 can attend the revision hours. Fair enough. So let's begin and probability. So this is, as I told you, let's go to the breakup of the weightage of the topics and you can see statistics and probabilities of 11 marks out of which the majority is a portion two statistics. So probability will have maybe two, three, you know, maximum three to four marks questions. Usually until last year, four marks questions, two plus two used to come or many a times one, one, two, one, two into one and one into two. That means two one marker and one, two marker question used to come. But yeah, so yeah, so you can't expect more than four marks any which way from probability. Okay, so there is no deletion. So whatever was there, it's still there. Though we in our regular classes, we did discuss about something called geometrical probability, which is not there anyway in CVC board. So you are, you know, kind of good here. So no deletion. Again, same breakup. So this has been now done so many times you must have by heartened. So let me not waste time in this anymore. Let's go directly to the content material. So what is probability? It is a concept. So we are basically intending to measure chances, right? So what are the chances of certain events to happen? We had certain basic definitions related to experiments, trials and all that. You have studied probability in grade nine as well. Can anyone tell me what's the basic difference between whatever you studied in nine and what is there in grade 10? What's the basic difference? In nine also you studied the same thing. You did apply the same type of formula, but what is the basic difference between the two? Anyone has any clue? Why did we do the same thing in grade nine? And why are we repeating it? We do for grouped data. Okay. And here we don't do for group data. So what's the basic difference? Yes, I need your observation is correct. But you know the explanation again, as in terms of the reasoning still needs to be a little fine-tuned. So we did probability in grade nine. All of you did probability in grade nine. Yes or no? Do you did or not? In nine we do actual experiment we just studied a probability. Okay, good. So hence RM is explaining now you're closer to whatever I was expecting. So basically probability is delta into approaches, right? So in ninth grade you did something called empirical probability. Empirical, isn't it? So what is empirical? So based out of results of experiments actually performed. Empirical, sorry, empirical. So it's not EX, it's E-M-P-E-R-I empirical. Empirical. Empirical. Empirical probability, right? And no, empirical spelling, same RN single E-M-P-E-R-I-C-A-L. Empirical, isn't it? This is, am I right or wrong? What is the spelling of this thing? Empirical. Okay. So anyway, whatever is the spelling, check it. Now, is it empirical? So you just let me know. Is it E-M-P-I or E-M-P-E? Whatever. The meaning is this. What is this? This is based on experiments, right? So check the experiments. That is, you perform actual experiments. So when you say you toss a coin, you actually toss a coin and you toss a coin n number of times, right? Certain number of times. That's like by, you know, you do actual experiment, you observe, tabulate it, and then find the probability. What is different in 10th grade is you do not do any experiment. You basically try your knowledge. So theoretical probability. So in your grade 10, probably very weak at spelling. Okay. Anyways. So what do you do Google? Okay. Okay. So empirical. Change it. Empirical. Empires empirical, right? It's okay. So the difference is understood. What is the difference between the two? So hence in the ninth grade, you would have seen tables made what R-N was, no, sorry, what Anish was mentioning. So you'd see grouped data or any type of data basically. So let's say head and tail, if you're tossing a coin, they will give you, let's say out of 1000 times, this was 490 and this was 510, something like that. Okay. And then you are asked to find out the probability. So used to do what number of times the favorable outcome, let's say head in this case comes divided by total number of observation. You recall this in grade nine, you did this right. So there was no theory involved in terms of, you know, what theory in the sense, can we do over with this experiment and still try to find out the probability of getting a particular outcome? So that's what we do in grade nine. So grade nine in, sorry, grade 10. So in grade 10, empirical based on experiment in grade, sorry, in grade nine based on experiments in grade 10, based on theoretical calculations. And for that matter, we did learn something about permutation and combination to facilitate learning of theoretical probability. I hope you remember those things. Okay. Now what is an experiment? An operation which can produce some well defined outcomes. Now this is very important. You cannot have vague or fuzzy outcomes. It has to be well defined. So you cannot have, you cannot have something between head and tail. So if you toss a coin, it has to be either head or tail, you cannot have something landing perfect, you know, perpendicularly on the ground, nothing that doesn't happen. So well defined. So you know, if you roll a die, it will be one, two, three, four, five, six, no 5.5, no quarter to five, no root five, nothing, one, two, three, four, five, well defined, right? If root five is also a well defined outcome, then that's also there. But in this case, rolling a die, we know only one, two, three, four, five. And in no case, the, the die is going to be like this. So this is the, yeah, so this is improbable, not happening. Okay. So hence this is not, you know, so clear cut definition of outcome must be there. Random experiment, an experiment in which all possible outcomes are known and the exact outcome cannot be predicted in advance is called a random experiment. So again, tossing a coin, drawing a card from a pack of cards or rolling a die, all our random experiments, because one, we know the set of outputs, set of outputs, already know, there cannot be any seven, there cannot be any minus 1.5, there cannot be any 6.7 in rolling a die. Set of output is known and there is fixed cardinality. Fixed cardinality means the set, the set will have fixed number of, the number of elements on that set of output is fixed. Here, if you draw that set, it will be one comma two comma three comma four, comma five comma six and no more. So we know there are six outputs. But we are also knowing that there is no one particular number which is favorable. Okay. Unlike in case of Mahabharata, have you seen Mahabharata, where there was a fraud, some scam was going on. Yep. What was that fraud? So Mama Shakuni is a very dangerous guy. Now he has skimmed and something he did and he made the dice which were there, then those were biased and he used to know what is the outcome before he used to throw the dice. Yes, die tampering, right? Like what cricket ball tampering used, you know, so those are wrong things to do, don't do it. So set of output is known, not known, sorry, a set of output is known rather, but exact output is not known. So what is going to be going to be there? We don't know. It could be anything. So all we say, all are, all outcomes are equally probable. Equally probable, probable, right? What is not equally probable? For example, Virat Kohli hitting a century is more probable. Let's now talk in terms of relative, you know, but Virat Kohli taking a wicket is kind of lesser problem because anyway, he doesn't bowl. So you now know equal problem and lesser problem, you know, so outcome is. So in this case, all are equal problems. So you cannot say that, hey, one has more frequency to be, you know, to come as an output or five has more frequency to come as an output. No. So all are, you know, until there is some bias, you know, deliberately put inside. So hence we are considering only those experiments which are random in nature. We don't know whether which output is going to come, but extent of output is known. Okay. What's an event? Now, as definition, the collection of all or some of the possible outcome is called an event. So let's say when you again, roll or die. Yeah, when you roll or die, then guys, can you just give me one second? I think there is somebody who's disturbed, who's not able to do anything. Just a minute. I will just come back. Yeah. Sorry. So it's done. Okay. Now what I was saying is, yes. So when you get a prime number when you roll or die, what are prime numbers will you get when you roll or die? How many prime numbers do you get when you roll or die? Well, getting a prime number on the die, right? So, well, how many prime numbers are there? All right. Come on. What happened? Guys, are you able to go? Am I on mute? Oh, no. Yes. So, well, tell me three. So two, three, five. Correct. So that's an event, right? So event is just like an event of, let's say, what is this guy's new Joe Biden, right? So Joe Biden, sir, can you explain the difference between equally likely and unequally likely event? Oh my God. Equally likely events are very, very clear. What are unequally likely, unequally likely events? Unequally likely event is that there are, you know, one particular bias. For example, you know, first of all, let's talk about equally likely event. Equally likely event is, let's say you have thousands of cards and in each card you have mentioned, or