 Good morning. Just want to paste the register on the chat. Please make sure that you are able to complete the register. I've just pasted the link on the chat. Welcome to your fifth session of the basic statistical literacy sessions. Today we are going to learn about the basic skills to solve basic concepts of probability. Today is the 14th. Time flies. Today is the 14th. The next following sessions we're going to cover discrete probabilities. Do you have any questions, comments, queries, anything you want to say to me before we start the session? I've noticed that on the videos that I have been posted, the previous videos, there are only two videos available, even though we've had four or five sessions. Yeah. If there are no videos there, please send an email to cityandtap at unisa.ac.za and ask about the statistics video. Just copy that email. Every time you have any technical issues regarding the sessions, the videos, the notes, and all that, send an email there and copy me in that email and probably they will address it, because I also don't always check if they have uploaded all the videos because I've got so many other sessions, but it would be helpful if you can send an email there. Okay, thank you. Thank you. Any other questions before we start? No? Okay. So let's start with today's session. It's going to be a very long one. I do have some exercises that we will do at the end of the session or closer to the end of the session, but in between, I will pop in some questions and I will pretty much like if you can also engage, ask questions because we're going to cover a chapter that can be distributed for more than two hours. We will try and fit everything in two hours. So I might rush on some of the things. If things are not clear, please stop me. Please ask questions so that we make it clearer for you so you understand the concept and you understand how to calculate them, right? Because I might run through some of this, some of the concepts. So the only thing you need, you don't need the formulas, you just need your oh yes, you do need formulas and you do need calculator to do your basic probabilities. By the end of the session, you should learn the key concept of basic probability. You should be able to learn the basic rules of probability by applying the general additional rule, multiplication rule, independent rule, mutually exclusive rules and so forth and also conditional probabilities. So since we're going to cover so many things, let's get to it. So with basic probabilities, you first need to understand some of the concepts. That relates to basic probability before you can even start to target to answer any questions. So you need to know how to define what the probability is and a probability is a chance that a certain event will occur and it can always be between zero and one. So when you're going to do some calculation later on, you need to be always mindful that your answer should be between zero and one. If you get an answer that is more than that, then it means there is something wrong that you are doing in your calculations. So we write probabilities in two ways. We can write them in a decimal form or we can write them as a percentage. If it's a percentage, we multiply the proportion by a hundred, then it converts it to a percentage. If it's in a proportion, it will be in a decimal format. So it will be zero comma one, zero comma two, zero comma two, five, zero comma three and one. If it is in percentage, it will be 10 percent, 5 percent, 20 percent, 30 percent. So it has to be between zero and one, which means it's any value between zero percent to 100 percent. When we calculate probabilities, therefore, there should be an event that happens. And when an event happens and we are sure about that event happening, we say that event is a certain event. For example, the sun will come up. That is a certain event that we know that always the sun will come up. And a certain event will always have a probability of one. An impossible event will be that event that you know that it might not happen. And when an event has no chance of occurring, then it will have a probability of zero. And you will see this in terms of mutually exclusive events. So those are events that cannot happen at the same time. For example, you cannot be married and divorced at the same time. So that is an impossible event to happen. Right. And that means it will carry a probability of zero. Now, in terms of assessing probabilities as well, there are three approaches that you also need to know about. But this one, I'm just going to rush through them because mostly we're not going to concentrate on them today. These are the things that your tutor needs to address with you, but I've just included them. So those three events, the three approaches that you can use to assess the probabilities are a priority, which means it's the likelihood of an event occurring when there is a finite amount of outcomes. And each is equally likely to happen. And you will notice that every time we talk about probability, we will talk about the number of ways an event can divide by the total number or which is the grand total. So this is a most commonly used probability calculation where you calculate the probability of an occurrence by using the probability will be given by the number satisfying that event divide by the grand total, which will be your t or your n, the sample size. Then we also have the empirical probability. And this one is the likelihood of an event occurring based on historical data. And this usually it's been used if you want to calculate, or probably when you're doing, if you know people who are doing machine learning these days, we talk about AI, data science, all that. But anyway, if you're doing things like regression, you're going to be using historical data to calculate phenomena. Probability, like if you're going to do a logistic regression, let's say, or a decision tree, the probability that you will get when you calculate those things, it's always going to be based on the historical data. And also it calculates the probability of an occurrence, it got less of that. And it's still going to be the number satisfying the event divide by the grand total, or which will mean the sample space or the total outcome. And the last one, and these are both of them are also called classical probabilities. The last one is a subjective probability. This one is used mostly for research, those research who do probably like your qualitative research more, because this is a subjective probability. It is based on your individual past experience, personal opinion, and the analysis is based on a particular situation. And you can use this also to calculate your probability or to assess how you're going to calculate the probabilities. This, like I said, I'm not going to go into detail in terms of them. Let's, we've been introducing concepts of events, event, an event happening, an event happening, but there are several events, different types of events. So it means when you look at questions asked in your exam or your assignment, you need to be able to understand what type of an event is given. So for every event, there is a possible outcome that comes of it. An event like gender has two outcomes, gender has male and female. In this instance, I'm not going to talk about the others. And so forth. A coin is a, if I toss a coin, that coin has two outcomes that it can land on. It can land on a head or a tail. There are two outcomes and so forth. A die, if I roll a die, I'm creating an event and that die can either land on one, two, three, four, five, or a six. That is an event when creating and that event produces some outcomes. A simple event, it is when you have only one event happening at that time. A simple event will explain a single characteristic. Like for example, if I have a calendar and I just want to know what a day in January is in 2014, that day in January is just that simple event. A