 Hey, welcome to this week's session where we're going to discuss introduction to basic probability. I know that in your module, you don't deal with probabilities in that extent, but there are a couple of topics or a couple of concepts that you need to be aware of and a couple of calculations that you need to be able to do as well. So we're going to try and see if we can get you to that point where you are able to read a probability question and be able to answer that. So by the end of the session today, you should be able to learn the key concepts of probabilities, when things are events, the types of events that are there, like your mutually exclusive events, your compliment events, your independent events, your simple and joint events, and so on, but also more specifically in terms of calculation, you need to know when to apply the multiplicative rule and when to apply the addition rule as well. Okay, so in terms of probabilities, we know that the probability is a term that is used for likelihood of events happening. So a probability can also be described as a chance that a certain event will occur and it will always be between zero and one. So a probability, it's a numeric value and you can represent it in a decimal format or in a percentage format. So in a decimal format, it's between zero and one, so it can never be a value in a negative or it can never be a value bigger than one. If we represent it as a percentage, it's also it's zero and 100%. So it can never be any value in the negative percentage and it can never be any value greater than 100%. When an event is certain to happen, for example, the sun coming up, that event, we call it a certain event and it's always going to have a probability of one. An uncertain event or an impossible event is that impossible, is that event that has no chance of happening and that always have a probability of zero. Now, when we assess probabilities as well, there are three different types of probabilities, the three different ways that we can assess probabilities. We can use the priority, which is the likelihood of an event occurring when there is a finite amount of outcomes that exist and we always use this. In most cases, the probabilities that you're going to be working with in this module are always the priority, they always use the priority approach and the probability that something will happen in priority, it's the number satisfying that event or that event divided by the total number of outcomes and we also have an empirical probability, which refers to a likelihood of an event occurring based on historical data. Also, the probability of occurrence for that empirical probability is the number satisfying that event divided by the total number of outcomes and both the priority and the empirical probability are also what we call classical probabilities because these are most often those probabilities, the way we approach probability concepts or the way we do probabilities, we use either the priority or empirical probabilities and then the third one is the subjective probability. This one is based on the researcher's opinions. It's a combination of individual past experience, personal opinion, analysis, and particular situation. So this one you use mixed method type of a way of assessing your probabilities because that you also have to put yourself in it, your opinion, your prior knowledge, or the way you view things. Okay, so we spoke about number satisfying an event and the total outcome and so on. What are these events? So for example, an event will be a process of something happening. That is an event. You create an event. So each event has a possible outcome that comes out of it. For example, if I toss a coin, a coin has two outcomes that it can land on. It can either land on a head or it can land on a tail. If I roll a die, which I'm creating an event of rolling a die, and a die can will have the outcome one, two, three, four, five, and six because it's got six sites. Those are what we call possible events. And we can define each event in terms of what kind of an event is that. Is it happening on its own or is it happening and it has other things around it as well, happening with others. For example, we do have what we call a simple event. It's just one event. A simple event describe a single characteristic which that single, so that one icon. So a single event will be the probability, though, sorry, an event that a coin will land on a head. That is one event. Landing on a head is one. Or all students registered in a particular module. That's single event. We also have joint events where things happen together. So for example, when you have two characteristics that define that. So a person who registered in a psych 3704 and is a male or a female. Those are two events coming from two different characteristics. One is the module and the other is the gender. Both of them happening at the same time. Those we call them joint event and they are represented by an end to combine the two events. Then we also have what we call a compliment event and a compliment event will be the event that is included, but not in the original event that you are looking at. For example, if I'm taking an example of a coin and I say I'm creating this event of tossing the coin and it must land on a head. A head is the probability is the event that I'm looking at. The compliment of the head would be the other outcome that is also included, which is the take. So a head, the compliment of a head is a take. A compliment of a take is a hate. And we always denote a compliment by using let us like a compliment with a copy or a copy or a compliment or not there must be subscript, superscript, a compliment and so on. You can denote your compliment event as such. Okay, so when we have a total of all collection of all outcomes, all those outcomes are called what we call a sample space. So for example, all sides of a dice create what we call a sample space. Those are the total outcomes. A coin, total outcome will be all the sides of that coin. Total outcome will be all number of students registered for psych. That will be the total outcome plus other characteristics that they have, but it will be the total outcome will be made up of all students registered for modules in a program. Let's say it's become degree or a BA degree in industrial psychology. Let's say that is the cost. So all students registered for that program, they do different modules. So we need all of them, all those modules, they when we combine all of them, they create what we call a sample space. Okay, so we can visualize probabilities and events or mostly events. We can visualize them to make sense of them as they happen. And we can use what we call a VIN diagram. And with a VIN diagram, it creates a joint event, single event of all possible outcomes that we have. Or we can visualize them in terms of a decision tree which branches off. It starts with the whole or the total outcomes and then you start branching out to say if I look at how many students are doing psych or other subjects. And then those