let's say, let me, let me rephrase. Let's say, no, no, I'm giving you another example. Let's say you take 100 cards, okay, 700 cards and in each 100 of all, let's say one in one set of 100, you write Sunday. In the next 100, you write Monday. In the next 100, you write Tuesday and likewise all days you write and mix them up randomly so that you do not know. I don't know whether you used to do in the childhood. Let's say when you used to play cricket, we used to write names in the chips, okay, and to divide the teams. Have you done that in your life ever? Or let's say instead of playing cricket, just let's say any team formation and you have to select teams, then what you used to do is you used to write the names of each of the player in a small piece of paper. We used to make sure that the papers are also of the equal dimension. You do this in Secret Santa and all that stuff when you play at school in school and all that. Have you done that where you write the name of your friends and all that and then and what do you do? You try to make sure that the dimensions are all equal so that you, in any case, you do not figure out who's or any let's say odd man out in that lot. So that's where you mix and all are equally likely then. Then you blindfold some person and ask him or her to pick a particular cheat or pick a cheat rather. So he doesn't know or she doesn't know what he or she is going to get. That's called equally likely. So all she can get any of them, any of the any of the chips. That's equally likely. But there could be cases where let's say sorry, Suresh, what are you saying? Now, yes, now let's say they're in the same example. If I do this, I do this two chips with same name. Yeah, that could be or rather let's say, the next level. In the first step itself, what I do is I change the paper quality of one particular person or couple of, or let's say I have some favorable people to be chosen. Then what I do is I write their name in different quality paper. Let's say one in a newspaper and another one, let's say in your A4 sheet. So by texture, it's different and you blindfold the person. Now you have changed the likelihood of selection of a particular chip. Do you get the point? Are you getting this example? The person who has, I think who has, aren't. So you are now saying that you know, not everything is and that also in that experiment also you change the number. So let's say only two with newspaper and the other one with let's say another eight out of 10 in different texture paper. So are all 10 equally likely now? Do you understand what I'm trying to say? I don't know. So once again, so let's say you have two names written. Question of the probability of basketball thrown into basket. Is that equally or not equally likely? What question? In NCRT, there is a question of the probability of basketball thrown into basket. Is that equally? What is the question? What is the event? What is the experiment you are doing? Throwing a basket onto a basketball. Is that, is that, is that it? Is that what you're saying? Yes. So the basket, you know, so throwing a ball in the basket, the outcomes are either it will go in or it will not go in. Is it? So hence, in this case, there is a 50-50% chance, 50-50% chance, no, it will go or not go. But the thing is, again, if let's say this is geometric probability, again depends on the context actually. So let's say if the basket is here, and you're throwing in this side, then obviously there is zero probability that it will go. Okay, there is no, right? And let's say if you're, this is the basket here, and you're throwing up, then obviously zero probability. But you are making sure that the person who is throwing is throwing with the intent of, as in, you know, it is going in this direction only. Yeah. So hence, yes, to an extent geometric probability, if they're considering that the ball thrown at the right angle with right velocity will land there, it becomes a very difficult thing to exactly predict the probability here. But for the sake of argument that, you know, okay, everything is fine, it was going there. So hence, you know, it could land up in, you know, whether in or out. But let's say, again, this is where the gap is. Yes, exactly. That doesn't, hence, I'm saying, see, why is probability called, wait a minute, let me, let me, let me no, no, hold on, hold on. Probability. And probability is in the world of indeterminate world, right? So it's as good as saying, will it be, what is the probability of raining today? What is the, what is the probability of raining today? Can you predict that in next hour, there will not be any rain? At least we cannot predict the IMD can definitely predict. But again, that's a probability. But all, all that probability, this indeterminate case is dependent on so many factors, the moment we know the factors, it becomes a determinate case. Is that understood? Okay, for example, you can also predict the, let's say if I give you the weight of the coin, you can try this demo. I give you the weight of the coin. I give you the air resistance, you know, information. I give you the amount of force which the toss the person who is tossing applied on that coin. So it becomes a mechanic mechanics cup problem right on also on what, what face it will land. So to that an extent, if I have all the information, I will be able to predict on what, you know, so hence the amount of force will guide it to how many turns it will go while it is going up. Then if let's say you remove the air resistance, then what happens? It will take the same route back and then land on this particular face. Is it? So hence if we have the knowledge of factors, the probability becomes determined or deterministic value. So hence we say probability only where and hence it is between zero and one. So one is happening, zero is not happening and everything else is a matter of chance because in between zero and one, there are lots of factors acting which I do not know the moment I know, I know one, I know, I can predict what is going to happen. So that's the, so wherever there's a knowledge gap, wherever we cannot explain stuff because there is a lot of, you know, other factors involved in and it is beyond our capabilities to analyze everything. For example, monsoon prediction, it's all on chances of probability, but there was a time when we did not have any factor and hence someone used to just look at the horizon and then he used to predict whether there will be a rain or not. Today we have pressure data, temperature data, wind speed data, pressure on the ocean data and XYZ thousands of data is there. We collect it and we'll try to make sense out of it. And then also we are not 100% sure because there are a few more things which we still need to discover. So till that happens, it is probabilistic in nature. So hence in this case, you know, if you throw a basketball, it depends on so many parameters to, you know, but still it is a probabilistic question. It's not that you cannot, you know, so let's say, if I say that I throw in exactly 180 degrees, what is the probability of the ball landing on the basket? So let's say if I'm throwing exactly opposite direction, will it land? What is the probability of it landing in the, on the, on the, or in the basket? Zero, right? Now you can counter argue. You can counter argue. No, sir, he has thrown in such a way that it takes full circle and then lands here. Right? You can, right? So hence you throw it in such a way in the opposite direction that it goes into orbit and comes back and then lands in the basket. Now all are extremely improbable thing. Right? So hence we talk in terms of something between zero and one. Okay, fair enough. So I hope this is clear. Now these are the common things, common experiments which you will be do, which is, has been discussed. There is no other gun here because their PNC has not been taught to you. So hence we cannot go beyond a realm of simplistic counting numbers. So hence tossing a coin, you know, two outcomes head and tail and mutually exclusive. What are mutually exclusive? This is what is this mutually exclusive? What does it mean? So in board exam either you are going to pass or fail. God forbid that you fail. But either or yes, not mutually exclusive for head. If head is there, the other will not happen. Okay. So if one has happened, the other will not occur. Right? So if one, let's say on tossing you have got head, there will not be any case. We will not get head. Sorry. You know, if it has happened, their head has happened. Right? Or what I'm trying to say is you cannot have head and tail together in the same one, one coin toss. So both the results cannot happen together. But there could be some exclusive mutually non-exclusive events also, not mutually exclusive women. Can you give me an example where there is a, there are events which are not mutually exclusive. So both the events can happen together. Maybe. Yeah. So any, any idea, any example, where there are two events which are happening together. So there's an experiment going on, outcomes are coming out and the outcomes are such that the two events are happening together. Any example, anyone? Any case where you see that, oh, sun during rain, need to understand the case. When you are experimenting with two coins, okay, Arayan, go ahead. So yes, when I experiment with two coins, what are the events you're talking about? So