joint event is when two events happen. Pushing and rolling a die at the same time will create two joint events. And a joint event can be described by two or more characteristics. A day in January that is also on a Wednesday for all the days in 2014. So there is January and there is a weekday day that I am also looking at. So there are two events. Then we also have what we call compliment event. A compliment is the opposite of the other. So for example, when I was talking about the coin, a compliment event of a head will be a tail and for a tail will be a head. Now, if I have multiple outcomes, let's say for example, on a die, a die has six sides. If I roll a die and it lands on one, the one, the compliment of one will be all other events or all other outcomes which can be either five, two, three, six and four. Those are your compliment of one. So what do we mean by compliment? So we have what we call also but I didn't cover a sample space. A sample space like a die. A die is a collection of all outcomes. That creates a sample space. Within a sample space there are outcomes. Oh, there is a, you create an event and then with those events there are outcomes that comes up. If I roll a die, so I toss a coin and it lands on a head, I've created an outcome a head and compliment of that head will be an event that didn't happen which will be a tail. So that is why a tail is a compliment of a head and a head is a compliment of a tail. I'm not sure if you do understand what I'm trying to explain here. So all events that are not part of A will be your compliment. Like I said, a sample space is a collection of all possible events for example, all these sides of a die create a sample space. When I roll, I create that event, I roll the outcome can be any of the sides. The same happens with the deck of cut. All 52 cuts of deck of cuts create, there are a sample space. If I pick one cut, I'm creating an event where I would pick an outcome of an ace of a diamond or ace of a hat or a clap and so forth. Now how do we work with this? How do we work with events? Organizing them and visualizing them, we can do that in three ways and I always prefer the last one that I'm going to show you. So we can organize events by using a VIN diagram. With a VIN diagram, it tells you this is your sample space, everything. Remember if I'm using all days in 2014, an event A can be all the days that falls on a Wednesday, and event B can be all days that falls in January. And where they both occur on the same, like a day is in January and also a day is on Wednesday, that we call a joint event. That will be where both of them are combined. That is one way of visualizing probability or events. The other way is by using a decision tree, which will give you a combination of different events that happen. So in terms of this, if we are experimenting with two coins, and we toss the first coin, it will either land, that coin will land on a head or a tail. The second time we toss the coin again, it will either land. So if the coin landed on a head, when I toss the coin for the second time, it can either land on a head or a tail. The same way, if it first landed on a tail and I toss again, it will land on a head or a tail because there are only two outcomes. And if I need to know what type of an outcome would I receive from tossing a coin twice, the first can be a head and a head, a head and a tail, a tail and a head, a tail and a tail. Those are the outcomes that can happen from tossing a coin that way, which I always like to use. This is my go-to way of visualizing probabilities. So if they give you statements and they ask you questions about probabilities, you can draw yourself a contingency table, which is the easiest way to visualize your information that you are given in order to facilitate the process for you to answer the question asked. So a contingency table, we've dealt with contingency table previously when we were talking, no, we didn't deal with it, but we will deal with contingency table when we look at Chi Square. But a contingency table gives you information. Mostly it will give you your simple events happening and it will also give you your joint events on the same table. So the values that you see inside the table, we call those joint events. The values on the total site, we call them simple event because the number of men or the number of police officers who have the status, oh yeah, the table is of promotion status of police officers over the past two years. So men who are police officers over two years, they are 960. That is a simple event. Women, they wear 240. That is a simple event. However, those who are promoted and they are men, they are 288. That is joint. Joint in terms of men and being promoted. Men who are being promoted, they are 288, which means this is a joint event. So you need to know where you calculate your simple event and where you calculate your joint event based on the contingency table. And the grand total or your T or your grand total, sometimes we use N, so it will depend. That will be, we have 1200 police officers, but this is how you visualize events. Now let's look at the examples of how do we calculate simple probabilities? A simple probability, like we said previously, it's where a simple event we explained previously, we said it's an event where there is only one single characteristic. So a simple probability will refer to the probability of any simple event of that one simple characteristic. And we calculated by saying the probability of an event A, if A is my event in this instance, the probability of an event A is given by the number satisfying the event divided by the grand total or X divided by N, given that table that we worked on previously. Remember that of police officers who were promoted or with their promotion status, so those who were promoted and those who are not promoted. And we also given it broken down by gender, which is men and women. And we know that there are 1200. So if I need to calculate the probability of promoted, that is irregardless of whether the person is a male or a female. I need to know how many of the police officers were promoted. And those will be 324 divided by 1200. Because irregardless of men or women, because there are 288 men and 36 females in this, we just add 288 plus 36 will give us 324. And we divide by the grand total, remember it's N, and it's the grand total, which gives us 0,27. I want you to calculate the probability of men. Why is the probability of men? That is your exercise. Okay, so the probability of men will be given by the probability of men will be given by the number satisfying men. How many? 960. 960 divided by the grand total, which is 1200, right? And that is equals to 0.8. That is easy, right? And that's how you will calculate. You can calculate also the probability of not promoted, the probability of women, and so forth. So let's move on, because we've got a lot to cover. Do you have any questions before we move? Is everything clear? No response? Then we move on to joint probabilities. In terms of joint probabilities, it refers to the probability of an occurrence of two or more events. That is, hence we say it's the probability of joint events. And that it is given by the number satisfying the joint event divided by the grand total. You can see that the same formula, right? Number satisfying the joint event divided by the grand total. If I need to calculate the probability of men and promoted based on the same table that we have, so I need to know men who have been promoted. Therefore, it means I'm interested in the 288. That will be 288 divided by 1200, which gives me 0.24. Calculate the probability of men not promoted. That is your exercise. Men and not promoted. That is the probability of men and not promoted. And I can see that you all have the same answer in the chart. You know that you don't have to read, what do you call that thing now? Type. You can always like the other persons, the one