A-level Math S1 | Last Class Before Exam | Q/A Session

Added: 2026-05-13

[00:00:40]Okay, Ch. Let's begin. I think it has also started. The recording is also started.

[00:01:22]Okay, everyone. Let's begin. I'm sorry for the delay. Uh let's start. Okay. So, first of all, if you have any general questions, let me know. I'll start with answering questions on Zoom first of all and then uh people on YouTube, I'll also be answering your questions. Then let's begin with some general questions in the beginning.

[00:01:42]Specific questions you want me to solve from past papers, we could discuss them as well.

[00:01:48]>> So okay.

[00:01:53]In questions involving reverse normal or inverse normal, what do we do if we are not getting an exact answer? Exact answer.

[00:02:02]For instance, if we need to add 1 15 to the number, but the third decimal place column has 1 4 and then 1 7. Uh, okay.

[00:02:10]So, in normal distribution, when we're doing inverse normal, in inverse normal, let's suppose you have something like this.

[00:02:28]So, you have probability 7080. Let's say okay probability 7080. So what you would do in that case would be you would look at the table see the value value that is immediately smaller than that that's 7054 right? So the first decimal place is going to be.5 second is going to be four. So we've got 0.54 from there. Now find the difference. It's 26. Right? Now you go to the right side and you are saying what if we don't get an exact value. In this case we are getting an exact value. We're not 26. So we have 26 there, right? There's 26 that we need to add more to it.

[00:03:06]In the table we have 24 and 27. What we do is whichever of those two numbers is closer to 26. That is what we use. So in this case 27 is closer to 26, right? So the third decimal place that we would choose in this case is going to be 8. So we write 0 548. That's the ZV valueue.

[00:03:29]Sometimes what happens is it's exactly in the middle. So for example, if you were looking at this part of the table 8 10 for instance and you ended up having nine there. In that case you're allowed to round off in any direction. You can round off to the left, round off to the right. You can take four or five. Both will generally be acceptable. Mostly in the marking schemes they round up. But it doesn't matter. You can use any of them.

[00:03:59]Okay.

[00:04:02]Zero was your question answered. All right. Let's move on to the next one.

[00:04:14]P3 65 or S1 20. Mosa. I'm sure you're joking, right?

[00:04:20]Aar dashes and stars method. Okay, I I'll go through that quickly. March 23. So, specific questions may be we will come uh in about 15 to 20 minutes in when the the more general questions are answered.

[00:04:39]Uh we will come to specific questions that you might want me to solve from any past papers.

[00:04:50]Uh Farha Farha is is this Faria from 55?

[00:04:57]>> Yes sir.

[00:04:58]>> Faria just a reminder I I I think I remember from your paper uh you were making some I'm sorry. Why am I pointing you out in public? But okay. So I I'll probably just mention that sometime in the middle or just leave me a message.

[00:05:16]Please leave me a message. Have you collected your papers? Have you collected your papers?

[00:05:20]>> Yeah, I did.

[00:05:21]>> Okay. So, make sure that you look at the small errors that I pointed out there.

[00:05:26]Okay. They're important because I was hoping that you would be getting a distinction in I hope you're on track for that. Uh not putting any extra pressure on you, but just putting it out there. Okay.

[00:05:41]Approximation question. Probability should be correct to some places in approximation questions.

[00:05:58]Which approximation questions? Nor binomial to normal approximation questions. Is that what you're talking about?

[00:06:06]May there's no separate rule for significant figures or decimal places.

[00:06:12]There's no separate rule. In general, whenever it's an exact value, you use the exact value. When it's not an exact value, you round off to three significant figures. When you're using a normal distribution table, for using the normal distribution table, for that you need the Z values to be correct to three decimal places and probability values to be correct to four decimal places. But otherwise, the general rule works everywhere else. Is that okay?

[00:06:42]So specific questions we will come to specific questions 2025 May June paper 4 paper 42 seriously are you talking about mechanics or is that a mistake paper 52 right we will come to that okay specific questions March 23 Hamza we'll come to that as well and just answer some general questions first okay for comparisons between data sets of box and whisker plots etc on what basis do we give the comparison Okay. Uh box incript. Let me go to one of those diagrams.

[00:07:26]Okay. In box and whisker plot or cumulative frequency curve based on that they often ask you to make two comparisons.

[00:07:37]Okay. Now, whenever they ask you to make two comparisons, what you do is one of your comments is going to be on some sort of average. It could be mean, it could be median, it could be mode.

[00:07:54]And the second comment is going to be based on variation in the data. That could be uh standard deviation range, interquartile range depending on what the diagram is. For example, we have this stem and leaf diagram or a box and whisker plot. Sorry, we have this box and whisker plot and they ask us to make two comments on the weights of the two groups. So, we we have made two box and whisker plots in the first part and now we need to make two comments in this.

[00:08:23]Now, what two comments? How do we make those two comments?

[00:08:28]One comment is supposed to be on central tendency. Now, what are your central tendency measures? You've got mean, median, mode. Which of those can you find from a box box in the script plot? Median, right? Median is relevant for this. So you'll make a comment based on the median. You can say you you can see on the diagram median for P is greater, right? As compared to median for Q. Now you're not just going to write that the median for P is greater and the median of Q is smaller.

[00:08:59]That is not acceptable in S1. When they ask you to make statements like this, make some comments like this, you have to put those comments in context, the examiner needs to understand that you that that you are able to interpret the given information and based on the context of the question, the comments should be you know related to that. So for example in this case they gave us weights of two groups of people one from country P and one from country Q. When we make this comment on median now we will say on average. So when we talk about central tendency measures we say on average weights of people in country P are more than those of people in country Q. So you'll have to relate t relate the median number to the data that is given.

[00:09:56]So something like this this is one comment that you can make. On average weight weights of 17 year old males in this case the data was for 17 year old males. In country P they are higher as compared to those in country Q. That's what you would write down there. The second comment has to be on variation in the data. Now there are different variation measures measures that we have. Which one of them is relevant for a box box in whisker plot? There are two that are relevant. In fact, one range and the other interquartile range.

[00:10:29]We can't find mean uh we cannot find standard deviation or variance from a box and whisker plot. But we can find interquartile range. These are the interquartile ranges and range of both the data sets. Now interquartile range for both of them is very similar. So we can't really compare that but range is it is clear based on the range the variation in Q is much greater than the variation in P. So what comment would you write then? You would say the variation of weights of those 17 year old males in country Q is greater as compared to that in country P. So again you have to relate it to the data.

[00:11:16]So something like this the variation in weights is more in one country as compared to that in other country. Okay, that's how you would write down.

[00:11:25]Okay, that's now you may have to make these comments based on a box incript sometime sometimes based on uh a cumulative frequency curve as well or a stem and leaf diagram. Just keep this in mind one comment on central tendency the other comment on variation. These are two comments that you make.

[00:11:48]Uh Fatima for the other question you have to give me a specific example. Give me a specific example. Okay.

[00:11:59]Aa in binomial questions where the number of successes we are finding.