which two events are mutually not exclusive or not mutually exclusive? You have to define an event, which I'll give you an example. So for example, I roll a die. Okay. So I say that event is, what is the event? Event is to, let's say E1. E1 is an event of getting a prime number, right? I roll a die and I say that event is, I must get a prime number. E2 is event of getting an even number. Okay. Now tell me, are these two events exclusive? Both of them cannot happen together or both of them can happen together also. Can even E2 happen together? Are you, yes, can happen? What is that case? What is that case? Two, very good. So hence, if let's say if I roll a die, I get two on top face, then this has happened. Yes, tick mark, prime number. And is this even number? Yes, tick mark, prime number, right? So hence, in this case, you'll see there's an overlap. Yeah. So overlap. So both of them are, so hence we say that even and E2 are not mutually exclusive. So when even is happening, you cannot say, you, yes, picking a diamond and an even number. Yes. So you cannot see, yes, very good. That's an example. So you can, you can pick a diamond or an even number or both, both are possible, right? There is an intersection in terms of set theory. So R and R people, you know that, you know, this is one set, another set, there is some intersection, right? So these are the outcomes of event one. So let's say in this case, how can I represent in case of a Venn diagram? So let's say even numbers are four, six, two, and this is three and five. So this is the outcome of even. And these are the outcomes of E2. And then we have an intersection here in terms of two. Okay. So is this related to an event being subset of another event? Event cannot be subset, subset again, don't use the term subset here because the moment you say set, then you have to have a collection of events, right? So event or singleton set of one event, you can, you can say, but let's not use the term subset here. Okay. So event is not a subset. Event is a different entity subset is a different different entity set of events itself is a set understood. So event cannot be subset set of events can be there. Okay. Okay. Now rolling a die, you know, and then card, all of you are well familiar with the deck of cards, 52 cards, we have 13 cards, each of four suits what spades, clubs, hearts and diamonds. We many times we say colors also. So instead of colors, so how many colors are there in two colors are there, right? Black, black and red. So cards of spades and clubs are black cards and cards of hearts and diamonds are red cards. Okay. Kings, queens and jacks are known as face cards. Thus there are in all 12 face cards. How many of you play card game here? Anyone poker? How many of you play poker? Anyone plays plays or bluff my one. Okay. Bluff. Okay. Anyone else? Anyone plays or blackjack and okay, blackjack. So Uno, anyone plays Uno different card game Uno, anyone for Uno here? Yeah. It's quite fun. No, I really love Uno, right? Uno, Uno. And I every time I forget saying Uno, you have to say Uno, right? Before you have left with your left with only one card and you have to say Uno, right? But every time I lose, I get penalty because I forget to say that. That is so disgusting. The person who has designed the game, I curse him like anything whenever I forget to say Uno. Anyways, I enjoy equally when others forget it. So that's quite fun. Anyways, so good game. So you also design a card game. Good. Now so now we come to probability mathematics of probability. What is it? So probability of occurrence of an event is denoted by P brackets E number of outcomes favorable to E. So what are number of outcomes favorable outcomes? So here is a question mark. So favorable what is favorable? Favorable outcome is something which I expected and is coming. So favorable outcome if I toss a coin I want a head and head actually happens, then we say that the favorable outcome has taken place. Okay, our total number of possible outcomes and hence we studied a little bit of PNC so that we can calculate without doing actual experiments. And that's the difference between the ninth grade priority which you studied, which was the table was given number of trials and experiment outcomes were shared with you and you had to calculate the probability. Can anyone also tell me what's the difference or if you have done these tossing of coin in ninth grade as well in tenth grade as well? What was the basic difference apart from the theoretical and experimental point of view? What was the difference as in you know can you comment on the values you used to get? Anyone? So in case of ninth grade if you do the same tossing thing you never got 0.5 as a probability of getting head. Do you remember guys? So there was a table let's say it says a head and it's a number of a thousand times you tossed it and you got a number 240 no not 240 for 480,000 times until you you get and yes correct and this was let's say 520 correct are in so if you see our theoretical probability is better oh no in terms of see theoretical probability and empirical probability are related to one thing and I don't know what is the criteria for theoretical probability or empirical probability to take towards theoretical probability. So you will see that when does that happen? So empirical probability is here EP and theoretical probability is here and empirical probability is always in the neighborhood of theoretical probability so let's say theoretical yes number of trials correct there is a lot of message in this so you know increase the number of trials you become profit go closer to the target isn't it? The deep message between empirical probability to theoretical probability you have to do multiple number of times to make sure that whatever you are getting observations you are going closer to the reality so hence do many many times repeat your exercises do many times repeat it so that if you are let's say pilot fly for more number of hours so you'll get a lot of experience you'll become perfect perfect perfect perfect right so hence okay so that's what so hence in the empirical probability you will get numbers which are off the target by some error let's say and that error is because you are experimenting and randomness of the experiment hence you'll get another you will study in the first few topics of 11th grade physics how errors creeping in all these data random errors are the what systematic errors or what and all that so so again there will talk about probability so see there is a logical connection very good now so you know number of outcomes favorable to e divide by total number of possible outcome is what is whatever reality theoretical is an ideal world yes my dear absolutely correct but everybody is changing an ideal world is it that you need a corrupt free corruption free prime minister though we ourselves are very corrupt we need you know everything should be good clean everywhere but you know um inside our mind there is a lot of garbage but anyways let's not go into philosophy uh yes that's true real world is little not perfect yep anyways so sure event what is sure event now even with probability one so something which is going to have give me an example of sure event folks give me an example of sure event and it's going to happen hundred percent sure india is going to win the world cup next when is our next world cup ticket world cup 2023 where is that isn't it uh no uh sure surety of you know i am saying india winning football world cup in 2022 true 100 percent problem what is the probability of that happening india winning football world cup in 2022 i think there is a world cup in 2022 if i'm not wrong is there a world cup in 2022 football world cup there is right so what is the probability of india winning the football world cup in 2022 zero see that thing is saying yeah okay all black balls in a bag probably picking black ball okay very good nice and um what else other other things tomorrow is a tuesday somebody said very good any other example getting all of you will fail in grade 10 what is the probability of that happening very good god willing you should no one should fail everyone should get what what is the target come on target is centum yes 100 on 100 very good so all of you should get centum but again there's a game of probability over there as well but god willing my probability of all students getting centum is one anyways impossible event the probability of an impossible event is zero give me an example impossible event example though ample unique on achieve the example should be unique are that cliched one red ball black ball these ball which are unique impossible event is zero they are getting a joker from the card words about one of those i mean real world forget about uh sun rises from west uh okay impossible okay so good anything else corona virus no no case of corona virus in next 10 days trump getting in peace for a third time can't be uh correct zero crossover yes good tree growing upside down okay uh okay uh okay anything else i thought creativity the higher come on what is it some more creativity zero absolute zero an impossible event is zero come on so so put some pressure on your brain come up with some good creative ideas of zero probability come on you are requesting so much people in this class having creativity that is between zero and one and you get something else yes come