person who posted first, if you agree with them, or you can put a hat next to it, or you can put a smiley, or you can put, let's see, a sad face, angry face. Nobody's angry in this session. A sad face if they got it all wrong. You can use that as a prize if it's something that you were not sure about and somebody did it, and a laugh if somebody commented, and a hat or a like. Okay, I was playing around in the chat function, so let's go back to our presentation. Okay, so probability of men and not promoted. Men not promoted. There are 672 divided by 1200, and that is 0.56. That's what you wrote in the chat. So if it's wrong, let me know. Happiness? Are you good or are you happy? Yes. There's nobody called happiness. Happiness is for me to ask if you guys are getting it or are you all right. It's like when I say happiness, it's like I'm calling someone no, I'm not calling anyone particular. So now we've learned simple probability, joint probabilities. Now let's go into marginal probabilities. So marginal probabilities are like your simple probabilities. So for example, marginal probabilities, we will calculate them by using, if we are not given the total, then we use the joint probabilities or the joint events to calculate your simple event, which are what we call marginal probabilities. So we add them all up. So if the table had three columns, you will just add all of them together. So it's like your simple probability. So let's say, for example, I want to calculate the probability of promoted. Probability of promoted, then I'm going to assume that I'm not given this column of total. Therefore, the probability of promoted will be 288 divided by 1200 plus 36 divided by 1200, because it is the sum of the joint probabilities. Can you see that? To calculate the probability of promoted, it will be 288 divided by 1200 plus 36 divided by 1200, and it will give me 0.27, because it is the same as 324. 288 plus 36 is equals to 324 divided by 1200, it's 0.27. Now calculate the probability of women. That's your exercise. Probability of women. Okay, I think now we've got the feel of this. Yay, E-P-E-P-O-E-P-A. All right, probability of women is given by, who wants to answer this? It's given by 240 divided by 1200. No, let's assume that you are not given that. How would you calculate that? It's given by 36 over 1200 plus 204 over 1200, which translates to 240 divided by 1200, which is equal to 0.2. There we go. So that is your marginal probability. Any question? Any comment? Then we move on to the next. Now let's talk about mutually exclusive events. So, mutually exclusive events are events that cannot happen at the same time, or they cannot occur simultaneously, or they cannot happen together at the same time. A randomly day chosen in 2014, if that day falls in January and that day is also in February, that will make that day impossible to happen, because a day cannot be in January and also cannot be in February. If it's in January, it's in January, and if it passed, it's passed. So this is a impossible event, and that is mutually exclusive events. Event A and event B are mutually exclusive, because they cannot happen simultaneously. And if we calculate the probability of this event, I'm going to be saying, remember that impossible event always has a probability of 0. So the probability, because this is a joint event, the probability of A and B, which is the other thing that I forgot to mention when we were doing joint event. So you can write the probability of A and B using the intersect, or you can use the probability of A and B. They mean one and the same thing. Will be equals to 0, will be equals to 0. They mean one and the same thing. So joint event, we use n, and later on we will introduce another symbol. But for joint event, event A and B, you can write it as A and B, or you can write A and B, doesn't like intersect B, and that will be equals to 0. Collectively exhaustive events. So now these are events that make up your sample space. It's everything. One of the event must occur, and the set of events covers the entire sample space. Let's use the same calendar, randomly chosen day from 2014. If A represent the day that is on a weekday, B represent the day on a weekend, and C represent the day in January, and D represent the day in spring. Can we say all these events are mutually collectively exhaustive? Yes, they are collectively exhaustive all of them, but there are also some of the characteristics. So event A and B and C and D, even A, B, C, and D are collectively exhaustive because they cover everything, the entire sample space. Because if I look at the A and B alone, they actually also A and B covers the entire sample space because every calendar has a day that falls on a weekday and a day that falls on a weekend. However, all of them are not mutually exhaustive, because a weekday can also be in January. There is a joint probability there, and a weekday can be a day in spring, but A and B are mutually exhaustive, right? Because they cannot be on a weekday and also on a weekend. And also, what else can we notice about this? Only that. I'm not going to go around checking, but A and B can be collectively exhaustive on their own, and event A, B, C, and D are also collectively exhaustive, but not necessarily mutually exclusive. And also B, they cannot not be mutually exclusive because a day on a weekend can either be in spring or in January as well. Probably you can use that. And January is January spring or summer. I always get confused with when does spring start, when does summer start. So January and spring are mutually exclusive because spring happens, August, September, October. I think I don't know the days of your autumn. When is autumn, when is spring, when is summer, and when is winter, when does winter start? So I guess January falls under summer. It's summer in January. I'm not sure when does summer start and end, and when does spring start and end. All right, let's now move on. So we now know simple probability, joint probability, mutually exclusive events, and collectively exhaustive events. We also know that there are what we call complement events as well. So in terms of what we just discussed right now, we discussed things that relates to this contingency table. So we discussed that if I have two events happening, event A and event B, they, we can have joint event. If I look at event A1 and event B1, I can have a joint event of the probability of A1 and B1 happening. Or I can look at the simple event of A1 and calculate that simple event by calculating the probability of A1. We can calculate joint event inside the table and we can calculate your marginal probabilities or simple probabilities by looking at the total. Right, let's summarize so far. We've learned that a probability is a numerical measure of the likelihood that an event will happen. Remember it can be either between zero and one. And an event must be between, or the probability of an event must be between zero and one, inclusively. If it's one, it's certain if it's zero, it's impossible and if it's in between, it's a 50-50. So what we mean already, we've said the probability is between zero and one for any event A. Now what we have uncovered is that the sum of all probabilities, the sum of all probabilities should be equals to one. So if I have event A, B and C, the sum of all their probabilities, their simple probabilities should give me one. And this is if A, B and C are mutually exclusive and collectively exhaustive. And we know about complement event. So a complement event, we also defined it as an event consisting of all points that are not part of A. So if I have a complement of A, it's all event consisting of sample points that are not part of A. And how we calculate complement event, the probability of A will be given by the complement of one minus the probability of a complement. So one minus the probability of a complement will give you the probability of A. Or the probability