[00:12:12]Okay. So the question is uh if we are in binomial distribution for instance let's suppose we have a binomial and the total number n is five. We when we want to find the probability of X= 5 is it necessary to write down the full thing 5 C 5 and then P raised to power 5 and then Q raised to power 0 it's not necessary if you want you can write P raised to power 5 directly as well and that's okay similarly if it's X=0 in that case if you want you can write down Q ra^ 5 directly as well that's okay there's no there's no no problem with Specific questions in a few minutes.

[00:13:12]Specific questions.

[00:13:26]Okay. Are there any general questions apart from this? Are the general questions answered? No other problems apart from this geometric distribution formula Q power Q to power R Q power geometric distribution distribution formula inequality distribution formula.

[00:13:57]Uh let me find that for you.

[00:14:04]Inequality formula geometric distribution. You need to make sure that you memorize that. That's not given in the formula sheet.

[00:14:12]Probability of X greater than R that is equal to Q ra. This formula is not given in the formula sheet. You need to memorize this.

[00:14:20]Equal case. Equal case.

[00:14:25]This one is given in the formula sheet.

[00:14:27]You need to remember both of them. Okay.

[00:14:29]Inequality value questions otherwise they become difficult.

[00:14:36]Okay.

[00:14:37]Aa concepts we are not doing in the session as in if you want me to explain a full concept uh this is not the time for that. You have the crash course playlist available for that. you can go through the complete concept. I can quickly go over uh any specific questions that you might have.

[00:15:10]Okay. So in cumulative frequency curves when they ask you to read values from a graph how much lenency is there?

[00:15:20]They generally give you a range of values in the marking scheme. However, uh the method mark you still get if your answer is correct from the graph that you have drawn. Okay, sometimes they can give you the answer mark as well. If the curve is smooth enough, but if the curve is very uh if the curve is not is not very smooth, in that case you might end up losing a mark on the curve. And then in the next part if based on your curve the answer is correct you will still get that mark. So they generally write that uh in the guidance on the right side of the marking scheme that the answer that you write it should be correct based on your graph. So make sure that you show evidence of where you're reading the answer from on your own graph. That is important. As long as you show that and the value that you write is correct based on your own graph that is going to be acceptable. Okay, let me go through this dashes and stars.

[00:16:23]Uh, let me remind you about the dash and stars method quickly. First remember this this distinction when they ask you for some objects that they should not be next to each other. So this is an example that you would have seen before.

[00:16:49]If we have 12 books, six of them are math books, four of them are physics books, two of them are chemistry books, 12 books in total.

[00:17:00]We want to arrange those books in such a way that not all physics books should be next to each other or not all chemistry books should be next to each other. How do we deal with problems like these?

[00:17:16]What we do is we find the total arrangements. In this case, if it's 12 books, that will be 12 factorial. And then we subtract from that the case when all the physics books are together. Now what this does is it excludes this particular case when all the physics books are next to each other. But the other cases are still allowed, right?

[00:17:37]where three physics books are together or two physics books are together, they are still included in the final result that you get because there's no condition on those other cases. This case is not allowed. We just subtract that from the total. However, when they say something like this that out of those 12 books, there are four physics books and what we want is that none of the physics books should be next to each other. None of the physics books that means that even two cannot be together, three cannot be together, four cannot be together. All of these cases are not allowed.

[00:18:18]So this method dashes and stars method we use this when we want to separate all objects of a particular type. Okay. Now how does this method work? You are going to start from those objects that you want to separate. Which object objects do we want to separate? We want to separate the physics books. They should not be together. We are going to represent them using stars. You make stars for the objects that you have to separate. Okay? And the rest of the objects then you represent them with dashes. So the way you do this is you say there are 12 books. Four of them are physics books. We represent them using stars.

[00:19:10]The rest of them are non-physics books.

[00:19:12]We represent represent them using dashes and we say okay we have these eight possible positions for the remaining books. For the non-physics books we have these eight possible places.

[00:19:29]Since we want to separate the physics books those physics books can come in the gaps between these places. We represent them using stars like this.

[00:19:41]Okay. Now in how many ways can we arrange them? Now for the eight non-physics books what we have is eight books eight places. So they can be arranged in eight factorial ways. They can be arranged in eight factorial ways.

[00:20:00]But then for the physics books now how many places do we have? 1 2 3 4 5 6 7 8 9. We've got nine places for the non-physics books. In those nine places, we want to place four physics books. Now, how do we do that? We can directly say 9 P4 and that will give us the result that we've got nine places.

[00:20:26]Arrange four books in them and that gives you the result. So, what you need to remember in such cases is we are going to use this method when we want to separate all objects of a particular type.

[00:20:38]The objects that we have to separate we represent them using stars. The other objects we represent them using dashes.

[00:20:46]We make dashes and then in the gaps between them we have stars. And then we just do something like this. And that gives us the result.

[00:20:55]I hope that makes sense.

[00:21:01]If there are if there are any specific variations of this that you're confused about, you can let me know about that.

[00:21:09]Let me answer some other general questions in the meantime.

[00:21:26]It's not a good idea to practice a lot of persons uh with your own hands right now at this point. You the last year or the last two years for some of you were for that purpose. You were supposed to practice a lot. Now it's the time for revision. Don't uh try many new questions at this point.

[00:21:46]No. Uh, Ma, you can do that afterwards.

[00:21:49]Uh, once you're done with your exams, revision, there is time for revision.

[00:21:55]You need to make sure that everything that you've learned until now, that's fresh in your memory. That's important.

[00:22:05]Just make sure to go through your notes.

[00:22:07]Make sure to go through your notes and also go through uh any questions that you got wrong earlier. Uh, that's important that you go through that as well.

[00:22:24]for the range questions. For the range questions, I uploaded a short video. So, I I would highly recommend that it's just going to take like 10 minutes or 12 minutes. There are a couple of short videos that I've posted. Uh if anyone any one of these topics is something that you are confused about, make sure to go through this. Let me quickly tell you about what cases I'm talking about. So this case in which in some cases you treat identical letters as identical which is what the standard way of doing such problems is. And in other cases where we treat the identical objects as different. This is one important uh confusion that this is an important concept that a lot of you uh get confused about. Make sure that you understand this.

[00:23:27]Whenever in combination questions, they ask you to find number of possible selections.

[00:23:32]In that case, if you have any identical letters, you are not going to do any selections from them. So for example, if you have two E's, you're not going to say 2C1 for this. If you have two L's, if you want zero L in a particular case, you are not going to do 2C 0 in this case. However, when you are finding probability in these combination questions, the condition for that is these three things should be happening together. It should be a probability problem. There should be some identical objects and it should be a selections problem.

[00:24:11]In this case, you are going to treat all of the identical letters as different and you will apply the combination function on those identical letters as well. In this case, you see in the first case we were not using combinations on E and L. In this case, when we have probability, we do use combinations on E and R as well. Okay, make sure that you understand this. I've explained the logic for this uh in a recent video as well. Go through that if you do not know this. This is another question that I keep getting close to the exams every single year. And that is when you have data sets like this, in what cases do we need to adjust the boundaries and in what cases do we not need to adjust the boundaries?

[00:24:57]This is the summary of that. This is the summary of that in case of histogram and cumulative frequency curve you are always going to adjust the boundaries.

[00:25:05]In case of cumulative frequency curve and histogram you always adjust boundaries no matter whether the data is discrete or continuous. You always bridge the gaps between the boundaries.