on coming back okay coming back to life there is a matman sitting next to you know your screen okay uh chateshwar puja having a strike rate of a cricket world ending tomorrow all right some positive one you are your name has sigh in it so some positive news please ending ending seeing alien life okay so you know look at the mirror everyone is alien and it would not be in fake news okay that is possible okay uh seeing alien life is possible is what aditi is saying aditi you have an experience with alien world from aditya to everyone be happy oh my god big philosophy oh dinosaurs coming back okay zero priority bro don't go that dark what kind of probability is this flunking exam okay Rohit not trying to hit a big shot so no cricket nothing it's sort of funny time travel it's possible it's not zero so that meeting doremon okay good so only few people are creative rest all are not enjoying probability at all zero probability zero probability so zero iron man coming back to life okay okay or india going back to africa you know zero probability uh mount everest converting back to tessit c zero probability and disobeying the law of conservation of energy uh actually it happens adit uh law of conservation of energy is not true in case of relativistic mechanics so mass okay so rawl on the earnings oh my god pretty political jokes not allowed sorry uh political priority in each year india may hit africa in the far future reaching light speed or let's say india india and pakistan reuniting tomorrow what's the probability what do you think guys india and pakistan becoming one country once again negative yes she has zero why do you don't want in india and afghanistan sorry india afghanistan pakistan bangladesh relanka nepal and all of them together as one country akhand bharat true not true zero problem negative means they have to so anyways so lots of zero and single digit one probability okay now complimentary event you know uh let boris johnson paying attention to him oh okay bhai no political this thing don't do that yeah so you will have to come to jail getting some gajar khalwa for me saying okay my teacher is jailed let's go and meet in there i somewhat karnagos i have still some life okay so no political either discussion okay chali complimentary event let e be an event and not e be an event which occurs only when e doesn't occur so it is usually stated or denoted by this e bar e bar is compliment that something is happening and uh you know um e bar is something not happening right so compliment of an event right the event not is called the complimentary event of e so clearly p e plus p not e is one and this is only when um okay anyways uh but so let's say if head was happening so p of h let's say event is head and uh is equal to nothing but one minus p of tail right so hence p of probability of getting head plus probability of getting tail is equal to one because either of them will have to happen right so that's complimentary this thing chalo but our sample we will log off now enough of gain in probability now this is the sample paper question there are only one i think only one mark two marks find the priority of getting a doublet in a throw of a pair of dice what is a doublet dice you know doublet so in these definitions you must be knowing okay otherwise happy about that a doublet is if you get the same number so one one and both of both of them so one one two two three three four four five five six six right so when you get both of them together on both the faces yes one by six very easy so what is the probability of not getting a doublet so this is a total number of outcomes so let's say you have to write like this boss don't just write one by six over there so e is event of getting a doublet okay you might not write inner throw of pair of a dice so n cardinality number of outputs of e is how much six and uh so hence total number of trials or total sorry total number of possible outcomes total possible outcomes in our role of today how many 36 okay so total is so n e upon 36 this is p e as you can you tell me if i change it to triplets what is the probability of getting a triplet in the role of three dice what is the probability of getting triplets in the yeah one by 36 so can you generalize it so let's say if i'm throwing n dice n dice so what is the possibility of getting n lit and let me just a doublet triplet four let five let six let so what is the probability of getting n lit if i'm throwing n dice one by six to the power what one by six to the power n minus one minus one is in the denominator or the numerator okay power menu understood okay yeah good yeah very good so it is nothing but one upon this six to the power n minus one that's what you're saying all of you agree this is the probability of getting an n lit if you throw n dice true false guys all of you agree or not agree to stress agree with stress yes no agree sarguli is saying different thing n by yeah that's same thing sarguli n n divided by six to the power n is nothing but one upon six to the power n minus one isn't it so n upon in this case in this case you'll get um n k any huh sorry you will not get this thing no no i is it okay guys so n and n lit is how many how many how many possible outcomes of n lit so one one one one one one one two two two two two so six only you know you'll get six in n lit case also you'll get only six possibilities yes or no six possibilities divided by six to the power n not n divided by six to the power n sarguli got it so it's one upon six to the power n minus one next find the probability of getting a black queen when our card is drawn at random from a well shuffled pack of 52 cards black queen black queen okay so let's play with this your card game okay black queen is one upon 26 why do you say that because if e is the event of getting a black queen event of getting a black queen right so what is n e how many black queens are there in the pack of cards so one is off spade another is off club correct spade and club yes or no spade queen and a and club diamond and hearts are red right so there are two and total number of possibility possible outcomes when you draw card is 52 so two upon 52 52 that is one upon 26 very good now tell me what is the probability of getting a red queen one by 26 what is the possibility of getting a queen what is the possibility of get probability of getting a queen one upon 13 perfect what is the probability of not getting a queen okay very good 12 upon 13 what is the probability of uh what is the probability of getting a face card face card you know king queen jack they are face cards right yeah very good so 12 by 52 should be reduced to three upon 13 very good okay now what is the probability of getting a even numbered diamond card so don't cut the don't count the face card in even not five by 52 even numbered diamond yes very good what is the probability that you don't get a diamond at all no diamond if you pick a card no diamond should not be diamond so say 39 by 52 is very good very good so you are thorough with that okay what is the probability uh now what is the probability that I draw two non-face cards oh wait a minute what is the probability that I draw two cards uh with replacement so listen carefully what I'm saying with replacement you are drawing one card and putting it back okay so let's say you draw one card and jot down the number so let's say you draw five of clubs so you write down the number five and then put this five back into the pack shuffle it draw one more okay and this time you got seven right on the card seven right so what is the probability that uh no it will be just question for you to follow it will become more deadly so question was this what is the probability that the sum of the two cards number is 12 so you draw one card get one number forget the face cards face cards are not included in numbering so one to ten ace is included let's say or ace huh whether it will be too difficult for you right now don't don't do that it will take some time a dice thrown once what is the probability of getting a number less than three one marker it is a dice thrown once what is the probability of getting a number less than three so less than three is one and two so one so hence outcomes e e ka outcome is a set of outcomes of e is one and two so n cardinality of e number of such possible favorable outcomes is two and number of outcomes possible outcomes is six so two upon six one upon three very good okay uh what is the probability of getting a one die is anyways very easy so no problem so let's go to the next one a letter of English alphabet is chosen at random what is the probability that chosen letter is a consonant what is the probability that the chosen letter is a consonant 21 upon 26 very good what is the probability that the chosen number is not a consonant 5 by 26 very good what is the possibility that uh what should yeah now let us pick two letters okay so will that will you be able to do that wait a minute okay no no you will actually let me just see let me see if you can do this from it is little stressed up it will not be asked in board exams but then just to make fun some bit of fun here so let's say you are designing a passcode okay three letter passcode okay what is the possibility that starts with a I don't know whether you'll be all of you are well versed with pnc have you done counting before anyone did not do counting before pnc