbers here. Maybe I'll leave-- no. All right. Yeah? Now of course, if it comes out 100 straight times heads, you've got to be really unlucky to have the fair coin. Well, that sounds pretty good. So A is the event the person has the disease. Description: Covers conditional probability and its applications to examples including medical testing, gambling, and court cases. FCS is the event that the applicant is female and applying to CS. All right. These videos are made by Corbett Maths. So to do that I got to figure out which sample points are in A and B here. And I'm going to do not the actual data but something that's represents what's going on. Sometimes it seems like this in Boston. And you could win or lose here. They'd get sued and get killed for that. For example, what about this one? Any questions about the definition of the conditional probability? And you can see why this makes sense when the picture-- say this is our sample space. So let's figure it out. Now nobody would do this. Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; September 4, 2019 corbettmaths. All right? Now in this case-- in fact, this is a question we had on the final exam a few years ago. Any questions? All you can conclude is that either the population is close to 60% Republican or you were unlucky in the 1% you sample. But of course, no matter where you're going you're better off with American Airlines. Venn diagrams and probability. Well, you can be positive or negative. A is the event you have the disease. It is what it is. The game was played. Given that the tennis player wins the second set, find the probability that she won the first set. Any questions about that? I'm going to see another example in a minute where it's even more clear which side to believe in. In fact, there is intersection because there's a chance I rolled all six-- all Ns. All right. All right. I'll get P over 2 minus P. So the probability with which I picked the coin to start with impacts the answer here. But we won't go through it. So that's a proof that we win with probability of 1/2. So it's 1/6 times 1/6 minus the 1/36. And it's just the product because the product rule saying product of the first probability times this one, which is the conditional probability of being here times this one, which is a conditional probability if these events happened before. But you don't know who won the first game. American Airlines is better, every single city. And B is the event that the person tests positive. What's the probability the test is right in general? Videos, worksheets, 5-a-day and much more. Conditional Probability - Venn diagrams : 1: 2: 3: Corbett Maths keyboard_arrow_up. You're very close. We're almost done. Where's the formula for that? 5-a-day … Do another example with that exact kind of thing is very important. At least one tail for this one. You know you won the series. » The fair coin is equally likely to be heads or tails, each with 1/2. Everywhere it's better by a bunch. Now, in general, you have something like the conditional probability that an event, A, happens given that some other event, B, has already taken place. We already computed the probability of A and B. 1/2. So the proof is really the same proof you use for inclusion, exclusion with sets. This video explores Conditional Probability, a key concept in IB Maths HL Topic 5: Statistics and Probability. So I get 1 over 2 minus 1. And then finally we have MEE is the event the applicant is male and in EE. they're based in Phoenix. So you cannot do those set rules on the right side of the conditioning bar. Why is this happening? And the probability of B is P over 2 plus 1 minus P. That's P over 2 up top, one minus P over 2, and that is all multiplied by-- what am I going to multiply-- 2 here. Our tips from experts and exam survivors will help you through. AUDIENCE: The probability changes depending on the result of the previous game. Back to Top. Let's draw that. All right? This would happen with 1 over 2 to the K. This would happen with 1 minus 1 over 2 to the K. So this is now p over 2 to the K. This is now P1 minus 2 to the minus K. Let's recompute the probabilities. Conditional Probability video tutorial 01:18:22; Conditional Probability video tutorial 01:56:16; Advertisement Remove all ads. And his proof is that three other mathematicians agreed with him. 0.09. And let's figure out the probability for each one. For example, the sample space for rolling a normal dice is {1,2,3,4,5,6} … Now say you select a random person and they test positive. We don't offer credit or certification for using OCW. The conditional probability that Rahul eats pizza for lunch, given that he eats a bagel for breakfast, rounded to the nearest hundredth. This you just would never make this mistake again seeing that example. What do you think? But in this case, the probability of A given B is less than 1. Well, the probability that I picked the fair coin is 0 over 2, which is 0. Here's American Airlines. It's just drawing out the tree and computing the conditional probability. It's historically been a difficult area. Electrical Engineering and Computer Science Say that for women in computer science, 0 out of 1 were admitted compared to the men, were 50 out of 100 were admitted. So the teams are matched 50-50 for the first game. Conditional Probability … The probability of A given B is not always the probability of B given A. And we'll figure that out next week. This information can be used to show a relationship between the two events. OK. Probability of Single Events: Videos. And she was denied tenure. Just don't even think about it. 