[00:25:16]So for example in this case this is 10 this is 11. There are gaps between the intervals. You will use 0.5 10.5 and so on. Right? These will be the boundaries that you use for histogram and cumulative frequency curve. For diagrams, you always adjust.

[00:25:32]But these are some cases in which for discrete you may have to adjust sometimes. For continuous you might not need to adjust or the other way around.

[00:25:41]For discrete you might not have you would not need to adjust. For continuous you will have to adjust sometimes. Now what exactly are those scenarios?

[00:25:49]Sometimes they ask you to find the greatest or least possible value of interquartile range. In case it's a discrete variable, discrete variable in your syllabus are just those variables that can take integer values. Continuous variables are those variables that can take any decimal value as well. For example, if you have length of something, length can take decimal values as well. If you have time, distance, height, all of these variables, they can take decimal values as well. These are called continuous variables. If you have number of something like number of people, number of chocolates, number of something, they are supposed to be integers.

[00:26:25]That's discrete. In discrete variables, we are not going to adjust boundaries.

[00:26:30]When we are supposed to find greatest or least possible values of interquartile range for continuous, we do adjust the boundaries. Now, before we move further, I'll show you a couple of examples of this.

[00:26:42]Look at this data. We have time in this data. We have time in this data.

[00:26:49]Time is a continuous variable. So when we find greatest range or least range, greatest interquartile range or least interquartile range, we are going to use adjusted values right after adjusting their intervals.

[00:27:05]When it's a discrete variable like this is a discrete variable number of chocolate for example. In this case when we find greatest or least possible value of interquartile range even though there are gaps between intervals we will use the original intervals. We are not going to use adjusted values in this case because it's a discrete variable. It cannot be in decimals. In this case the treatment for discrete and continuous is different. Okay. In discrete we do not adjust. In continuous we do adjust in case of greatest to least possible values of range and interquartile range.

[00:27:40]Then another condition uh situation is when they ask you to find mean or variance for data sets like that generally you do not need to adjust in this case whether the data is discrete or continuous you can find the midpoints directly because even if even if you adjust you end up getting the same midpoint so for example if I have this table now in this table if I adjust the values or don't adjust the values In both cases you can see for all of these intervals since the adjustment is the same on both sides. We are adding 0.5 on the right side subtracting 0.5 on the left side in each interval the midpoint remains the same in all of them. So it doesn't matter whether you adjust the intervals or not. But there's one exception and that is when you have a continuous variable. When you have a continuous variable for example like this it's a continuous variable that is length of something that means it can take decimal values as well.

[00:28:44]In this case when you adjust the intervals since the first interval is starting from zero. The first interval is starting from zero.

[00:28:56]When you adjust values, you have 9.5 here, 14.5, 19.5 on the upper side as well. 30 becomes 30.5.

[00:29:07]But in the first interval, we cannot take the left side to negative 0.5 because length cannot be negative.

[00:29:14]So the lower limit of the first interval that is still going to start from zero.

[00:29:19]And in this case now if you find the midpoint from the original interval as compared to the adjusted interval the midpoint is going to be different.

[00:29:33]In this case it's necessary to adjust the interval first. So you adjust the interval 0 to 9.5. And now the midpoint that you get from that is 4.75. the mean calculation that you would do or the variance calculation that you would do they have to be based on this number they have to be based on this number okay so that's one exception in case of midpoints for stating the interval that contains the median upper quartile and lower quartile in this case we never need to adjust whatever the interval is whether it's continuous data or discrete data we will just write down the original interval given in the question so for example Example in this case interval containing median lower quartile upper quartile they are these intervals we write the original intervals okay so that's what you need to remember for adjusting intervals inter quartile range for ungrouped data for ungrouped data remember do not use these formulas do not use these formulas they are not reliable they will often give you the correct results but sometimes they will not give you the correct results when n is odd they will give you correct results. But when n means the total number of values when that is even these formulas are not going to work. Okay. So make sure that you do not use these formulas for ungrouped data.

[00:30:57]If n is odd it gen it it generally is going to be safe but when n is even it does not give you the correct result.

[00:31:03]What would you do in that case? uh the method that we have discussed median of the lower half is lower quartile where did that go median of the lower half is lower quartile uh is lower quartile median of the upper half is half half is upper quartile okay you need to make sure that you remember this as well and this is one other thing that I recently posted we you must have done questions like these already just go through them when they ask you to find the probability of getting your getting a third success on a particular ular attempt or a second success on a particular attempt right how to deal with problems like these make sure to go through that as well I hope that helps let me answer some other questions physics books and those in that example that I showed earlier they were not identical so they were all physics books but they may be uh a book for P1 P2 P3 3 P4. So even though the the subject was the same but the actual books were different. They were different books.

[00:32:10]Okay.

[00:32:19]Questions questions and this is answered.

[00:32:36]For all diagrams, please make sure to use a pencil. For all diagrams, use a pencil. Use a pencil for cumulative frequency curve, for histogram, for boxing plot. Stem and leaf diagram is also a diagram, right? We also draw that using a pencil. Make sure not to use a pen because a lot of times what happens is you end up making some mistake somewhere. In stem and leaf diagram it's very common that you when you first time when you go through the data at first you end up missing out on one particular value and then if you try to adjust that later on you will end up losing a mark because the gaps between values have to be consistent. So you need to make sure that you are using a pencil so that you can erase something when required. Let me show that to you quickly as well.

[00:33:26]Where was that S1 revision representation of data?

[00:33:34]So when you're drawing a box and brisk uh stem and leaf diagram for example, remember it's necessary that the gaps between values are consistent.

[00:33:44]These gaps between values they have to be consistent. The values need to be vertically aligned like this.

[00:33:50]If you use a pen, if you end up writing a wrong value somewhere by mistake, then you'll have to adjust another value in the middle and that will make you lose a mark. So do not use a pen. Do not use a pen.

[00:34:07]Okay. So when you are starting when you're starting the graph from another value, let me show that to you.

[00:34:15]Representation of data. It might be on this as well.

[00:34:21]Okay. So when your graph is starting from a value when the horizontal axis is starting from a value other than zero, you can just start from that value directly like this. It's not necessary.

[00:34:35]It's not necessary that you make that zigzag thing here.

[00:34:40]If you do that, that's also fine. But in recent papers in recent papers they generally just in the recent marking schemes they generally just start from that value directly. So it's better to just do this directly. So instead of zero start from another value. Okay.

[00:35:16]if the median is 5.5th value uh how do you find lower quarter and upper quartile I explained that there as Yeah. So when median is in the middle here. So for example in this case it was 6.5 value. Right? This was 51. So median is not one of the values in the data.

[00:35:46]It's actually in the middle here. This is the median. So to the left of the median we've got this data. To the right of the median we've got this data. We can find the lower quartile and upper quartile by finding medians of those two halves.

[00:36:00]When median is part of the data. So for example in this case it was the sixth value in the data it was 42. In this case we exclude 42 and we say to the right we've got these values. To the left we've got these values. Find their medians and they are going to give you the lower quartile and the upper quartile.

[00:36:18]Okay.

[00:36:20]when median is in the middle in that case we just consider the right side of that and the left side of that.

[00:36:27]Okay.

[00:36:37]Important point for box in the script light. Uh I will show that to you.