guys hello otherwise your grade 10 probability is just like a monotonous repetition of same thing can you do that I don't know if you are it is too tough for you anyone what is the probability that no no probability I'm asking can letters be repeated no letters can't be repeated you have to get three letter passcode it has start it has to start with a what is the probability that starts with a how about 26 so yes can you explain how so the total number of possibilities of passcodes are 26 into 25 to 24 because letters can't be repeated and the number of ways so we are fixing a in the first in the first place and the other two places can be filled by 25 and 24 ways so 25 by 26 into 25 to 24 but this is a conventional I was I thought the moment you said 1 by 26 I was like I jumped out of my seat because I was expecting some other quick answer there's a quick answer I so I the way I did it is I just considered the first box so the number of possible letters that can be in the first box is 26 out of that is one option so one by perfect so that's all yes good right so you don't need to yeah one upon 26 is good because any any letter says did you understand so any letter you put there in the first box you know so a can be you know in one of the 26 such cases because I'm not worried about what's happening to the next two boxes but your method was also that was conventional method how we do it but here also even if let's say I don't know anything I know that there will be 26 possibilities here so 26 yeah so that is the conventional method of doing it right you count and all that yeah so instead of that I'm saying let us say forget these two anyways whether it is a whether it is b whether it is c whether it is z whether it is z whatever it is there in the first all of them have same set of the other two yes or no so that this part is common to all all the letters okay so hence even if it is not there I don't care so as are in single was mentioning so it is one upon so there will be so in the first box there will be let's say a is there in the first box and there are let's say n number of words out of them let's say now you put b you will get another set of n number of words out of them you put c in the first box and you'll get another n likewise likewise likewise you put z here you will get another n so all these are same and same in the sense count is same they're not same exactly in the form but the count is same all of all these will be n only for any choice of a b c d whatever so hence a can be one of 26 wins probabilities did you understand so you have to even even if you don't know counting let's say p and c it's okay so you could have thing you could have thought this way as well all of you clear on this is this particular question this will not be asked definitely in words don't worry but now we have to give you that favor okay p and c is very important chapter in grade 11 very very important and it is involved in binomial theorem it is involved in let's say complex number it is involved in trigonometry it is involved in probability so hence very very very important okay actually next if the probability of winning of game is 0.07 what is the probability of losing it 0.93 1 minus this you can do when they are mutually exclusive is it so someone who's winning cannot lose right so that that then possible but but no problem so hence let's say there is a pole going on a pole is going on this is let's say so if there is a probability of draw then not happening so hence there you can't write 0.93 right so let's say there is but then you have to also say what is the probability of draw is it is draw equally likely for example in a test match winning losing or draw what is the probability that some team will win what is the probability theoretical probability that some teams let's say all are equally probable let's say again there will be a great discussion on how do we say that equally problem how can you say let's say let's let's take this example let's say test cricket can you say the loss and the win and the draw drawing the test are all equally probable yes or no how many of you say yes aren't single says no no no why on what basis do you say no so how will you decide what is the probability what will you decide intuition no you have a better better power are in better than intuition skill of team may are or be better you have learned this that's where the bridges has to be you know parted now you please go back and remember what you have done in your life previous match results perfect experiment yes so you have historical data right now historical data and that's what becomes empirical probability historical data so out of let's say thousand tests done so far 400 ended up on the let's say you know so 400 were or yeah then you will you will say that there were 600 cases where there was a result 400 was no result that is draw okay now you'll say what is the probability now can you say actually let me give you this data then with result 520 without result out of thousand that is 480 now can you say that winning losing and drawing all are equally probable yes no below now what i'm saying is winning losing or drawing are now are now equally probable our intendant says still says no okay anyone else who says yes now winning or losing is no are in singa also says no okay anyone else who says yes no one says yes how can it be equally probable when 400 80 times 480 out of thousand times it is a result in a draw so obviously for equally probable of win lose and draw what should be the ideal value 0.33 0.33 0.33 then you could have said all are equally probable yes or no but what is the probability of draw here 0.48 so that means 50 chances approximately is there for a draw and the other 50% what happens there it's different thing oh again uh are in see the everything for that matter is dependent on external factors okay so hence you know hence we deal with probability hence we are talking uncertainties here hence we are measuring uncertainties right so hence if you toss a coin wind blows takes away the coin only okay that is also a factor external factor right but do you consider that if you toss a coin hard enough and it goes and strikes the roof so all those conditions are you know right so hence here we are talking about you know how to uh so in such case obviously there will be certain you will you will not just say that okay you know this is a this is a chance this is a let's see if you have to bet or let me let me rephrase this question let's say if you have to bet someone is saying that uh you know if the draw happens then I will give you one lakh rupees I don't do this is illegal in India so let's say you are betting on drawing so what is your chances of winning forget about external factors you winning the bet what is the chance understand you are betting on draw that in the if the match is drawn I win I get some money what is the probability of you winning it how will you calculate you will look at the previous data and CA every other match is being drawn so there is a this is a calculation I am doing so hence my chances is 50 50 percent here almost not exactly 50 50 percent so there is always a possibility that it can go into a draw yes but obviously you'll have to also see let's say uh uh uh the recency is involved for example let's say India playing Australia and Australia is pathetically bad for the last six months then you know that the local probability here the recent recent recency is having some influence then you might influence your betting behavior as well so keeping that aside let's say everything is fine and then you know so hence you can historically you can say that this is the chance okay now tell me another example from cricket only in India versus Pakistan World Cup matches who wins who wins India India has a brilliant track record of not being beaten a single time by Pakistan in a World Cup match any World Cup match okay now still don't you think people people are so confident that they stop watching only or I think I know so many people start praying when India India Pakistan match happens in especially in a World Cup despite the fact that India has beaten Pakistan in all games 100 probability right but still there is some you know uh in either India or park but what Aniket I didn't understand but India in past however many years didn't understand your your your statement I didn't understand I'm saying in the in the World Cup India has beaten Pakistan every yes so hence theoretically we have 100 probability of winning it but then that external factor keeps us you know nervous on that day anyways let's go to the next question what is the probability that a randomly this is a good question take a leap here has 52 Sundays this is so can we say that theoretical probability is not applicable where the events are no no no I'm not saying that moment I am saying theoretical probability is something where you don't need to perform for example in in in case of cricket you do not have any way of evaluating the theoretical probability of winning and losing understand there is no formula given so let's say you can't do two to the power n here right you have two so another another example I'll tell you let's say um uh and you would have done this uh um so so I didn't intend just a minute let me finish with momitas this thing so let's say you momita and all of you please pay attention to this