of a complement of A is the same as one minus the probability of A. Let's see if you understand that concept of complement. The probability that it will rain today is 0.7. What will be the probability that it will not rain? You don't have to type in the chat, you can unmute and give me an answer. It's 0.3. It will be 0.3 because if I know that the probability of rain, if I know that it's 0.7, therefore the probability of no rain will be given by one minus the probability of rain. And that is one minus 0.7, which is equal to 0.3. That's how you find complement event. Happiness, I must find another way because my happiness doesn't feel like you guys are feeling it. Are you fine? Are you okay? I'm going to use the English words now. Are you fine? Are you okay? Is everything clear? Okay, you are happy. All right, so now let's get into the probabilities. So we've learned about the basic things. So now let's apply what we have learned in terms of how do we calculate additional rules or further probabilities if we're looking at the rules of probabilities? So the first one is the general addition rule. And this is the probability of disjuncture or the additive law. What does that mean? It says, if I need to find the probability of an event happening or another event happening, that is the probability of that simple event plus the probability of another simple event minus the probability of their joint event. So finding the probability of A or B happening, it's given by the probability of A plus the probability of B minus the probability of a joint event of A and B happening. However, if they tell you if and if A and B are mutually exclusive so it means if A and B cannot happen at the same time then the probability of A and B will be equals to zero so therefore the probability of AOP will be given by the probability of A plus B you always need to remember that if AOP are mutually exclusive then the probability of A or B will be the probability of A plus the probability of B if they didn't tell you or you don't see them as mutually exclusive then the probability of A or B is equals to the probability of A plus the probability of B minus the probability of A and B. Now when you look at this probability of A and B let's say they give you the probability of A as 0 comma 1 and the probability of B is 0 comma 2 please do not take this and say substitute here and say 0 comma 1 plus 0 comma 2 minus and put it in the bracket and say it is 0 comma 1 plus 0 comma 2 this is not the same as that they should give you the probability of A and B as 0 comma 0 5 if if possible they are different because this is a joint probability this is a simple probability this is a simple probability you cannot add the simple probability to find the joint probability but you can add joint probabilities to find a simple probability remember that right so now let's get an example of how we use this if I need to calculate the probability of men or promoted I need to use this formula probability of A or B and the table we're going to continue using the same contingency table so I need to find the probability of men or promoted and that is given by the probability of male or promoted so I just simplified this is P and this will be my NP and this will be M and this will be W in order for me to write them short on this instead of writing main women they are long with so I can just use the first let us so the probability of men or promoted is given by the probability of men plus the probability of promoted minus the probability of male and promoted and that is given by the probability of men I need to go to main it's 960 plus the probability of promoted 324 minus the joint probability of men and promoted and that gives me 0,83 on that note that we do not double count because if you look at 960 it includes 288 because it's 288 plus 6 72 gives us 960 and also 288 plus 36 gives us this so we just remove only one of them and not double counted your exercise calculate the probability of women not promoted women or not promoted calculate the probability of women or those who are not promoted are we winning okay who wants to do it probability of women not promoted is given by the probability of women plus the probability of not promoted minus the probability of women not promoted probability of women is 240 divide by 1200 plus probability of not promoted it's 876 over 1200 minus the probability of women and not promoted which they are 204 over 1200 did you get the same yes okay so 240 plus 876 minus 204 divide by 1200 gives us 0.76 that's what you have right it's what you have on the chart is it right is it complicated when we do more exercises then it might be more clear so let's move on to what we call conditional probabilities my if you don't understand something before we move you need to let us know so that we can help you don't be quiet okay so conditional probabilities are probabilities of one event given that another event has already happened so if we want to calculate the probability of an event happening given that another event has already happened that is what we call conditional probabilities the key word here is given that if you get it in the sentence in this introduction but no then they tell you that given that an event X has happened what is the probability of Y you just need to know that you're dealing with conditional probabilities given if I have event A and B happening and they tell me that I must find the probability of event A given that the probability B has already happened that will be given by the probability of joint event of both A and B divided by the probability of the given event B so it means this given the statement when it starts with given will be the one you divide by and the given is also denoted by this line that divides between the two so this is the conditional probability of A given that B has occurred or you can write it vice versa the probability of B given A has occurred will be given by the probability of the joint event divided by the probability of the given event and that will give you the probability conditional probability of B given that A has occurred so how do we then apply this? Let's say our statement moving on it says of the people hired has 27% have been promoted and 80% are main and 24% are both main and promoted so now we are given any contingency table but we can take this statement and convert it into a contingency table because they have given us a simple event 27% have been promoted that is a simple event promoted they also gave us 80% are main which means this is a simple event or simple probability of main and they also gave us 24% are both main and are promoted that is joint event probability of main and being promoted so we can draw up a nice contingency table for ourselves and have the top headings main and women for gender promoted and not promoted you can also swap the two doesn't really have to be main and women at the top I could have written it the other way around as well now let's start including our data that we have 0.27 it says they are promoted that will be a simple event remember of promoted goes into the total column and 80 into the total column of main 24 will be the joint event of main and promoted and then the rest we can fill it up because we do have 0.27 we can calculate probability of women by saying 0.27 minus 0.24 and that will give us 0.03 or we can calculate not from main not promoted and that will be 0.80 minus 0.24 and it will give us the answer what we also know is remember the sum of all probabilities should be equals to 1 that's all what we know and we can also already calculate this probability because then it will be 0.2 because 1 minus 0.8 will give us that so these are some of the things that we already know and some they gave us remember the sum of all probabilities should give us one and that is why I have that one so let's populate the entire table 1 minus 0.27 is 0.73 1 minus 0.8 is 0.2 and 0.27 minus 