[00:36:45]You can use pencil to do rough work. uh there's no there's no problem with that.

[00:36:49]Of course, you can do that.

[00:36:53]For tree diagram, if if the question is asking you if in the tree diagram, if the question is asking you to make a tree diagram, do that with a pen. Do that with a pen. But if you're just doing it for your own cells, if you're drawing it with for your own cells, in that case, uh you can use a pencil as well, and that's okay.

[00:37:19]All right.

[00:37:23]Uh, no more. I don't know him.

[00:37:29]There was uh something else that I missed here as well.

[00:37:37]No, let me get rid of the spammers.

[00:37:55]Are we allowed to use logs in S1? Okay, let me actually just go through that as well. That was one thing that I had in mind uh because it hasn't come in recent papers. Uh but you need to know about that.

[00:38:07]might be covered somewhere here.

[00:38:10]Discrete random variables binomial distribution.

[00:38:17]Okay, let's look at this question. Let's look at this question. Uh I'll give you maybe 30 seconds to read the question.

[00:38:22]Look at part B of this the second part.

[00:38:25]Part two. Read the question statement.

[00:38:29]Part two of this in particular.

[00:38:33]Ohas at Nawas. Yeah, I actually don't teach A2 there. Only A1. So I'm not sure as in this year I was teaching A1. So I'll be teaching A2 next year.

[00:38:49]So he's from another batch.

[00:38:52]He is the head boy now. I remember. Yes.

[00:38:55]Five boys, five girls alternate. I I'll have a look at that. I'll show that to you.

[00:39:02]Coded mean and variance.

[00:39:04]Search for linear transformation in the crash course playlist. You'll find you'll understand that.

[00:39:11]Okay, there were some other people asking about that as well. What I'll do is I'll just go through that in a couple of minutes as in what you need to do in those problems. But for more detailed explanation, you can have a look at the concept video for that.

[00:39:37]Okay.

[00:39:43]So, a survey of a large number of customers found that 42% of the customers rated the logo as good. So, probability of the logo being good that is 0.42.

[00:39:55]Now they're saying a random sample of n customers is chosen. So we are selecting n customers. We want to find the smallest value of n for which the probability that at least one person rates the logo as good is greater than 0.995. Now how do we think about that?

[00:40:13]Now first of all focus on the probability that they're that they're talking about probability that at least one person rates the logo as good.

[00:40:23]Okay. And now what I'll do is I'll just uh copy that working here maybe so we can just focus on that.

[00:40:39]Okay, let's focus on the on the second part of this.

[00:40:45]42% rated the logo as as good. So we can say it's a binomial distribution because we've got a total number of times we're repeating something that is n probability of success and failure they're constant right if we call this success rating the logo is good we have two possible outcomes good or bad right we can call one of them success the other failure the probability that they're talking about is that at least one person rates the logo is good so that means probability of having at least one success so we could say it's a binomial distribution with these probabilities of success and failure.

[00:41:21]Probability that X is greater than or equal to 1. That's the probability that they're talking about. How do you find this probability?

[00:41:28]You would say it's the same as 1 minus the probability of X equal to 0, right?

[00:41:33]Because X greater than equal to 1 means 1 2 3 4 all the way up to n. Now, we can't find all of those probabilities.

[00:41:41]We just subtract the probability of zero from the total and that will give us that result. So what we do is we find that probability and we say that is 1 minus probability of x=0.

[00:41:53]Now they told us that this probability is greater than 0.995.

[00:41:59]So what we do is we put this probability number greater than.995.

[00:42:04]So we have an inequality like this. Then we put this greater than.995 and we get this inequality. Now think about it. NC 0 what's that? That's just one. 0.42 42^0 that's also 1. All we are left with is 0.58 raised ^ n from this term. Okay.

[00:42:29]Now we solve this inequality. We solve this inequality. How do we solve the inequality? We simplify everything. This is what we end up getting at the end.

[00:42:37]0.5 that is greater than.58 raised to power n. Now how do we solve an inequality like this? N is in the power, right? N is in the power. Musa, I think everybody else is going to think this is a paid actor in class who uh keeps pasting that message again.

[00:43:02]Please focus on what we are doing. 0.5 greater than 58 raised to power n.

[00:43:11]Uh how do you find how do you find the possible values of N from this? N is in the power n is in the power.

[00:43:23]Sorry, please no distractions. N is in the power. How do we bring N down? How do we bring N down? We use logs for that.

[00:43:35]Those of you have done paper 3, we use log on both sides. We any base doesn't matter ln lg any other base doesn't matter we take ln for example in both sides simplify this now you have to be careful ln of 0.58 in this case this is a negative number log of number is less than one that is negative so don't just divide it on the other side as it is because you may end up doing a sign error in in case of that And that so what you should do instead is you should first of all evaluate this number from your calculator and that will clearly show you it's negative. So when you divide it on the other side make sure to flip the inequality sign now and that gives you n greater than this. So the smallest possible value of n since n is supposed to be an integer that turns out to be 10 in this case.

[00:44:33]Okay. Now those of you who are in as those of you are in AS and have not done logs there's an alternative method for this as well and that is once you reach this point once you reach this point you have an inequality like this you can just use trial and error so write this inequality on the side 05 raised to power.58 raised to power n once you reach this point just try inputting some values of n on your calculator you have to show working for this you'll have to show working for is write down some values of n and what results you get from that write them down that is important so for example in this case you could say 0.58 if I were to write it the other way around.58 raised to power n that is less than 0.5 try inputting some values of n on your calculator so 0.58 raised to power Five for instance. Now is that less than this? It's not. Try another value. 10 in the power. What does that give you?

[00:45:44]Is that less than that? Yes. Now this is less than that. Now is it the first value that is less than that? You need to check that. So you can try nine as well. See what you get. Nine. 9 is greater than this. So that means n= 10 is the first time the number becomes less than 0.5. And you would say therefore the smallest possible value of n is 10. So if you do this trial and error method, make sure to show a few values of n and the results from them and then it'll be okay. Okay. So this is another thing that you may have to use sometimes.

[00:46:24]Okay.

[00:46:37]Okay, so this is the alternative for AS students. This is the alternative for AS students. If you don't want to use logs, use trial and error like this. But make sure to show some working for that.

[00:46:47]That's important.

[00:46:50]Okay.

[00:46:53]Uh this question is coming again and again.

[00:46:58]Uh this is very similar to a question that you may have seen in a recent paper as well.

[00:47:05]Uh is Fatima here today? Fat Fatma Javeed can you taking full names? Okay. Fatima uh can you please uh remind me about that question from that recent paper where where they said two things had to be alternating. Yeah, I think it was consonants and vowels, right?

[00:47:28]Consonants and vowels. Do you remember which paper it was? Consonants and vowels.

[00:47:41]Regenerate.

[00:47:44]Was that the uh word in that question?

[00:47:48]Regenerate.

[00:47:53]Yeah, this was the one. Okay. So, uh, the boys and girls problem that that that you're asking about, it's very similar to this one, right?

[00:48:02]It's very similar to this one. Mention 24 paper 52. Yeah, this is the one. So, now they're saying the consonants and vowels, they have to alternate, right? They have to alternate.

[00:48:16]Now, a common confusion that people have about this is this that I have on the side here as well.