let's say you are uh what um yeah airbag manufacturing company airbag you know airbag that car safety airbag is there right in the steering wheel there's an airbag right so momita is there yeah all of you are there understand do you understand what is an airbag all of you would have seen that god forbid you see that ever but you know that there is an airbag inside that steering wheel and when there's a deep let's say great impulse let's say unfortunately someone's vehicle ramps into someone some tree or something and then that airbag will just get inflated and you will be saved right so because of that so let's say you're an airbag manufacturing company how do you calculate the probability of this airbag opening up when the real impact happens how do you think the companies will be calculating all these values how do you know or let's say um uh have you seen asbestos sheet uh temporarily they make asbestos sheets so how do how do we do quality testing of such thing yes so that means you have to you know basis experiments fake crash multiple times and exactly or fake fake crash also has to be in the simulated environment yeah so hence you have to do a lot of so you could have argued why can't I just know the strength of the material get the velocities and uh let's say all data there and then try to calculate through theory and uh you know I don't know if you have watched this there's a funny I would just like to show you this just spare some minute it will be entertaining you also because you also have been in so much of a world I don't know if you have seen engineering where engineers way of playing a basketball have you seen that and and have you seen anyone any engineers way of playing playing basketball playing basketball very funny I want you to this one is it I don't know if I have the is the audio shared just the audio shared is it all is it I can't hear the audio share what is the probability of so poor guy how much before it starts how many of you've seen this tell me how many how many of you've seen this okay I can see stress rn ralph jharnia okay when I've seen then I will not show it to you so if if everyone has you know uh anyways I'll just do a faster this thing and so that speed speed increase right okay so question how many of you think he is going to put it in the basket so he's taking an attempt do you think he's going to put it in the basket now he's converting a probabilistic model to a different model difference between theoretical probability now you understood what is difference between okay okay so this is what was um what do you say you engineer's way of doing uh so yes coming back to this question what is the probability that a randomly taken leap year has 52 Sundays so answer found out two by seven okay two by seven everyone two by seven what is the probability that some rn random was saying something some issue was there randomly taken leap year has 52 Sundays so how did you solve it randomly how many days are there in uh you know any any leap year how many weeks are there 52 right is it complete 52 weeks and two extra days very good so hence you have two extra days so 52 Sundays are anyways there so what is the probability that a random randomly taken leap year has 52 Sundays right 52 Sundays are there no what is the probability that 52 Sundays are there every leap year will have 52 Sundays one yes is it yes or no can any any leap year have uh no no no sorry sorry sorry sorry sorry sorry sorry no not at all sorry wrong wrong wrong no so they are there yeah correct so hence how what is the probability that 53 is not there so hence you have to check correct Prashin good sorry it was not only 52 Sundays so definitely there but then it can have so they have mentioned uh uh uh little vaguely because 52 Sundays are anyways there okay so yes question is so hence one one is the answer for this because 52 Sundays are anyways there but I think the the question is this not only 52 Sundays so let me change it too so they should have framed it like this what is the probability that only so only 52 Sundays are there what is the probability that uh otherwise let's say there's a question like this what is the probability that a randomly taken leap year has 52 Sundays what would you infer so friends my recommendation would be if you get any vague let's say if you are considering uh you know if you're assuming that there is some kind of ambiguity in the question you can write your assumption and you know very well assert that that you know if you it is talking about 52 Sundays and it has 52 Sundays uh no no uh no sense wise has 52 Sundays means only this only 52 Sundays okay so that it doesn't mean doesn't mean 53 Sundays if it has said at least 52 Sundays then one yeah someone was mentioning that right yes at least 52 Sundays then the one there is no possibility of having 51 Sundays in any leap year so it could be 52 and 53 both so only 52 Sundays that means the leap year uh so 5 by 7 is it it the two extra days which are there should not be Sunday okay so it can be so hence the two two extra days should not be Saturday Saturday Sunday or Sunday Monday in this order right others are possible right so hence all others or there will be five more here what all Monday Tuesday Tuesday Wednesday Wednesday Thursday Thursday Friday and Friday Saturday so these are the total combinations these are ruled out they should not be there so five out of seven leap years will be like that good so hence my recommendation would be you if you are as you are let's say you know uh thinking that there is some kind of ambiguity you can do that next if a number x is chosen at random from the numbers this then find the probability of x square less than 4 x square less than 4 answer is 1 2 3 4 5 6 7 3 by 7 perfect yep only this candidate this candidate and this candidate their square is 1 0 1 square square is 4 square square is 9 square square is 4 again square square is 9 so these are ruled out ruled out ruled out ruled out why it is great just check the sign it is less than 4 has that been less than equal to 4 be very very careful then probability would be different right so in this case 1 2 3 out of 7 so 3 upon 7 very good next ah very easy these are all board papers yeah very easy paper probability you should get 100 marks a die is thrown once what is the probability of getting a prime number okay actually let me change this our die is thrown twice one is very easy you can throw once it is one by two why because you'll get two three and five out of total six so three upon six no problem right now I'm saying throw the dice twice what is the probability that the sum is prime number now this could be a question again you're throwing a dice twice what is the probability that the sum is p rank number p rhyme 7 by 36 how are in how many such cases did you count the cases some highest number of prime number as a sum would be in second case also 1 by 2 prishim this is first answer second answer first answer is half what about the second answer second one you throw it twice sum is prime how many such our intendance is 1 7 by 36 sorry others please try so you're throwing the dice twice or die twice and you are noting down the sum so what all possibilities or how many prime numbers are there either you can do the manual work of adding all so let's say one one one two and all that so it will go from two three four five six seven if first is one then if first is two you can get two plus one three five and then two plus three two plus four six seven eight you can keep counting like that and now it is three so only till five yes you have to add all right instead of that you should make a table I would have made a table right so if you have to count only otherwise I will tell you another method maybe that is also there okay so let's say I have one two three four five six and one two three four five six then this is one two three four five six one two three four five six so two three four five six seven then three four four right so two three four five six seven eight then four five one two two three four five six seven like that and four five six seven eight nine then six seven eight nine ten seven eight nine ten eleven and this is eight nine ten eleven twelve how many are the prime numbers card one two three four five six and seven eight nine ten eleven twelve thirteen fourteen I got fourteen I mean fifteen fifteen did I miss something seven aha yes okay thanks a lot so fifty good 36 is there any other way out without counting because if I do a three three dice throw then I can't make table that will be too hectic job there is another way do you think there is another way of finding out the prime numbers yes or no so there are another you know other ways like that so like this what are prime numbers do you know two right but two can happen only in one combination one comma one let's say three this is one comma two two comma one right let's say five one four not do not two three then three two that's it then seven five two three four then eleven two three five seven eleven eleven what will happen four one ha ha the individual should also be prime number correct correct correct yes then this is also true one four four one that's what I was not counting I was thinking that four is also a not a prime number yes correct we have to be both sides of the angle three four four three anything else possible one six also so one six six one