who are doing psych, we can look at their agenda and look at whether how many are doing, how many are female, how many are male. And from those ones, we can also look at other characteristics, how many are in terms of race, how many are black, color, white, Indian, and so on. So you can start branching it out in terms of that to look at the possible outcome. So if we look at the coin, a coin has two sides. So if I toss the coin, the first coin, the coin will land on a head and a tail. If I toss that coin again, if it first landed on a head and now it can either, when I toss it the second time, it can either land on a head or a tail. And you can continue with that to create different scenarios as well. And you can use this to create your scenarios and answer the probability questions based on that. And we can use some of these examples when we look at the exercises as well. You can also use a contingency table, which makes it easy for you to visualize events, especially where you have joint events and simple events and you want to make use of a contingency table. So there's nothing stopping you from using also a decision tree. But for me, contingency table, it's much easier to use and interpret. Okay. So from a simple event, we can calculate a simple probability by using observation satisfying that event of a simple event divided by the sample space or the total outcome. So let's say, for example, this is our data of events happening. So we've got gender and the module register. So if I want to calculate the probability that a student is registered in an STA 1610, then regardless of what kind of agenda they are, I'm using simple evented means I'm only looking at the module. There are eight of them. And the total outcome is the sum of all the other values that are there, which is your sample space. There are 20 outcomes. So I'll take eight divided by 20, and I get 0,4. If I need to calculate the probability of a main because the students are registered in different modules, I'm disregarding that I'm only interested in the probability of a simple event main, regardless of which module they are in, the total outcome satisfying that simple event, they are five. And I divide them by 20 to get my simple event, probability of my simple event main, and you can do it for any other question that you want to answer, especially if you're only interested in simple events. So you need to be able to recognize that this question is asking you to calculate a simple event or it asking you to calculate the joint event. So in terms of joint events, these are events that happen at the same time. So it comes from to calculate the joint probability, you will calculate it from a joint event. So two events happening at the same time, two or more, it can be more, actually. So that is also given by number satisfying that joint event divided by the sample space or the grand total or the total outcome. Looking at the same example, so when do we see joint events, STA 1610 and female, there are seven of them. So it means students registered for STA 1610 and there are female, there are seven of them. Those are joint events. So this is where you calculate the joint event. So let's see if we calculate a probability of a male and statistic or STA 1610. So a male STA 1610 event satisfying is one, there is only one male doing STA 1610 and out of 20 people that are registered for this program. And we say one divide by 20 and that gives us the probability and that is the joint event. Marginal events, I'm not going to touch on the marginal events, it's just the addition of the joint events to create a simple event as well. So I'm not going to touch on that. Then we also have what we call mutually exclusive events. And mutually exclusive events are events that happens simultaneously or at this, sorry. Mutually exclusive events are events that cannot happen at the same time. They cannot exist at the same time. So it means usually this refers to joint events. And we know that the probability of a joint event is number satisfying that event divided by the sample space. But for mutually exclusive event, for example, if I need to choose a random student registered for a program and I choose a student from site 3704 and student from QMI 1501. If on that dataset, all of them, it's individual students, they don't repeat courses. They all do different courses. So student A is from site 3704, student in QMI 1501. Both of them cannot happen at the same time because you cannot be in site and be also in QMI. At the same time, you will be in one. So these are what we call the mutually exclusive event and the probability of A and B. And you will notice that I use not an end end, but I use the intersection end, or you can use the end. It means the same thing, and B. So if I need to calculate this probability, the probability of A and B, that probability will be equal to zero. That is mutually exclusive event. Collectively exhaustive event. These are events that complete the sample space. One of the events must be okay, but they also need to cover the entire sample space. Remember on a coin, there is a head and a tail. So for them to be mutually exclusive, head and a tail needs to happen some way, some amount in that. So let's assume that we randomly choose a day in 2014, a day A representing all the weekdays, B representing days that falls on a weekend, and C representing days that falls in January, and D represents days that falls in spring. Weekday and weekend and January and spring, they all create a collectively exhaustive event because they all include all the days of the calendar. But they are not mutually exclusive and weekday can also be in January and can also be in spring and a weekend as well. So if you think about it, collectively exhaustive means all of them together, right? They complete the sample space, but they can also still not be mutually exclusive because mutually exclusive events are events that cannot happen at the same time. So A and B are mutually exclusive because a day in weekday cannot be a day on a weekend, right? So A and B on their own as well, they are also collectively exhaustive. So what have we done so far? Up to now, we learned that we can calculate from using a contingency table, we can calculate the joint probabilities and we can calculate the simple probabilities. And so far what we learned was the probability, it's a numerical value, and it measures the likelihood of an event happening, right? And we learned that the probability has a value between zero and one, inclusive because it can either be zero for an impossible event or one for a certain event. The sum now, you need to also know and remember that the sum of all events or the sum of all probabilities should always be equals to one. And if and only if A and B are mutually exclusive and collectively exhaustive, then the sum of all of them would always be equals to one. The other thing we also learned is the complement event A is defined to be an event consisting of all possible points that are not included in that original event A. So we can use the