[00:48:24]They're basically confused. Why can we not use the dashes and stars method in this? Why can we not use this method on this particular question?

[00:48:33]Okay, look at the question again uh if you haven't seen the question before. So we have 10 letters, five of them are consonants and the other are vowels. So five each. What we want is we want the consonants and vowels to alternate. So no two consonants should be next to each other. No two vowels should be next to each other.

[00:48:54]Now since we want to separate all the consonants and separate all the vowels in this case a lot of people think that we should be using the dash and stars method in this but there's a problem with that. Let's understand that problem. Let's suppose we say represent consonants with dashes and represent vowels with stars.

[00:49:19]Now we could have five dashes for the consonants. We have these five dashes for the consonants. Now the consonants could come in these five places.

[00:49:29]In how many ways can we arrange them? We could say five factorial but then two of them are identical. So we divide by two factorial. Nothing wrong until this point.

[00:49:39]But then we say how many places do we have for the vowels now? How many places do we have for the vowels?

[00:49:45]1 2 3 4 5 6. We've got six places for them.

[00:49:51]Now, if we do 6 P5 here, what we're saying is out of these six places, we want to put the wobles in five places. Arrange five wels in those six places. Now what would happen would be in some cases for one possible arrangement of WS for example could be that they come in these positions 1 2 3 4 5. Now if they come in these positions that means there's no vowel in this position. So there's no star here. And that's a problem because now we have two consonants coming together.

[00:50:36]And that does not satisfy this condition that they have to alternate. So the dashes and storage method in this case is not going to work because there's a specific arrangement that they have told you about that they have to alternate.

[00:50:48]They have to alternate. So we'll have to consider that exact arrangement. We say okay we have 10 places like this. We can make boxes for them, dashes for them, doesn't matter.

[00:50:59]One possible arrangement could be that we are starting with a consonant and then we have a vowel and so on for the rest of the boxes. And we could say now the consonants can be arranged in five factorial over two factorial ways and then the rest of the letters five factorial over four factorial because four of them them were identical.

[00:51:19]Now this is one possible arrangement.

[00:51:21]Another possibility could also be that we're starting with a vowel instead of a consonant. So we have a wall on the left side constant on the right side.

[00:51:30]So this could be another possible arrangement.

[00:51:34]It will have exactly the same number of possible ways.

[00:51:38]So what we can do is what we found for the first case. We multiply that by two and that gives us the result. That's exactly the same thing that we will do in case of five boys and five girls as well. if they were supposed to be alternating with the exception that we are not going to divide them by two factorial or four factorial because any two people any two humans can not be identical. Okay, let's move on to the next one.

[00:52:34]for upper quartile, lower quartile, median examiner does not require any calculations.

[00:52:41]They will only check the answers. Okay.

[00:52:43]So the the process of how you are getting to that median, lower quartile, upper quartile that is not required by the examiner. They're only concerned with the final answer.

[00:53:00]Okay, Zion, that's a long statement that's going to take me a minute to read. It'll be probably easier if you just speak verbally.

[00:53:22]like uh selecting uh select number of people from a group like if you need to select men or women from a pool to make a team at least two men we can say two men four men up till we fill that pool combinations consider like if we're saying that there are two men. So then do we add those at least two men in that combination or are they already built into the case that we're assuming? That's what confusing >> if they okay if there are two specific people. So if they say two specific men have to be there. So if we know who they are then we don't do any selection for them. But if we do not know who those men are going to be, for example, if we have five men and we want to select two of them. So in a particular case, two of them have to be there. Now if you don't know which two of the five have to be there, then you have to do 5C2. But if you are given that these people A and B, they are going to be there in that case.

[00:54:27]In that case, you don't do any selection because there's only one way to select A and B if that what that's what your question was. or you can just tell me uh an example a particular example from past papers I can open that and that might make more sense but I think that's what you were referring to if not let me know yes siri thank you >> quick question specific question uh there's another question here as well stem and le conditions this is very important again a lot of people don't pay attention to representation of data thinking topic A lot of people end up losing a lot of very good students end up losing a lot of marks in this because they think there's a lot of small detail in representation of data that you need to be careful about. So if you have time I would highly recommend that you go through the marking schemes of recent representation of data questions.

[00:55:28]There's a lot to learn from those marking schemes, the guidance that they provide on the side. Uh on what small things they will be deducting marks.

[00:55:37]Make sure that you take care of units.

[00:55:39]Make sure that you take care of labels in representation of data. If you have any diagram, you need to make sure that you label the diagrams properly. Units should always be mentioned.

[00:55:49]Uh for a stem diagram, these are small things that can make you lose marks. The vertical line, drawing that vertical line is important. The rows have to be sorted. You have to show a key. Leaf can only have one digit. There should be equal spacing between numbers. It's a diagram. Don't don't just write numbers randomly. You have to equally space them. There should be vertical alignment of the numbers. You cannot use any commas etc in the diagram. Commas, full stops, they are not allowed. Okay? So you need to be careful about these things.

[00:56:30]Okay.

[00:56:33]Linear transformation transformation.

[00:56:49]We have talked about uh can adjusting intervals and uh cumulative frequency curve histogram. We always adjust intervals. We always adjust intervals.

[00:57:07]Spammers they will be uh blocked. So don't spam. I can ask question like this question at least at least one.

[00:57:38]So, I'm really sorry to interrupt.

[00:58:27]Uh, Fatima, is it from last 5 years? Is it from last 5 years?

[00:58:46]25. November 25. Or let me open that.

[00:58:49]Actually, I was looking at the wrong notebook.

[00:58:56]November 25 paper.

[00:59:10]For exact answers, write down the full number. for non-exact answers, decimals.

[00:59:19]Okay.

[00:59:22]53.

[00:59:39]Okay.

[00:59:41]uh number of arrangements in which there's a D at each end and the two Fs are not next to each other. So you fix the D's, you fix the D's on the sides. What letters are we left with? A F F O I L S.

[00:59:58]You find the total subtract from them the case when the Fs are next to each other. That will be the method that will work or the dashes and stars method that is also going to work. So both of those methods for two letters both of them will work right these are fixed we don't have to do anything with them for these seven letters either do total minus together or dashes and stars both of them will work now your question is why is that not not going to work what you are saying there let me do it using dash then stars method so apart from these If we are left with the other seven letters, what we can do is we can say fs are supposed to be separated. They should not be next to each other.

[01:00:46]So for the two fs, for the two fs we could say let's use stars. They're the ones that we need to separate. And then the rest of the letters, how many of them do we have?

[01:01:08]Five.

[01:01:10]We use dashes for them. So what we say is we have five places for the letters apart from the Fs. Five places for them.

[01:01:20]Now what are those five letters? A, O, I, L, S. So in the dashes space we have these five letters. In how many ways can we arrange them? Five factorial because they're all different.

[01:01:37]Now those two fs that we have got they can come in these places. They can come in these places.

[01:01:46]Okay. How many places are these? 1 2 3 4 5 6. There are six places. We know we need to place two Fs in them.

[01:01:57]We could say 6 P2 arrange two fs in those places. However, since both of them are identical, divide by two factorial as well.

[01:02:09]That is the same thing as saying since fs are identical, we could also say it's 5 factorial into 5 C2. that will be the same thing as well. These are the same things.