0.24 is 0.03 0.8 minus 0.24 is 0.56 and the last one will just be 0.2 minus 0.03 will be 0.717 or we could have said 0.73 minus 0.56 it will give us the same answer if I add 0.8 and 0.2 will give us 1 0.27 plus 0.73 will give us 1 if I add everything that is inside the contingency table it will give us 1 so 0.24 plus 0.03 plus 0.56 plus 0.17 will give me 1 now we can answer the question if I need to find remember the formula that we can use we can always rely on so if I need to find or calculate the probability of main given that they have been promoted and that will be given by the probability of main and promoted divided by the probability of promoted so you go and find the probability of main and promoted which is 0.24 divided by the probability of being promoted which is 0.27 now my question to you is calculate the probability of women given that not promoted calculate the probability of women given that not promoted what is the probability oh sorry given that not promoted what is the probability of women it's one and the same thing as what is the probability of women given that they are not promoted given that they are not promoted what is the probability of women and remember that that is the same as the probability of women and not promoted divided by the probability of not promoted. Hi, seem like you guys are on the roll on the chat. So what is the probability of women and not promoted? Not promoted is 0.17 and the probability of not promoted that will be the simple probability 0.73 and that is equals to 0.23. All right, any question? We are almost almost almost done with introducing some of the concepts and then we just going to go into exercises. If there are no questions then I guess you guys understand how to calculate conditional probabilities then we can move on into the other rules. Then we also have what we call multiplication rule. Multiplication rule for two events A and B. Remember the previous one when we did without with addition rule it says the probability of A or B, right? Now because we have introduced conditional probabilities let's go back to that. Remember the formula for the conditional probabilities that is the formula we can use to calculate the multiplication rule because with conditional probability we can calculate the probability of A and B. So I can write the formula conditional probability says the probability of event A given B is given by the probability of event A and B divided by the probability of event B. Now if I need to find this probability of A and B and because this is dividing I can multiply by the probability of B this side therefore it means I must multiply by the probability of B that side then this side the probability of B and B cancels out and I'm left with the probability of A and B is equals to the probability of A given B times the probability of B. That is the multiplication rule is it from conditional probability to a multiplication rule. Now if and only if event A and B are said to be independent then the conditional probability of probability of A given B will be the same as the probability of B because the probability what happens to the probability of A and the probability of B they are independent they do not influence one another so given that B has okay has no bearing on what happens to the probability of A this is if and only if A and B are independent and how would you know that you will know that A and B are independent if they give you a statement then they say A and B are independent what is the probability of A given that B has okayed and then you will know that now you are working with independent event however here is the thing in your assignment or exam they might ask you questions to prove that event A and even B are independent and how would you prove that you have to use the conditional probability because what we're saying is if I can prove that the conditional probability of A given B is the same as the probability of a simple event of probability of A then event A and B are independent I guess you understand what I'm saying if you don't let me repeat again in your exam or in your assignment they might give you a statement and say event A and B are mutually sorry are independent what is the probability of A given B and you just need to calculate that or you need to calculate the probability of A and B or something like that however sometimes they might want you to prove that event A and B are independent and to do that you need to first calculate the conditional probability of A given B and if it's equals to the simple probability of B then it means event A and B are independent and if that is the case therefore the probability of A and B the probability of a joint probability of A and B it's given by the multiplication of the probability of A times the probability of B if they are not independent then the probability of A and B is given by the probability of A given B times the probability of B now here's the other thing that you always need to remember we are we referring to the probability of A and B remember in the beginning right at the top we spoke about event A and B being joint event and we said the joint event of A and B is equals to the number satisfying that divide by N when do you use this scenario where you have to use conditional probabilities it is if they give you conditional probabilities or they talk about independent events then you can only use these two formulas if not then you use the number satisfying the event divide by the grand total if you're given conditional event somewhere where they tell you that A given B or they tell you about independent event then you're going to use these two formulas to calculate the probability of A and B if you are not given that then you need to calculate the probability of A and B is given by the normal probability of event satisfying the probability of A and B divide by the sample space I've explained it enough in even I'm going if I try to explain it even further might confuse even more okay so what are we saying two events are independent if and only if if and only if the probability of A given B is equals to the probability of A and the probability of B given A it's equals to the probability of B event A and B are independent when the probability of one event is not affected by the other fact that the other event has already occurred and that is for independence you must always remember that let's summarize what we just learned up to so far we have learned independent event is when an event happens and has no effect on the probability of the other happening we learned mutually exclusive events when one event okay or one event concludes the occurrence of the other we've learned about mutually exhaustive or collectively exhaustive events which means all set of all representing all a set of events representing all possible outcomes we've learned about addition rule which when events are mutually exclusive they will be equals to zero and the additive rule will state that the probability of A or B happening will be the same as the probability of A plus B and if they are mutually if they are not mutually exclusive then the probability of A or B will be the probability of A plus the probability of B minus the probability of the joint event of A and B we've learned about multiplicative rule which also includes the conditional probability that the probability of A given B is the same as the probability of A and B divided by the probability of the given B if A and B are independent therefore the probability of A given B is the same as the probability of A that's all what we've learned so far we can put that into practice later on when we look at exercises so in summary we've learned so far up to now the key concept of basic probabilities we've learned the rules we've learned general additional rule independent rule mutually exclusive rule conditional probability multiplication rule and so forth and complement event now let's