 Hi everyone, it's MJ and this is question 4 of the September 2017 paper. This question is on probability and it is very difficult. So I would highly recommend that you give it a go first before watching this video as this is brilliant, brilliant exam practice. So pause the video and have a go. Okay, you've decided to continue watching so let's get into it and let's see what we can do here. Okay, we are looking at an airline and it's analyzing the punctuality of its scheduled flights. It measures departures and arrivals and classifies them as early, on time or late. From its records, no flights depart early, 85% departs on time, arrivals are early 10% of the time and late 20% of the time. And the very first question is determine the probability that a flight arrives on time. Okay, and first thing that we need to do is let's set up some notation. So let's have the following notation. Let's say that the probability that our arrival is early, oh yeah, let's use this as our notation. Arrival can be on time and the probability that arrival can be late. So I'm using AE to mean arrival is early, and we're going to do the same for the departures. So I want to know what is the probability that our departure is early, the probability that our departure is on time and the probability that our departure is late. Okay. And the question is asking what is the probability that our arrival is on time. So this is actually the answer for question number one. And we can use the results that the probability of arrival, whether it's being early, late or on time, is going to be equal to one. Okay, and the probability that departure given all those other ones is equal to one. So it's an assumption that none of the planes blow up in the sky or that none of them fail or crash. I mean, you don't really get a mark for saying that, but it kind of is implied. So what we have from its records, no flights depart early. So the probability that the departure is early is equal to zero. On another way, what we can do is just eradicate that and almost have departure as just being two things. And we see that the parts on time, 85% of the time, which means one minus 0.85, we're going to have the parts late 0.15. Arrivals are early, that is 10%. Late is 20%, which means arrives on time is going to be 0.7. Okay, cool. And that is the answer for question one. So it looks easy. This looks like it's an easy question. It's not too bad. Well, hold on, it does get quite tricky because yeah, we then given further information and the trick with this is not just to read it and then be like, oh, okay, that's interesting. No, try and write out these pieces of information in the notation of that we've been using. So the probability that none of the flights, none of the flights that depart late are early arrivals. So the probability that our arrivals are early, given that the departures late is equal to zero. Now this can also give us a whole bunch of other information because what we know from this is we can use all of our relationships. And I think there's some, some relationships, I want to show you the probability. This is using, you know, when conditional formulas, probably of E j given a is equal to the probability of E j and a divided by the probability of a that's conditional probabilities. We then also have the following relationship E j and a is equal to the probability of a and E j, which is therefore going to be equal to the probability of E j, given the probability of a given E j, which itself is also equal to the probability of a probability of E j given a. Okay, these are, we feel like where these come from, these are the stuff that we looked in the course. These are our results that we found. And of course, there is the law of total probability. Probability of a is equal to the sum of the probability of a union with our E i's given in i equals one, whatever like that, law of total probabilities. So we can now use this, use this information to, to make some other things. So we're going to be told that the probability that we have an earlier rival, given that the partial later is equal to zero. This is going to mean that the probability that we arrive early and D L is going to be equal to zero. Okay, now why that is important is because if we use the law of total probability, we see that the probability of arriving early is equal to the probability of arriving early and D zero plus the probability of arriving early and departing late. Now, if this here is equal to zero, it means these two are going to be equal to each other, which means we also have this result here. So see how we've taken this information and we've been able to figure out this using this rule over here. And then we've used this rule over here to figure out what this thing is over there. And this is what makes it difficult. If you're like, whoa, what just happened? It is because this is a difficult question. Okay. This is not supposed to be easy. What we are doing is we are using the rules of probability to find out more pieces of information. Okay. And that means when we look at our question, okay, and this question is saying, determine the probability that we will arrive early if it departs on time, given that it departs on time. Okay. That's what we want to figure out. So probability AE, given DO, we know from number one over here that we're going to have probability DO given AE divided by the probability of DO. Okay, that's what we have. We know that DO is equal to 0.85. How do we know that? Because of that over there. And we've just shown over here that we know this value over there due to the fact that that other one was equal to zero, which means 0.1, which means we get our lovely little answer of 1176. And doing that does give you three marks, and that is a difficult, difficult, difficult question. That's like I said, you're using conditional probabilities formula along with the law of total probability to come to that result. So well done if you got that. If you did this question beforehand and you got that, then wow, that is