[01:02:29]This makes more logical sense. However, since we're arranging, a lot of people get confused. Why are we using combinations? So, it'll be easier to think of it this way. 5 P2 / 2 factorial.

[01:02:40]If you're doing 5 factorial into 6 into 5 6 into 5 because you're saying we have six places. So, six options for the first F, five options for the next F, that is actually the same thing as 6 P2. That's the same thing as 6 P2.

[01:03:02]That will arrange those two Fs. But since the Fs are identical, we still need to divide by two factorial in this case as well. So, all of these will be correct. This will be fine. This will be fine. This will be fine. All of them mean the same thing. Okay.

[01:03:20]marbles if if you can tell me the question number uh as in the paper number question number I'll I'll open that okay now I'm going to do specify questions questions, specific questions, in probability questions. In probability questions and permutation questions, make sure to slow down a little. A lot of times statements are very confusing.

[01:04:00]You need to make sure that you understand them correctly. Otherwise, a lot of times what happens is people come out of the exam center and then realize they understood a question wrong. Right?

[01:04:10]The question was saying something else and they understood it differently.

[01:04:14]So make sure to slow down in probability questions and permutations and combinations questions and leave them for the end. If it's a difficult probability question, sometimes probability distribution questions for example, they tend to be quite tricky the first parts of them and some probability questions as well. Uh and permutations questions, leave them for the end. Do the rest of the paper first.

[01:04:33]Try to save time on that. standard questions like binomial geometric distributions there isn't a lot of complication that those questions have generally if you have covered the recent papers cases all the different types of questions in recent papers they don't do a lot of new things in those questions representation of data questions normal distribution questions if there is nothing uh very complicated generally the first two parts are going to be straightforward and in the last part sometimes in very uh rare cases they introduce something uh that's complic complicated but generally that's an easier question as well. Try to save time on those questions so that the more difficult questions that you have then permutations and combinations uh and some probability or probability distribution questions depending on the paper leave enough time for them so that you're able to think about them at the end. Okay, generally time is not an issue in S1 papers but uh you need as in the working is not that long. However, a lot of times people get stuck at a particular question in the middle. They keep thinking about it and that makes them miss out on some easier question at the end. So, make sure not to get stuck in the middle of the paper. That's extremely important. Leave permutation combination questions difficult parts uh probability questions different difficult part for the end of the paper.

[01:05:58]uh May June 2020 paper 52 C. So let me open that. June 2020 I don't have that here.

[01:06:11]June 2020.

[01:06:31]25 paper 53 of November 25 uh or June 25.

[01:06:39]Let me answer this question first. June 2020. June 2020 paper 52. Let me open this.

[01:07:01]Just like the rock.

[01:07:11]Okay, I shall open I'll do that as well.

[01:07:16]June 2020.

[01:07:35]Press the number.

[01:07:50]So you want me to compare two questions?

[01:07:54]June 2020 paper 52. Okay. Which question are you talking about? Tell me the question number.

[01:08:11]I'm sorry, Mus. I did not read your messages.

[01:08:40]Question number six.

[01:08:49]Okay. Uh G what's the problem in this is question part is part C. Okay. Four letters are selected from this different selections. Let me copy this. Let me copy this. So we can compare this with whatever you're saying there. Part C of this question. Okay.

[01:09:10]Copy that here.

[01:09:16]Add it here.

[01:09:29]A with uh what was the other question?

[01:09:32]June 21, paper 52.

[01:09:35]June 21, paper 52.

[01:09:41]Question number is question number just here. I'm here for some time.

[01:10:07]Okay.

[01:10:12]Which part of this?

[01:10:18]Which part of this? Hammed part C.

[01:10:22]Hammed, are you seriously asking about this now?

[01:10:26]probability.

[01:10:36]Is that what you're asking about?

[01:10:46]up.

[01:10:49]We have talked about it so many times discuss when we're talking about number of selections.

[01:10:58]We talk about number of selections and when we have to find probability of a certain number of selections is uh different read when we're talking about number of selections in this case we are not going to select from the repeated letters so M's and E we are not going to apply the combination function on them and in this tomorrow question we are finding probability of a certain selection Probability questions may when you have repeated objects in combinations you have to treat all letters as different. So you will be selecting from the Rs and the O's as well in this case. Okay.

[01:11:43]Okay. So I I've made separate videos for this very recently as well. I made some short videos on YouTube reals on Insta.

[01:11:51]Uh yeah that tells me you're still not following me and you're still still not subscribed to my channel. That's unfortunate. But yeah, you can find that there.

[01:12:01]Insta, that's actually very good. Do not install that. It's a waste of time. Uh YouTube, I just posted that uh yesterday as well.

[01:12:11]You can have a look at that.

[01:12:15]YouTube Chrome.

[01:12:25]Uh, could be up here.

[01:12:50]March 23. March 23.

[01:12:53]questionational student right. Okay, Mosa, I'm sorry. I did not understand what you were asking about. Ali Assad, I don't know who Ali Assad is.

[01:13:23]Why you bringing back to class?

[01:13:27]S1 BPS. Uh it's March 23. March 23. Question five, paper five to go.

[01:13:51]March 23.

[01:14:01]Okay, this looks like a weird question.

[01:14:05]Okay, somebody else was asking about this.

[01:14:14]Alisa, we've already talked about that.

[01:14:15]Pencil discuss diagrams pencil tree diagram and distribution table. I I would recommend you do them with a pen. Tree diagram and distrib distribution table.

[01:14:29]Representation of data value diagrams.

[01:14:31]Representation of data value diagrams.

[01:14:33]Those you do with a pencil. Okay.

[01:14:37]Aa uh March 23.

[01:14:41]What about this question?

[01:14:44]This one and question number three, part C of of the same one.

[01:14:50]Okay. All right. So, a number of people are asking about this. Let's go through this. Marco has four boxes labeled K, L, M, and N. He places them in a straight line in the order KM with key on the left. Okay.

[01:15:03]He has four colored marbles. One is red, one is green, one is white, and one is yellow. He places a single marble in each box at random.

[01:15:15]Okay.

[01:15:20]Uh, single marble in each box at random.

[01:15:28]The white marble is in either box L or box M. The red marble is to the left of both the green marble and the yellow marble. These these are the two conditions that were given.

[01:15:39]Determine whether or not events A and B are independent. A or B independent.

[01:15:44]Independent events.

[01:15:50]Independent events show the two events are independent.

[01:15:56]This is the condition for that. Right?

[01:15:58]Probability of their intersection that should equal the product of their individual probabilities. Condition meet independent events.

[01:16:12]Independent events probabilities.

[01:16:19]Sorry.

[01:16:21]This is how you can do it. You can say there are four boxes, right? There are four boxes.

[01:16:28]The boxes are labeled K, L, M, and N.

[01:16:30]So, I've labeled them here. K, L, M, and N.

[01:16:34]Now, what are the conditions that are given?

[01:16:39]K is on the left side. Uh, marles boxes already. Now the marbles are placed at random. Event A is the white marble is in either box L or box M. So white marble can either be in box L or box M. So if white is in box L, this is one possibility. What are the num number of possible ways we can have to fill the boxes in that this case? So if white is in box L, we have three other boxes. In those three boxes, we have to place those three marbles. And they're all different colors, right? So three marbles, three boxes. In how many ways can we place them there?