put that into practice looking at exercises yay we almost there exercises here is your exercise and we will come back to it this is what you need to calculate there are three questions i'm going to pop one question at the time on the screen now i'm going to give you two minutes to answer this question if the probability of A is equals to zero the complement probability of B is zero comma five and the probability of A and B is zero comma one find the probability of B maybe actually i must give you one minute but because i'm nice i'm going to give you two minutes to answer number one but why not continue also answer number two no answer only one sorry let me not confuse you what is the probability of B we do have the answer the probability of B is one minus the complement of B which is one minus zero comma five which is zero comma five that is the probability of B so how do we find the probability of A or B that is the probability of A plus the probability of B minus the probability of A and B do we have an answer we still calculating 0.8 let's see if it's 0.8 so the probability of A is 0.4 plus the probability of B we calculated it previously 0.5 minus the probability of A and B which is 0.1 and the answer here is 0.4 plus 0.5 is 0.9 minus 0.1 which gives us 0.8 happiness the next question if A and B are mutually exclusive find the probability of A or B if they are mutually exclusive find the probability of A or B so we know if they are mutually exclusive the probability of A or B will be the probability of A plus the probability of B and that is 0.9 0.9 0.9 see from the chat therefore it's just 0.4 plus 0.5 which is equal to 0.9 is it right so if this in um I removed this from one of the past tutorial letter or exam paper or whatever I just removed the answers and I just put them like this so in in your assignment or in the exam this will be options that you're going to have to validate because when you write multiple choice questions like you answer or you're writing an exam or an assignment with multiple choice questions you have to validate or use the process of elimination to answer every statement and most of the time because it's like this it's double the wake especially when you are in the exam is double the wake because every statement requires you to do the calculation in order for you to check whether that statement is correct or incorrect multiple writing multiple choice questions is not an easy thing but if it was just a question like this you know that you are done once you have calculated the answer but with multiple choice questions and later on we'll get to that you will have to choose which one is correct therefore it means you have to calculate each and every one of them to find that correct or incorrect answer okay so let's look at other exercises how do we calculate questions relating to probabilities probability of b is equals to 0.2 and the probability of a complement is 0.7 and the probability of a given b is 0.9 find the probability of b given a the first thing you need to do with this question actually as well is minus minus i also don't like multiple choice questions it makes you wake double time or triple time okay find the probability of b given a so the first step that you need to do is to write the formula for this question that they have asked you it's always important that you need to first write the formula because i now know what the question is asking and they i know what i'm given because everything i i'm given it's already there i don't have to rewrite it it's not like it's in weight format that i can go again and say let me rewrite the probability of b um is 0.2 so i've given that i'm given that i'm i'm asked to calculate the probability of b given a so therefore it means the first thing i need to do is to write this formula the probability of b given i can also even why am i writing this this it will be given by the probability of b and a or a and b we can use that weight it's given by the probability of a and b divided by the probability of a given event so since i know that this is the formula i need to use looking at the information given you need to always remember that some of the things you can derive from this information that you are given for example i'm not given the probability of a but i'm given the probability of a compliment i'm not given the probability of a and b but i'm also given the probability of a given b so what do we know about this probability of a given b we know that the probability of a given b is equals to the probability of a and b divided by the probability of b so now if i know that the probability of a given b is 0.9 i can calculate the probability of a and b because i'm also given the probability of b so then i can use that into this formula to calculate my probability of b given a that is the process that you need to use when you look at questions find the formula look at what you are given is it sufficient if not what else can i use from the formula to find the answer the first step let's find the probability of a we can do this one together you don't have to i don't have to give you more time what is the probability of a it's one minus 0.7 which is equals to 0.3 right oh you guys are so quiet it feels as if like i'm here alone so now i have my probability of a the next step is for me to find the probability of a and b because that's what i need to calculate there but i cannot find the probability of a and b if i didn't write this formula so now i know i can use my multiplication rule because i can multiply the probability of probability of a given b multiplied by the probability of b because that will give me the probability of a and b right and that is equals to the probability of a given b is 0.9 multiplied by the probability of b is 0.2 and that is equals to what is it equals to 0.18 it's equals to 0.18 now i do have the probability of a and b i have the probability of a i can find my probability of b given a and which is given by the probability i'm not going to rewrite the formula again because we have the formula there the probability of a and b which is 0.18 divided by the probability of a which is 0.3 and this is equals to 0.6 any question any comment any query okay here guys we only left with 30 minutes and you don't want to talk to me sure it's gonna be a long long day a long 30 minutes me alone talking to myself no it's just Lizzy we are here we are just understanding and digesting like what you're saying oh is that too complex okay so let's let's revisit back our previous questions now we know what the probability of a is we know what the probability of a and b is we know what the probability of b given a is now we can answer the following question because it says use the previous answers as a hint if you're given so for example this will be option one option two option three and then option four and option five in the same question and they're asking you find the incorrect answer or find the correct answer depending on which one they messed with and this is the statement that they give you and they say our event a and b independent what do we know about that so first let me write out all the answers that we got from the previous question so probability of a we said it is 0.3 probability of a and b we said it is 0 comma 1 a and the probability of a it was b given a right the probability of b given a we found that it was 0 comma 6 now here they say our event a and b independent we need to prove that event a and b are independent how do we find that we can use the probability of a and for independent event we can prove that the probability of a given b is equals to the probability of a or we can prove that the probability of b given a is equals to the probability of b that's all what we need to do right now so let's prove that what is the probability of a given b is 0 comma 9 is it the same as the probability of a probability of a is 0.3 so 0.9 is not the same as 0.3 therefore