this is another way of doing it and eleven one ten two three four five four six five five six six five that's it okay no like that now count how many one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen like that this is also yeah so hence I don't need to sum all and then check rather than I take prime numbers and then find the combinations and then do it okay yes now if you have try this for let's say three if you if you roll out three dice then you have to now come into diafintine equation solving and all that right so basically finding our integer solution to sum of three numbers adding up to a prime number let's say if you roll out three let's say let me give you a question this like this if you roll out roll three dice what is the maximum possible prime number you'll get sum all maximum possible prime number is when you roll three dice bull up seventeen but in this case now you have to solve this equation what equation x plus y plus z is equal to seventeen this is what we're going to learn in next level so what x plus y plus z team integers are there positive integers three natural numbers x y z is adding up to seventeen how many solutions are there again pnc come into picture okay so using permutation and combination you can find out right so it's like let's say there are there are seventeen you know sticks one two three one two three four five six seven eight nine ten these are ten sticks then eleven twelve thirteen fourteen fifteen sixteen seventeen seventeen sticks are there so basically you have to divide these seventeen sticks into three three groups so how do you do do that so you are let's say you're putting a larger card or a card board between the group of sticks so let's say if I put here one cardboard and here one cardboard then I get this is x this is y and this is x so I'm just clustering them so how many ways can you put these two sticks these two cardboard these one out of how many empty spaces are there there are 18 empty spaces isn't it so empty spaces here two three four five six like that till here there are 18 empty spaces so basically the question is reduced to how many ways can you pick two empty spaces out of 18 empty spaces what is the answer those who have a studied combination they know it how to how to pick two out of 18 what is the formula you have to how to pick a two out of 18 18 c2 right now so you know those many solutions are there for the first equation isn't it so these many come these many so you can put anywhere for example let's say you put the cardboard here and here you'll again get three solutions like that x y z three values so you don't we don't know we don't need the values per se we know that we know or we want the number of values so you can use permutation combination there so this is the next step of probability next so right now in grade 10 you're throwing two dice in grade 11 you will be throwing maybe three dice and then you will require the art of counting which is which we discussed in our classes p and c to find out the so this is just one case there has to be next case let's say for example adding up to 13 then to 11 and all that yes suddenly what are you saying below so anyway i i just got i gave you some idea around it this one the probability of that it will rain tomorrow is 0.85 what is the probability that it will not rain tomorrow 0.15 no brainer here so let's move ahead i hope this is clear to everyone right now this is again on the leap year question find the probability that a leap year selected at random will contain 53 sundays and 53 mondays now so you know it's like a previous question remodeled 53 sundays and 53 mondays 1 upon 7 perfect because these these has to be so the extra two days you know leap year it's a leap year do leap year so extra two days will be there right so extra two days you can have sunday monday monday tuesday likewise all that you need sunday monday only one case is there out of seven cases so you can see monday tuesday tuesday wednesday wednesday thursday thursday friday saturday saturday sunday saturday sunday so out of this only one so one upon seven next this will one marker last year last two last year please be very very careful while you're doing this see three red five seven white ball is drawn from the bag at random probability that the ball drawn is not black two by three okay they are very smart no someone if if they don't put attention don't pay attention to this then there were options are one by three two by three you're also there okay both options are there so if you don't look at it so right so bag contains three red five seven white what ball is drawn from the back at random the probability that ball drawn is not black that means though two ways of doing it either you draw red slash white anything right so probability of drawing red slash white red or white that is is nothing but three plus seven upon 15 three plus five plus seven so this is two by three or you find out the probability of drawing black which is five by 15 which is one by three so p of not black both ways you can do this one some time what is the probability that the sum of two numbers appearing on the top is 13 so what is the problem if i take you what is the probability of probability that the sum of two numbers appearing on this on the top is 11 yeah yeah again just let me see yeah done this guess this this is done 11 two dice are thrown and uh what is the probability that the sum of the two numbers appearing on the top is 11 one upon 18 yep so it is five six or six five agent it agent it okay in the previous case when we were dealing with this there is a constraint also on this guys this was just an idea it is not it is not complete the idea was that x is less than equal to six y is also less than equal to six and z is also less than equal to six this is a constraint isn't it x yz are definitely positive but they are less than equal to six also so hence here some bit of limitations are also there okay so you can't really have one one and 15 so we have to rule out so there is again that that question is still one more step left there so hence don't think that it is 18c2 only you have to go watch much more one step there more there you have to eliminate few of the results from them from that anyways we will cross the bridge when it comes right now this is yeah so arian has posted a query let me just take it let me just finish with this probability of an event that is sure to happen is fill in the blanks very very very simple and not worthy enough to be so let me not say that probability of an event that is sure to happen is one that is not sure to happen no surety zero die is thrown once find the probability of getting a number which is a prime number we did this half lies between two and six the number point number two lies between two and six how much what is the second property okay so this is done four so two kebab uh three and four are the possibilities so this is three and four so what are the possibility in case two events are this three and four and five so hence m e is equal to three so this is will be three upon six half okay this is the answer okay next one probability of selecting a blue marble at random from a jar that contains only blue black and green marbles is one upon five probability of selecting a black marble at random from the same jar is one upon four if the jar contains 11 green marbles find the total number of marbles in the jar is this done probability of selecting a blue marble at random from a jar that contain only blue black and green marbles is one upon five so probability of blue then black probably of green so if probability of getting black is one upon five of sorry blue is one upon five no one upon four so it is one upon four oh then it is not correct so this is one upon four black getting black is one upon four so probability of getting green will be one minus one by five minus one by four right and then 11 green marbles are there so hence what is this nothing but 11 upon total number of marbles and isn't it so hence simple calculation now 20 right so this is 20 minus 4 minus 5 so this means 11 by 20 is equal to 11 by n n is 20 okay can you tell me how many blue marbles are there blue marbles how many how many blue marbles are there yeah blue four how do i know so let's say blue blue is b b by 20 will be one upon five and black is bl so bl by 20 is one upon four so black is five and blue is four so black is five blue are in blue is one by five yeah four and five right so blue is four okay good a bag contains 15 balls out of which some are white and others are black probability of drawing a black ball at random from the bag is two by three find how many white balls are there in the bag very good these are very very easy questions again bag contains 15 balls out of which some are white and others are black if the probability of drawing a black so p b is two by three so since there are only two types of balls so two p w will be one upon three one minus two by three right now let us say total number of black balls or total number of what how many white balls are there in the bag so total number of bag is sorry ball what 15 total number of balls in the bag is 15 so number of white ball let us say w is equal to what now w upon 15 is the probability of being a white ball is equal to one upon three so w is five five white balls here to everyone any anybody has any difficulty so far