[01:17:32]Three factorial.

[01:17:34]That's the number of ways of filling the boxes. If white marble is in box L. Now another possibility is that the white marble is in box M.

[01:17:45]If it is in the box m again we can say this is the second case now it's in box m the rest of the marbles they have three other boxes they can be arranged in three factorial ways so the total number of ways of arranging in case of event A that is 12 out of the total what will be the total number of possible arrangements if there were no conditions there will be four factors If there's no condition, four marbles, four boxes, four factorial. So what's the probability of a then? The probability of a trans to be 1 /2.

[01:18:23]That's the probability of one event.

[01:18:25]That's what you want to wanted to calculate. Okay.

[01:18:30]Yes. So once we've got this, let's think about the other event. Now the other event is the red marble is to the left of both the green marble and the yellow marble. That's the other condition.

[01:18:44]Red marble should be to the left of both the green marble and the yellow marble.

[01:18:49]Now what could be the possible positions of the red marble? Now what could be the possible positions of the red marble? Now let's talk about event B.

[01:19:01]Since green or yellow marbles at least position position Since green and white or green and yellow, green and yellow have to be on the right side. Green or yellow has to be on the left side.

[01:19:56]So green or yellow two factorial. So that's the number of ways of arrangement. That's the number of arrangements that we get for the first case.

[01:20:13]boxes arrangement marbles arrangement in those theme boxes in those C boxes that's three factorial add them up together that's eight and now the probability of B therefore that's 8 / 24 and that's 1 / 3 probability a probability intersection probability intersection probability intersection probability.

[01:20:45]Let's figure that out.

[01:20:48]intersection.

[01:21:06]White marble box mble.

[01:21:09]Mos right.

[01:21:28]Let me actually show that diagram to you.

[01:21:35]I will just encircle them. White key possible positions.

[01:21:57]Red factorial intersection probability four out of 24 that's 1 out of six intersection probability probabilities put them in that equation condition satisfy 1 is equal to 1 /2 into 1 3 that's quite difficult for three marks but and an important thing to understand from this is this is easily going to take like 8 to 10 10 minutes standard selection question that's very few diagrams that I've drawn here standard four letters between two letters you should know this Easy marks.

[01:23:36]So, make sure to not get stuck in the middle, please. These were three marks for a lot of working.

[01:23:51]Just make sure.

[01:24:02]Make sure to leave that question for some time.

[01:24:06]Do the rest of the paper. Come back to it again.

[01:24:11]Okay.

[01:24:13]Difficult questions playlist. Just search S1 difficult questions math prep and you'll find that playlist difficult questions revision revision video. That's important for more people. revision workshop that covers everything that you need for S1 along with the different types of questions as well.

[01:24:47]Okay.

[01:24:48]All right. Uh I love to everyone who's leaving. Uh if you have any questions, you can let me know. Otherwise, whatever you think is better use of your time right now, revision, make sure to do that.

[01:25:16]last five years attempt to there's no point just go through your work in again drop already questions attempt okay 39 to 43 marks that's good uh depends on what you want u that's good for an A for an A star you probably need a bit four.

[01:25:46]You're aiming for that grade.

[01:26:01]Uh 3C question number three part C 3943 Sorry. I'm sorry. Hamza Hamza.

[01:26:34]Seventh person is the second person not in favor of the leisure center. Not in favor.

[01:26:43]80% of the residents are in favor. 80% are in favor. Not in favor. 20%.

[01:26:52]20% in favor. Right.

[01:26:55]probability.

[01:27:00]Let me do it slightly more efficiently on the side here.

[01:27:05]25 paper 53 question.

[01:27:22]Probability of success success criteria not in favor of the leisure center not in favor probability of success 0.2 in this case that's the probability of success okay you'll say 1 2 3 all the way up to six and the first six you want one success so that will be 6 C1 into 0.2^ 2^ 1 into 0.8 raised to power 5 and then the seventh one has to be a success probability 0.2 and that will give me the result.

[01:28:03]Okay.

[01:28:06]Make sense?

[01:28:10]Questions recent papers.

[01:28:14]probability that the seventh person asked is the second person who is not in favor of the legis center. Actually, let me look at the question statement again.

[01:28:26]80% are in favor of the center.

[01:28:31]20 residents are chosen.

[01:28:35]Find the probability that more than 17 are in favor of the legis. Okay.

[01:28:42]Fifth person asked is the first person who is not in favor.

[01:28:49]And then the seventh person asked is the second person who's not in favor.

[01:29:04]This is fine. It should work.

[01:29:07]It should work.

[01:29:10]What are you getting from this? 6 into 0.2 into 0.8 raised ^ 5 into 0.2 2 that's 0786 that's 0786 that's correct say author so this should be fine grouping while questions uh grouping while questions video where they ask you to make groups groups out of a certain number of uh number of people permutations or combinations difficult questions videos there are three four videos from recent papers difficult questions from last six years questions question so questions next line paper March 23. March 23 questions 25 questions.

[01:30:29]Uh you have a question March 25.

[01:30:38]I can't find that anymore.

[01:30:44]the virus send please.

[01:30:50]So that's May June 23. May June 23.

[01:30:57]June 23, paper 52, question number six, part C.

[01:31:16]Was it 52 or 53?

[01:31:23]53. Okay.

[01:31:47]In a random sample of 365 days, on how many days would you expect the mass of grave soul to be within 1.3 standard deviations of the mean? Is that what you're asking about?

[01:32:00]Is this what you're asking about?

[01:32:04]So they can whenever they ask you about probability of something being a certain number of standard deviations above or below the mean rightvation.

[01:32:34]one. So within 1.3 value 1.3 plus 1.3 no matter what the data is eventually number of standard deviations you will get this eventually. So you say probability of Z that is betweenative - 1.3 and positive 1.3 right because Z standard deviation the 1.3 standard deviation means now how do you find this we have done a standard rule for this right this is one case I recommend that you remember directly Z less than 1.3 plus probability of Z less than 1.3 minus one And that will give you the result.

[01:33:22]You multiply that number by 365.

[01:33:27]Okay.

[01:33:30]Okay.

[01:33:35]June 25, paper 52. June 25, paper 52.

[01:33:40]Let's look at that now.

[01:33:45]So I did not even need to read the question statement. Number of standard deviations June 52 Okay, Okay. June 25 paper 52 is say questions I write question three and seven questionain You can have a look at that part I explained that difficult question solution you can have a look at that post you'll find it there part C June 25 paper 53 question most explained in detail I've explained this okay 52 is the question Number seven.

[01:36:49]Let let me come back and answer your question.

[01:36:56]What about this question seven?

[01:37:05]concept art. So apart be specific which part of this people on zoom specific question otherwise it's a good idea to do your own revision then maybe you can hang around and see what questions other people are asking otherwise Please revision. Make sure make sure that you go through the questions that you got wrong the first time and have good sleep after that. S1 permutations questions probability question does what whatever your name office.

[01:38:20]Don't I also make dua for everyone in Crystals are birds whose adult wingspans are normally distributed with mean 78 74.8 and standard deviation. This how many of these 120 crystals would you expect to have a wingspan between 72.4 4 and 76.3 first part second part 20% have mass greater than202 probability of X greater than 2022 that is equal to 0.2 28% have mass less than this number. So probability of mass being less than.185 that is equal to 0.28 into it.