they are not independent that is the first one you can use let's look at the second one we keep we want to use that is b given a so if b given a is 0 comma 6 is it the same as the probability of b which is 0 comma 2 no they are not so they are not independent event a and b are not independent so we have proved that can we prove that they are mutually exclusive then what do we know about mutually exclusive mutually exclusive event says the probability of a and b should be equals to 0 that is the rule that that we use remember that that is the rule the the rule says the probability of a and b should be equals to 0 so what is the probability of a and b the probability of a and b for this question was how much a and b was 0 comma 1 8 are they mutually exclusive that's your question are they mutually exclusive no they're not no they are not mutually exclusive not mutually exclusive if the answer here was 0 then we would say they are mutually exclusive if the answer for the probability of b given a was 0.2 then we would say they are independent if the answer of a probability of a if it was 0.9 we would say they are mutually exclusive but they are not independent sorry they are independent because they are not independent they are not mutually exclusive and that's how you will answer the question in the exam so let's move on and look at how other ways do they ask you questions in the exam so as you can see you given the statement of the majority of employees are waking from home during the lockdown a team of mental health expects reported the following probabilities of primary givers waking from home versus those waking away from home and the long-term impact of their children attachment style secure or insecure attachment and they have the table of contingency table values and they've got secure insecure home and away and they've given you the joint probabilities and they gave you the actually they gave you the the other thing I need to also mention is that you might be given events or you might be given probabilities these are probabilities already if you see the the answers or the values in decimal or in percentage you must know that those are probabilities so there is no need for you to go and divide by the grand total there are probabilities you just plug and calculate so they've given you the joint probabilities of home and secure of 0.38 away and insecure as 0.29 away away and secure as 0.18 and home and insecure is 0.15 and they gave you the the grand total what they didn't give you is the simple probabilities so you can calculate your simple probability so let's do that before we answer any of the questions so simple probability for home take a calculator and give me the values 0.53 0.53 of away 0.47 0.47 and secure 0.56 0.56 and for insecure 0.44 0.44 so now you have your joint probabilities and your simple probabilities the first question is asking which one of the following question or statement is correct so we need to find the correct statement but in a way we're going to calculate all of them right now so that we just go through a process of elimination the probability of home and secure is 0.15 is that true or false you don't have to calculate this one just look at the contingency table tell me home and secure is this is not true it's incorrect the probability 0.38 can see that that is that probability you're looking for the probability of home or secure now this one you need to calculate as long as there is an or you will have to calculate so the probability of home or secure it's given by the probability of home plus the probability of secure minus the probability of home and secure so now you have all the probabilities we just substitute and calculate so what is the probability of home that's for you to work it out with me I want this one we work it out together I'm not going to give you time what is the probability of home is 0.53 what is the probability of secure 5 6 minus the probability of home and secure then open my 38 and just calculate and give me the answer and that answer will determine whether that is correct or incorrect 0.71 so therefore this is also not correct right the next one we need to evaluate is following c will be the probability of home given secure so we also need to calculate this one home given secure it's given by the probability of home and secure divided by the probability of secure plug and calculate what is the probability of home and secure home and secure is 0.38 right is the joint probability of both of them home and secure what is the probability of secure is divided by 0.56 so calculate and tell me if the answer is correct what is the answer there 0.68 0.68 and this says is 0.38 therefore that is incorrect so we're done with that let's move on to the next one next one says the probability of secure given home probability of secure given home it's given by the probability of the joint home and secure divided by the probability of home what is the probability of home and secure is 0.38 over the probability of home is 0.5 0.53 calculate and let me know if that is correct 0.72 that is 0.72 therefore that is the correct answer and the probability of home home is 0.53 so that wouldn't be correct so you can see with multiple choice questions you work double time you like you work you have to some of them you have to calculate some of them you you can just use as they are but every statement means calculation as well okay using the same information as previously I'm just going to copy the numbers you have them right you will give me the numbers for the contingency table so we have 0.53 0.47 0.56 0.56 and 0.44 now which one of the following statement is incorrect now we're looking for the incorrect one and this is another way of how they can ask questions they can ask them in a simple format like asking giving you already in a formula type of question or they can ask you in a sentence way like this so you just need to convert the word into a understandable formula that you can use as well a the event secure and insecure are complement what are they saying is event secure is it a complement or event insecure is that correct that's correct that is correct you always need to remember like for example secure and insecure think of it as a coin another site is secure another site is insecure home and away it's also think of it as another coin and the other site is home other way is away so they are complement of each other a home is a complement of away away the complement of home so this is correct but we're looking for the incorrect statement the event home and away are mutually exclusive there is home and away can they happen at the same time no they can't they can't so the statement is also correct because they cannot happen at the same time so this statement is correct they are mutually exclusive event home and secure are not independent event home and secure are not independent now this is where you need to think about the independent events so they say event home and insecure are not independent so we need to prove that event home given secure is the same as the event home probability of sorry we need to prove that probability of home given secure is the same as the probability of home. So what is the probability of home? Probability of home is 0.53. We have that. So we just need to calculate the probability of home given secure. The probability of home given secure is given by the probability of home and secure divided by the probability of home, which is home and secure 0.38 divided by the probability of home 0.53. I'm not sure if we did calculate it previously. What is that probability? 