any difficulty so far please tell me so a child has a die which which whose six faces show the letters given below the die is thrown once what is the probability of getting a now if you see this is not theoretical probability this is on the basis of experiment right what is the probability of getting a or without experiment also you can figure out so hence yes to an extent theoretical yes theoretical only because you're not doing performing any so what you're doing is you're just finding the right outcome divided by total number of outcomes possible so no experiments but then you have special type of die which you have created so theoretical only in this case okay and b what is b b is one by three got it very good next cards marked with numbers a i'm just going through all the questions in a bit faster manner so hence if you anyone of you are not understanding please let me know even after the class you didn't understand any questions stop stay back we will deal with that next card mark with cards marked with number 5 to 50 one number on one card are placed in the box and mixed thoroughly one card is drawn at random from the box find the probability that the number on the card taken out is a prime less than 10 are marked with number 5 to 50 how many are total cards it cannot be 45 46 yes please be very very careful total number of cards are 46 total number is 46 not 45 first one is 1 upon 28 are in tendon is saying 1 upon 46 there are two prime numbers less than 10 5 and 7 right so in this case first first case e set will be having 5 and 7 less than 10 right and there are how many number of cards is 46 right number total number of cards is 46 so hence p e is 2 upon 46 that is 1 upon 23 yep 1 upon 23 correct are in tendon 1 upon 28 2 upon 46 are then number which is a perfect square perfect square less than equal to 50 are what 9 16 25 36 49 yeah 5 upon 46 very good easy so p e in this case a dice thrown once find the probability of getting a composite number one answer is one answer is first is 1 by 2 prime number is a prime number is 1 by 2 prime number is also no versus 1 by 3 hey how come 1 by 2 1 is not a prime or composite so composite numbers are only 4 and 6 so hence it will be 2 upon 6 and prime number there are 3 3 by 6 what all 2 5 and 3 and this one composite numbers are only 4 and 6 1 is neither prime nor composite so 1 by 3 1 by 2 right right answer everyone is comfortable 1 by 3 1 by 2 no problem 1 do not count 1 as composite or prime find the probability of drawing a card which is neither a spade nor a king neither a spade nor a king so count the number of spade count the number of kings but they are not mutually exclusive my friends you have to be careful so there are 13 spade spade and 3 more kings because one king is a king of spade only so total number of such cards are 16 so 16 cards must not be there right so that means the required probability is so what is the probability that you draw spade or a king is 16 by 52 so what is the required probability p of e will be simply 52 minus 16 upon 52 am i right well 36 upon 52 which is 9 upon 23 sorry I am a 13 clear 9 upon 13 oh this is done fair enough so we come to the last slide wherein this is how see how people have responded to the probability question so event they have mentioned this particular person has mentioned dice is thrown outcomes are 1 2 3 so you write possible outcomes favorable event composite number 4 and 6 priority of getting a composite number number of favorable outcomes divided by total number of outcomes the formula has been given so 2 by 6 is equal to 1 by 3 always reduce to simplest form reduce the fraction to simplest form okay then prime number of case made 235 probabilities number of outcomes favorable to the event divided by total possible outcomes which is number of prime numbers divided by total of outcomes total outcomes 3 upon 6 is half okay then this one is another question in this you know so numbered 7 to 40 are chosen total possible outcomes 7 8 9 c it is 34 so you be careful here number of cards counting favorable event is 7 favorable outcome 5 cards so hence like that you know so write each and everything elaborately so probability of selecting a card multiple of 7 so don't write just p e and all that if you are writing then you have to write mention here let e be the event of you can write like that let e be the event of whatever then you can write p e directly no problem is equal to this by this x by y whatever but try to give all the details of whatever you are assuming and on those 3 to 4 marks which are scheduled or which are there which are there now there's a question from arian let me take that question and solve okay so here is the question let me go to so this is something like this all right this is the question correct there are 60 students 60 students in a class among which 30 are boys okay in another class there are 50 students among which 25 of them are boys very good if one from each class is selected what is the probability of both being girls okay so total case a how many girls and boys so we have b is equal to 30 plus 25 and girls in both sections put together 30 plus 25 isn't it 50 50 60 student in the class among which 30 are boys okay so 30 will be girls in other class there are 50 students among which 25 of them are boys so other 25 are girls if one from each class is selected what is the probability of both being girls both being girls is 55 four two are selected yeah what is the problem both uh one is selected from the first one okay one girl what is the probability of half into half is the answer one by 25 is it one by one by four sorry one by four why because probability of selecting a girl from the first class is one upon two one selected probability of selecting girl from the second one is also one by two so hence the total probability will be half into half is that understood uh what is the probability of having at least one girl at least one girl so at least one girl means two girls one girl from the first section and or one girl from the second section so case one case two case three correct or you find out the total probability of both boys together so both boys are there how what is the probability that both of the boys are there one by four again so what is the probability that at least one girl is there three by four all right so probability that probability of both boys both are boys how much is this one by four again same probability 30 by uh sorry uh one by two into one by two so probability that both are not boys this is what is at least one girl is three by four am i right or you can do it from here also two girls both are girls so one by four probability only one girl from the first section so that is uh half of you know half is the probability of selecting one girl uh from the first section and one is the probability of what selecting a boy from the other section let's say that's that's the case of one one girl and no wait a minute one by two one by two sorry one by two into one by two again so this is the probability of one girl and one boy and this one again one by two into one by two so total is one by four plus one by four plus one by four this plus this plus this so three by four are in did you get it both questions are in yes no confirm hello not confirm me yeah once again they go what is the what are three two ways of doing it one is at least one girl is not both boys right these are same condition no at least one girl mean not both boys both boys should not be there correct so both boys top probability is what so probably that both are boy is nothing but again one by two from the first one by two from the second so one by two into one by two one by four one by four is the priority of both boys are there so hence probability that both boys are not in there are in both of them are not boys it's nothing but one minus one by four three by four clear this boxes this part is clear The second part is case by case. Let's say both girls are there. So both girls have probabilities 1 by 2 into 1 by 2. One girl and one boy. One girl from the first section probability is half. One boy from the second section probability is half. Then this one. One boy from the first section probability is half. One girl from the second section probability is half. So total probability is 1 by 4 plus this probability 1 by 4 plus this probability 1 by 4. Why am I adding? Because these are mutually exclusive. If this happens, that is both girls are being selected then one girl one boy is out of question, right? So both are mutually independent, isn't it? So this doesn't impact this. This doesn't impact this. Hence add all of them. Okay. So total probability is 1 by 4 plus 1 by 4 plus 1 by 4 is 3 by 4. Is that understood now? Both. Yeah. Fair enough. So thanks for your time guys. We'll close the show here. We'll meet again day after tomorrow. We are going to start triangles now. The only most crucial chapter left. Then there is circles. I think circles we did. Construction is there and then we have to do the real numbers for the regular batch and matrices and rational expressions for the R and R batch. Okay. So we are maintaining a good pace. We will be thorough by first week. Okay. Fair enough. Thanks a lot for your time. Bye bye. Take care. Have a nice evening. Bye bye. Take care guys. Enjoy. Go take a walk. Go for cycling, skipping, whatever. Do some physical exercise. Enjoy.