[01:39:57]This is the process. You can take snapshot of this standard inverse normal process if you want standard deviation.

[01:40:13]Uh third point 10 are selected probability that fewer than three of mass is greater than this the binomial distribution greater than this probability 20% is probability of success 20% 0.2 N value distribution probability of X12 that was not very difficult 513 sorry.

[01:40:57]So your question is why can we not do 5 C2 into 1 / 6 raised to power 2 into 5 / 6 1^ three.

[01:41:33]Find the probability that a three is obtained for the second time before the sixth row.

[01:41:41]How does that make sense in this?

[01:41:46]Uh probability first five may say two successes.

[01:42:01]Let me do it like this. 1 2 3 4 5 attempts. 1 2 3 4 5s.

[01:42:19]Why is that not the same as this?

[01:42:23]Because five successes, right?

[01:42:38]S S FFF for example success success and then failure failure failure.

[01:42:45]That does not make sense in this scenario because all we care about is second success.

[01:42:57]second but we don't care about that for example second success attempt we don't care about Right.

[01:43:40]This does not account for that possibility. So we don't care what happens after the two successes.

[01:43:52]You'll have to do it like this. This is one way of doing this.

[01:43:58]First success, fifth. Second success, fifth attempt, third is possible.

[01:44:19]Okay.

[01:44:33]explain short video. I'll not go through that.

[01:44:40]Allah is Aisha. All the best for the exam. Everybody else as well.

[01:44:47]Uh what is the condition in cumulative frequency curve where we have to look for the value by subtracting from the total probab course uh representation of data cumulative frequency curve Something up Yeah.

[01:45:47]It's for some data like this rooms corresponding frequencies.

[01:46:00]Estimate the number of days on which more than 80% of the rooms were booked.

[01:46:06]Vertical AIS rooms frequency rooms or sorry horizontal axis room or vertical ais number of times number of days number of days estimate 80% rooms booked 80% rooms booked total Number of rooms 90 90 80% it will be 72 72 right more than something greater than something Horizontalis 72 corresponding 184 200US 184 that gives you 16 more than something right side or vertical axis. I hope that makes sense.

[01:47:38]You could have a look at that. All the best.

[01:47:51]Okay. M 24. Uh M 24 question 3a. Okay. Let me go through that. Otherwise your feet leave concept.

[01:48:34]less than November 25, paper 53. Questions, November 25, paper 53 or June 24?

[01:48:50]November 25.

[01:48:58]Paper 53 part C of question number last last question actually attempted those before let's do it within 1 minute of the mean time to walk to work 5 days a week for 45 weeks in a year. So times taken by him to walk to work each morning are normally distributed.

[01:49:33]Mean is this and standard deviation is this. Find the probability that a random chosen day achieved on a particular day takes more than 15.6 6 minutes to work to work.

[01:49:57]Okay. The first part each mean and standard deviation.

[01:50:12]He works to work 5 days a week for 45 weeks in a year. 45 weeks. 5 days. On how many days in a year would you expect him to take within 1 minute of the meantime to walk to work?

[01:50:34]Within 1 minute mean.

[01:50:40]So 14.8 within 1 minute plus minus one right so plus one right side 15.8 or left side 13.8 to probability of X being between 13.8 and 15.8 find this probability convert it to Z convert 13.8 minus 14.8 8 divided by the standard deviation which was 1.5 and 15.8 - 14.8 / 1.5 probability 1 over 1.5 that is going to be 667 negative or 0.667 667 positive Z less than 667 plus probability of Z less than 667US probability number expected value 5 days for 45 weeks 5 into 45 I think that's 225 right multiply that number by 225 and that will give you the result expected Take okay June 24 June 24 paper 52 binomial to normal approximation to mark scheme That's necessary.

[01:52:30]I can have a look at that.

[01:52:36]June 24, paper 52.

[01:52:40]Okay, we're almost ending now in Shara.

[01:52:44]Paper 52 is a concept person 3A.

[01:52:57]What's the problem with this?

[01:53:06]A large orange weighs at least 184 g.

[01:53:11]So you say probability of X greater than equal to 184 you find convert it to Z find the probability write that as a percentage are you sure you're asking about this question this is June 24 paper 52 June 24 paper 52 Asan Okay. All right. Ch. All the best for your exam then. Make sure that you have a good night sleep. That's extremely important. S1.

[01:54:07]So, make sure that you sleep well.

[01:54:09]That's important or last year was for that detail. You make sure that you are not missing out in any small detail that may be tested. Then you should be good to go and exam. Make sure that you do not get stuck at any particular question. That's extremely important for any exam in general. But exams Uh this video startain look at the first 10 15 minutes or 20 minutes.

[01:55:11]First 20 minutes explain slow maybe you can have a look at that video start binomial to normal approximation that's what I'm saying correction bin to normal Bin normal normal normal normal distribution should say Omar odd or even different medcliff even or odd up.

[01:56:21]Even number of values, number of val odd case median is going to be one value in the data. For example, 11 values 42 was the median median lower quartile or upper quartile.

[01:56:58]Okay.

[01:57:12]How many yearly should you have done?

[01:57:14]Now there's no point of that question.

[01:57:16]Just do revision now.

[01:57:21]Do what is in your control.

[01:57:25]Make the best use of this more likely.

[01:57:34]Max it depends on the question representation of data questions normal distribution standard question and then save time for the more difficult questions You do the rest of it.

[01:58:16]This is one note. This is one note.

[01:58:21]Lower quartal or upper quartal positions. Decimals decimals.

[01:58:35]Just medal, right side median.

[01:59:00]57.5.

[01:59:18]I'm from Lahar. I'm not from Karachi.

[01:59:21]March 24. Paper 52. March 24. March 24.

[01:59:27]Uh, Mars 24 C.

[02:00:01]Okay. Members stand in two rows. Six rows. Six on the back. Four on the front. There are six men, four women, 10 members in total.

[02:00:13]Okay. So, uh these 10 people are supposed to stand in two rows with six in the back row and four in the front row. So, back or front rows.

[02:00:23]These two siblings, they have to be next to each other in the front row.

[02:00:42]do xent.

[02:00:51]Okay.

[02:01:12]row.

[02:01:23]Okay. Out of the remaining eight people, select two for those two places that are left.

[02:01:31]86.

[02:01:39]Now you just have to arrange those people.

[02:01:52]for that six factorial and that gives you the result. It wasn't that complicated.

[02:02:02]Continuity correction bin to normal approximation bin to normal approximation representation of datent.

[02:02:12]You can have a look at that.

[02:02:40]Okay everyone, all the best for your exam then all the best for your exam.

[02:02:43]Make sure not to get stuck please on any one question. That's the most important thing that you need to remember. Do the questions that you know first how to do combinations lastly.

[02:03:10]Make sure to do them first and then you have uh enough time for the remaining questions to be able to think about them uh a bit slowly. Take that's it. I'll see you in again in the next session and then inshallah after you are done with the S1 those of you are in com taking composite exams I'll be starting P3 uh classes from tomorrow in P3 session from tomorrow. I love all the best.