0.38 divided by 0.53. It's 0.72. So now can we prove that they are independent? Are not independent? Yes, we have proved that because they are not equal. If they were equal, then we would say they are independent because we wanted to prove that they are not independent. We can clearly see that 0.72 is not the same as 0.53. So this statement is also correct. Oh, actually I used the wrong thing. Wait, do it, do it, do it, do it. We were supposed to prove in secure. Oh gosh, in secure. Sorry, my bad. Our event. I don't want to remove it because the next one. Okay, let's think of it this way. We have solved number G. So let's go to number D. D says event home and insecure are independent. We've proved that that they are not independent. So therefore this is the correct answer that we are looking for in terms of incorrect answer. For this one, the home and insecure, we also need to follow the same. So we're going to prove that the probability of home given in insecure should be the same as the probability of home. So let's see if they are the same by proving the probability of home given insecure, which is the probability of home and insecure divided by the probability of home. The probability of home insecure, home and insecure, home and insecure is 0.15, 0.15 divided by the probability of home, which is 0.53, 0.5, 0.15 divided by 0.53 is equals to 0.28, 0.28. So we have proved that they are not independent because this is 0.28 and this is 0.53. They are not independent. So that is correct. Apologies for the first one. Okay, so we are left with only nine minutes. Let's see if we can look at one last question. So sometimes they will give you events in a contingency table. Consider the following cross tabulation of the shopping habits of dirty grocery shoppers. So they shop at different stores and by gender, made of female. What is the percentage of checker shoppers? What percentage of all checker shoppers are female? What percentage of all checker shoppers are female? So what they are asking you is calculate the probability of checkers and email. And that is observation satisfying divided by the grand total. How many? Seven divided by 10. Seven divided by 10. Nope, grand total. No, but is this not based on the specific store and not the total of the shoppers? Nope, nope. We need to calculate the probability of checker shoppers that are female. So remember, probability of checkers and female is given by observation satisfying divided by how many there are, which is equals to 0.2333. If I multiply that by 100, then it will give me the probability of checker shoppers who are female. When you calculate probabilities, think about it. If you are given, if in your statement you have words like given, then you can use. Given that all checker shoppers are female, then you will use the given statement where you will take seven divided by 10 because that will be the probability of a joint event divided by the probability of a given. So the statement is saying, calculate the probability of checker shoppers that are female, which it says all checkers that are female are those ones. Remember, all of them will mean both of those. If it's all checker shoppers, but now we only need that are only female. Probability, remember, always go back to the basics. The probability of a joint event, the probability of a simple event is observation satisfying that event divided by how many there are, which is the grand total. Let's look at another example before it's almost time. So now on this one example, if you look at it, they have given you some information. So yeah, the employees of a company will survey on the question regarding their educational background. And yeah, their educational background is around college degree or non-college degree and also their merit status, which is either single or merit. Of the 600 employees, 400 had college degree, 100 had a single or 100 were single, 60 were single college degrees. So it means I can also come here and create a contingency table that will help me answer the question. Single, merit, college, no college. I'm told there are 600 of them. Now I'm also told that 400 had college degree. I'm also told that 100 were single. I'm also told that 60 were single college degree. And therefore this is 40, this is 340, this is 200, this is 160, this is 500. See how quickly it is to calculate total. Now, since I have all my total calculated quick, quick, the questions asking us the probability that an employee of a company is merit and has a college degree. We need to calculate the probability of marriage and college degree. Observation satisfying, divided by how many there are. What is the probability of marriage and college degree? Marriage and college degree, there are 340, right? And the grand total is 600. And then the probability is 0 comma. I must leave it to four decimal or three decimal. I think the majority of them are on three decimals. So let's see with the first three decimal, we get 0 comma 5, 6, 7 if I leave it at three decimal. So therefore only option two is correct. And that's how you answer questions on probabilities. Some comes easy, some comes like. You need to evaluate every statement, do a process of eliminations, some you need to work out some things first before you can answer the question because it might not be as straightforward because at the moment they didn't give you all information that you are able to use. For example, they didn't give you in that statement how many are merit. So you needed to go and find that out. So you can use other mechanisms because if I know that merit and single are complement of one another, I didn't have to do the contingency table based on the information because that is what they are asking. I could have just said in terms of merit and single, there are complement of each other. If I have 600, I need to know how many. So I have 100. So it would have been that merit would be 500. But that doesn't help because I'm looking for the joint probability of college and merit. So you don't have a choice. College and merit, if you don't have other information, you cannot calculate it because you need to know how many are college and single. There were 60 and you need to know how many there are from college. We needed to know there are 400. So we can calculate merit and college by using 400 that we have here minus the 60 that we have here. And that would give us 304 which is that value that we calculated. And then the grand total they gave us 600. So you just substitute 340 divided by 600. But if you had multiple questions or answers that will require you to calculate the given, calculate A on B and you will need to convert the information or the statement given into a contingency table to help you answer every single option. Okay. So I think we had one last question. I'm not going to ask you to do it, but you can always check on the notes. The notes are in the folder and you can also come back to this. You can take a screenshot of it and do your answer. We are right on top of it. As to recap, I've already re-kept previously, but we've learned about basic probabilities. We've learned how to calculate the event probabilities. We've learned how to calculate simple event, joint event, to use the rules, conditional probabilities, compliment events and so forth. So you just need to go and practice and practice and look at different type of questions and how they ask you in your exam papers or your tutorial letters and workbooks and all that. Find more questions and exercises to go through. That concludes our session for today. Any question, any comment, any query before we leave, absent of any of those. Have a lovely, lovely, lovely weekend. See you next week when we look at discrete probabilities. And please next week, bring tables because next week we start using the tables every week when we come. Calculator tables, you can bring them from your tutorial letters or from your prescribed book, not prescribed book, your tutorial letter study guides, your, yeah, your prescribed textbooks and so forth. If you do have those and if you have past exam papers, you can also bring those at the back of the past exam paper. There are tables. Otherwise, I will see you next week. Thank you. Enjoy your weekend. Thank you, Slyzy. Thank you, Slyzy.