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. Much better than the test you paid a lot of money for. So that's an example where they would be different. But the medicine for it you take for six months. Good. And now, game three-- well, it doesn't even take place here. Previous Negatives: Addition and Subtraction Video. Yep. Won the first, lost the second. I could roll all sixes and then each of these would be a winner. Contents. So I've got 7/18 over 9/18. There's a disease out there. For example, what if I picked the unfair coin for sure? You use the same mathematics. Any questions about that? Game one, you can win or lose. All right. In fact, this is the logic that we've just debunked over there-- exactly that claim. Now, this is a nice example because it shows how a rule you had for computing the cardinality of a set gives you the probability. And you just say, it's going to rain. OK? 1/36. May 2020 This video shows a method for solving conditional probability problems. OK. Any questions about that example? Like, we just had an election. Now once we've conditioned on that, all this stuff outside of B is no longer possible. It's just the product rule. So instead of a single heads I get K straight heads and no tails. So I got all the probabilities. What does that tell you about the population of the country? And which argument would you buy now? Yeah. And then same thing here. Probability-- there's no intersection. Made for sharing. And in each case they're going to expose a flaw in our intuition about probability. That's this one and this one. Yeah. So we've now expressed their arguments as conditional probabilities Any questions? Now in general, there's a very simple formula to compute the probability of A given B. Because I picked it with 50% chance, what does the information do for you? Very simple to do, which is good. Everybody understand the example, how it's clearly not always the case that probability of A given C union D is a probability of A given C plus probability of A given D? That's one. PROFESSOR: They could double up. Suppose you have to calculate the chance of a person meeting with a road accident. So this sample point we'll call win-win. 1/3, because I just lost the last game. Well Berkeley took a look at that and said, nothing good. PROFESSOR: You're right. OK? Conditional Probability … 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. Use OCW to guide your own life-long learning, or to teach others. Menu Skip to content. In fact, let's figure out. Game two you can win or lose. So we're going to do the same thing over and over again because it just works for every problem. What do you think? And that's just follow straightforwardly from this definition. Can anybody look at that and tell me? What do you want to say? Yeah. Which one? Visit Art of Problem … And then here I just won a game. And it will look a little different each time maybe. You can do them on the left, not on the right. All right. We haven't talked about independent. A couple people like 1/3. Back to Top. And of course what I want to know is the probability that I chose the fair coin given that I saw a heads. OK. So the way it works is as follows. And here we're going to use the general product rule to get it. You don't want to get it. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. Conditional probability occurs when it is given that something has happened. All right. In fact, imagine nobody had the disease. blsce B suq C couscc poxse. So this is sort of paradoxical that it looks like a pretty good test-- low false positive, full false negatives, but likely be wrong, at least if it tells you have the disease. And this is called a false negative, because the test comes back negative but it's wrong, because you have the disease. 72%. 2.!Two fair six sided dice are rolled. And the same thing needs to be done here. In fact, all the set laws you learned a couple weeks ago work for probability spaces the same way. And that's what's doing it. We know that if I swapped all these, it's true. This video is accompanied by two exam style question to further practice your knowledge. Is this true? Well, in fact, the reason we have that rule is because that is the same as the product rule. Might be I've drawn it as 1/3 third if it was uniform. So I've got 2/3 and 1/3. The following content is provided under a Creative Commons license. Assuming the probability of B is non-zero than the probability of A given B is just the probability of A and B happening, both happening, divided by the probability of B happening. All right. American Airlines got a lot of flights in Seattle where the weather sucks and you're always delayed. Many years ago when my wife was pregnant with Alex, she