very, very good. Otherwise, don't be disheartened. Like I said, this is tricky. This is very, very tricky. Okay. Done question one. Question two. Listen, I look at question three. Okay. Question number three is show that the probability that a flight, see now, what I love is actually giving us the answer. Show that the probability that a flight both departs on time and arrives on time is 0.665. So we're looking for the following. Both arrives on time and departs on time. Okay. So this is what we are looking for. Okay. Now, what are we going to do? How are we going to figure this one out? Well, we know that the probability that it departs on time using the law of total probabilities is equal to the probability that it arrives on time and it departs on time plus the probability that it arrives early and it departs on time plus the probability that it arrives late and departs on time. Okay. We know this from, once again, the law of total probabilities and we know that the DO over here is equal to 0.85, AODO is what we're trying to find out. We're trying to show that that is equal to 0.06, but this AE and DO, oh hello, we've got this from the previous question. So we know that that is 0.1, but what is this AL union DO? That is the tricky one. And what we can do, what we can do is we can say, hang on, hang on a second, let's look at our question. Let's look at our question over here. And this is 10% of the flights that depart on time are late arrivals. Okay. 10% we've got depart on time and we've got late arrivals. Hmm. Hmm. Maybe we can use this. What we know is that the probability of AL intersection with DO is going to equal to the probability of AL given DO times the probability of DO. How do I get that? I'm using conditional probability number one just in a different format and I'm also using number two. Okay. What we know is, well, hello, they just gave us this answer of 0.1. We know that this is equal to 0.85 and we see that we have 0.085, which means we now have plus 0.085. So what we can do is we can say that the probability of AL union DO is equal to the probability of DO minus the probability of AE and DO minus probability of AL union DO. And that's going to equal to 0.85 minus 0.1 minus 0.085. And that is going to give us our answer of 0.665, bam. And we've got those two marks. If you got number three, you probably got number two. If you didn't get this one over here, you know, yeah, this because you kind of need this one going forward. But then they were nice. They give you this answer so that you could sort of continue. Okay. The number four is worth three marks. So this one's also going to be a little bit tricky and says determine the probability that a flight will arrive on time if it departs late. Given DL, okay. So this is what we want to do. Probability of arrives on time given DL. We need to see where else have we used this. This is going to be equal to probability AO union, I always say union, sorry, intersection DL divided by probability of DL. Now we know what the DL is equal to and do we know what AO DL is, okay. And I believe we do know what it is. Hold on, do we know what it is? No. I was thinking that we could use this, hold on, don't get confused. Don't get confused over here. This is arriving late and departing on time. This is arriving on time and departing late. There are two different things, which means we need to first work out what this one is. So if we're arriving on time and we're departing late, this is going to be equal to us, so the probability of us arriving on time, okay, arriving on time and then hold on, hold on, we can use this result, we can use this result because, because, hold on, this is how we do it. This is how we do it, okay. If in doubt, use the law of total probability. So we know AO is going to be equal to the probability of AO and DO plus the probability of AO and DL, okay. We know that because of the law of total probability. We know that this is therefore going to be AO is equal to 0.7. This is equal to 0.665, which means this guy over here has to be equal to 0.035, tada. And then we can now use it back here, now that we've figured this one out, we can use this back here to have 0.035 divided by DL and we have DL, what is DL, there we go, 0.15, 0.15 and we're going to see that the answer is equal to 0.233, whoo, okay, that one was a little tricky because in order to find this value, we had that value, we had to use the law of total probability, gosh, this question is difficult, I'm struggling here, but determine the probability, this is the final one, that if a flight arrives late, it departs on time. So we want to know the probability that if it arrives late, it departed on time, this they try to trick us with the semantics or the way of the words, because we do want to write it like this and not the other way around, it's tricky because they say it arrives late first, you might throw it around. The probability that it departs on time, given that it arrives late, and this is just appreciating that our probabilities can be flipped around like this, so let's do it. The pot's on time, given that, or if it arrives late, it is going to be equal to the probability that it arrives late, that's the probability, and we know this guy over here because it is this one over here, which is the 0.085, 0.085 and AL, what we know AL is 0.2, and there we get the answer 0.425, and there we go, we are done, we are done, this is a very difficult question, very very difficult question, it gave me some troubles as well, so if you are struggling with the try find another exam question that gives you more practice or even just try and attempt the question again on your own and see if you can do it, like I said, it is difficult and you are using the probability theory that we discussed in the earlier videos to a large degree, this was a tricky question, but I hope you did well, if you've got any questions and I'm expecting there's going to be quite a few, please use the comment section below, thanks guys so much for watching, cheers.