was exposed to somebody with TB here at MIT. It should be one option all heads and another option at least one tails. All right. For example, what if I hadn't told you the probability that I picked the fair coin? You don't get symptoms right away. All right? And so they look worse on average because so many of their flights are in a bad city and so many of America West are in a good city. So it's the same math. No. Now what's the principal you used two weeks ago when you did cardinality of a set-- cardinality of a union of sets? FEE is the event that the applicant is female and applying to EE. The waiting. OK? And even that's a terrible game to play. Conditional probability occurs when it is given that something has happened. So I got 0.09 over 0.36 is 1/4. I picked the coin, 50-50, behind my back. What's the probability of A given B in this case? 5-a-day GCSE 9-1; 5-a-day Primary; 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. occurs when it is given that something has happened. So we normalize. It is not always true. How many people think it is a fair game-- 50-50? This now becomes-- the probability of A given B is this share of B weighted by the outcomes. Now we're conditioning on the fact that B happened. Now the key here is we knew the probability of picking the fair coin in the first place. The unfair coin, guaranteed to be heads, probability 1. Then you subtract off the double counting from here, from here, and from here. That seems bizarre. Like if K is 100 I got a big number down here. It's not 1. 0.36. Now, there's a nice rule called the product rule, which follows from the definition very simply. Well now, instead of 1/4 I get P over 2 up here. So the math is the same with a postieri probabilities. 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? That's all I'm conditioning on. Last time we began our discussion on probability with the Monty Hall game-- the Monty Hall problem. So to do that we plug in our formula. Actually, the woman won the lawsuit. It's a fancy name for just saying that things are out of order in time. The probability of winning any particular game is influenced by the previous game. And 9.9 you don't have it. For example, say you're trying to predict the weather for Seattle. So let's do another simple example that'll start to maybe be a little more confusing. Right? What's the probability of B? Game one is 50-50. AUDIENCE: [INAUDIBLE] The options you have, is it all heads or is it all tails? This number here is so small. Now we can compute each sample point probability. There's no randomness in the population. And you've got 6,520 over 7,260 for 90%. So it gets very unlikely that I've got the fair coin here as K gets big. Just say it's going to rain tomorrow. First, represent the information on a tree diagram: From the tree diagram, the probability of winning the second set = \(\frac{27}{50} + \frac{10}{50} = \frac {37}{50}\). And there were several of those in homework that you just had the last problem set. So she asked the doc, what's the probability I really have the disease? That's 1/9. Say that-- let's do it over here. Well this was 7/18 over 9/18. All right. So if we took A and B as defined up there, that's the probability that Carol places it in box one and the contestant chose it. And the same thing with designing the poll if you're way off. Same thing here is 1/9. So let's fill in the other probabilities here. If you don't have the disease there's a 30% chance you test positive. The probability that a tennis player wins the first set of a match is \(\frac{3}{5}\). And it's called a false positive because it came back positive, but you don't have it. Not plus, union. If I'm in C I'm in A. A1 intersect A3 minus probability of A2 intersect A3. Therefore there's a 90% chance you test positive. Maybe it's worth writing down what happens if that's a variable-- sum variable P. Let's do that. You can go through each individual law but it all comes out to be fine. So they're fair dice. 30 here and 70% percent chance you're negative. 1/3 because what you would do is add up these three sample points. The probability that I lose the second given that I won the first, that's 1/3. The probability of having the disease, what is that for a random person? The probability of heads is 1. So the probability of A given B is 1/3. If she wins the first set, the probability that she wins the second set is \(\frac{9}{10}\). It really is one of the very best websites around. Send to friends and colleagues. And A intersect B, That's the only point. Download the Activity Sheet here. Here's another example. So the probability of A given B is the probability of A and B is p over 2. Can you sort of see why this seems contradictory? So what's one place conditional probability is entering into this problem? So let's figure out the probability that you won the first game given that you want the series. All right. We'll start with an easy one. Sometimes when it says you've got the disease you really don't. that the stat. Now, I'll give you a clue. And it came back positive. And you see heads. So we're going to look at the following events. Probability … (Hint: look for the word “given” in the question). Probability Video Videos; probability; Post navigation. And you've got C here and D here. What's the probability the series lasts three games? There's an intersection here. Right? Let's look at that case. So now we're looking at the event that B is K straight heads. OK. Now the data for American West looks something like the following. All right? These videos are made by the award winning Hegarty Maths. 86 versus 77. And all we care about is the probability that A happens inside this new sample space. In LA it's 700 out of 800 for 87%. Does that sound right? OK. It's easy to do given the tree method. So what went wrong with the intuition, which you didn't fall victim to, but people often do, that it shouldn't have been possible given that? What's the probability that you tested positive? When probability A equals probability B. Anybody notice anything unusual about that answer here? All right? What's the probability-- is this going to do what I want? the second time will be the same as the first (i.e. Welcome; Videos and Worksheets; Primary; 5-a-day. 3/4, 1/4 plus 1/2. Yeah? What if I do the same game? Yeah? OK. Therefore, given that she wins the second set, the probability she won the first set is \(\frac{27}{37}\). All right. What makes them equal? The doc doesn't know. Both independent and conditional probability are covered. But like most medical tests, they're not perfect. And then you would always be wrong if you said you had it. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. So it looks like an easy game. And the proof, question mark, is you let Ai be the event that the i-th die comes up N. And i is 1 to 3 here. But I've got huge weightings on these guys, which changes the numbers quite dramatically. Back to Top. 35 is less than 75. 1/4. So the test is pretty good. And that is the probability of A1 plus the probability of A2 plus the probability of A3. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers. All right? Now the bad thing-- TB's a bad thing. Number on time, number of flights, and the rate. But it really depends on the numbers as to which one you might, if you had to vote, which way you'd vote. And that is then 1/2. If you're in B-- wait. And let's say they're fair and mutually independent. Well, it's just the probability of B and A over the probability of A. There's no signup, and no start or end dates. 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. Which airline would you fly looking at that data? But it could be. And just for fun, let's make A be here-- include all of them. A is the event I chose the fair coin. All right. How many people think that it's a least a 70% chance you got the disease? Further Maths; Practice Papers; Conundrums; Class Quizzes; Blog; About; Revision Cards; Books; June 15, 2013 August 21, 2019 corbettmaths. Come up. Videos, worksheets, 5-a-day and much more. Probability of B given A. But really, if you just do this is it's going to be very easy. 72%. It's those three sample points and they better add up to 1/2 because we sort of said, the probability of the first game's 1/2. Anybody think it's not a fair game? Is it 1/2 or is it 1/3? And you can have a test for it. Share: Share on Facebook Share on Twitter Share on Linkedin Share on Google Share by email. So I want probability of B given A now for this example. And then there's the test. Visit his excellent website for more videos, textbook exercises, exam questions , 5-a-day, and ... Venn Diagrams Conditional Probability Hegarty Maths keyboard_arrow_up. It's the same as the answer over there. And we use our formula. Somebody talked about the series lasting three games. All right? In fact, we could make a simple example to see why that's not always the case. So let's see what happens. The probability of tails is 0. All right. And here's the data they showed for the two airlines. So as ratios, 70% is less than 100%. Now we get the sample point outcomes. All right. So the female professor's argument effectively is, the probability of being admitted given that you're a female in CS is less than the probability of being admitted given that you're a male at CS. So you can't-- Carroll didn't apply. Now, conditional probability comes up in two places in this problem. Back to Top. Video lessons with examples and solutions to help High School students to understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. And to compute that I got to figure out the probability of A and B and the probability of B. You see a heads. Now, part of what makes this tricky is I told you I picked the coin with 50% probability. 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. That's the other case. 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Only a 1 in 6 over 2 this Share of B weighted by the probability a! Predict corbett maths conditional probability video weather for Seattle on Google Share by email this to work out the probability. A hub of 2,200 for 86 % of him to meet with accident. Not have the best of the fair coin Francisco, and court cases union A3, is the unfair --. Really the same as the product rule, which coin is picked probabilities any questions as conditional any...