formula. If we want to find the probability of A and we want to find the complement of it, we can use one minus the probability of the complement or we can find the complement event, the probability of A complement will be given by one minus the probability of that event. They will all give you the same. So if I've got a complement, I can calculate the probability of the event or if I have the probability of an event, I can calculate the complement and going to learn all those things. Then we also have what we call probability rules, and one of them is calculating the event that either or the two of them happening at the same time. So if I need to calculate the probability of A or B, which can also be denoted by the probability of A union B, you can either use the two. It's given by the probability of the simple probability of A plus the probability of B minus the joint probability of A and B. Only if and only, if A and B are mutually exclusive, then the probability of A and B is equals to zero because we've learned that the probability of A and B, the joint probability of A and B, if it's equals to zero, therefore it means the events are mutually exclusive. So only and only if A and B are mutually exclusive, then the addition rule states that the probability of A or B, the probability of either A or event B will be given by the probability of A plus the probability of B. That is only if they are mutually exclusive. We also have what we call conditional probabilities, and these are probabilities of one event given that another event has occurred. And that is given by the probability of an event A given that the probability of B has already happened, is given by the joint probability of A and B divided by the probability of B. And you can write it vis-a-vis-a as well. When events are independent, then the conditional probability states that because of the two events being independent of each other, it means one event does not have any influence on the other one happening. So when one happens, it does not affect how the other one is happening. So two events are independent, if and only if the probability of a conditional probability of A given B will be given by the probability of A because B has no effect on what A is gonna be. And we can do the same. So this one is a repeat of that. Event A and B are independent when the probability of one event is not affected by the fact that the other event has already occurred. And that is independent. So then how do we then find the joint probability of A and B if we've got independent events? So, and that's when we introduce what we call a multiplicative rule or multiplicational rule for two events. So when two events are independent, we know that the probability of A given B from the conditional probability, let's put it this way. We know that the probability of A given B is equals to the probability of A and B divided by the probability of B. This is for normal probabilities. Now, if I'm given the conditional probability and I'm given the simple probability, I can then in a mathematical format, we say we cross multiply. In order for us to remove the probability of B, this site going to multiply by the probability of B, whatever you do on one site, you must also do on the other side. Therefore, this one will cancel out and you will be left with the probability of A and B is equals to, I'm gonna start with this one. First, the probability of A given B times the probability of B as you can see there. That is what we call a multiplicative rule. That's where it comes from. However, if we're dealing with independent events, now if we're dealing with independent events, we learn that the probability of A given B is equals to the probability of A. Now for independent events, and if and only if A and B are independent, then the probability of A and B will be given by the probability of A multiplied by the probability of B because we're going to replace the conditional probability with the probability of A because the conditional probability of A given B for independent events is equals to the probability of A because B has no effect or does not affect the outcome of the probability of A. Okay, so we've learned the multiplicative rules. Now, let's do more examples. But before that, in summary, just to conclude as well. Independent events, remember, is when occurrence of an event has no effect on the probability of the other event happening. Mutually exclusive event is when occurrence of one event precludes the occurrence of another. Or we can also say, mutually exclusive event is when two events can never happen at the same time or cannot happen at the same time. Exhaustive events are a set of all events representing all possible outcomes of a sample space. The additional rule is when events, additional rule occurs when events are mutually exclusive. So you must ask yourself events mutually exclusive and am I calculating an event of either or? Then I'm going to apply the additive rule. The law of conjunction deals with independence. So it meant to go into ask yourself this one, am I dealing with conditional probabilities here? Does event A and B independent from one another? Or are these two events independent from one another? So when they are, then we're going to use, especially for joint event, we're going to use the multiplicative rule because the probability of A is equals to the probability of A. Probability of A and B is equals to the probability of A plus the time, the probability of B. Always remember that. Okay, so let's look at activities. I'm going to do the activities with you. We'll do them together, so. Let's see. In the population, there are 450 people of whom, 150 smoke. What is the probability of randomly selecting a non-smoker? So if this is our sample space N and if this is our X and if non-smoker is X complement, or let's not make it an X way, let's make them A and B. A and non-smoker is A complement. So we need to calculate the probability of an A complement. We are told we can calculate the probability of A which is your X divided by M, which our X is 150 events satisfying, which is the smoking, divided by 450. And that gives us 150 divided by 450, 0,33. 0,33. Therefore, to calculate the probability of A complement, which is non-smoker is one minus the probability of the event. So which is one minus 0,33, which is 0,67, which will be option one. That's one way of doing it. The other way of doing it is, I know that there are 450, 150 are smoking. Therefore, A complement will be 450 minus 150 and that will leave me with 450 minus 150 leaves me with 300. So in order for me to calculate the probability of non-smoker or I can say it's the probability of A complement because I use the A's, I'm gonna use 300 divided by 450 and 300 divided by 450 is equals to 0,67. So you have two ways that you can use based on how we answered the question. So you can either answer it using the probability concepts or you can just go ahead and calculate the event and just do that. Easy, easy. Moving on to question number two. Which of the following does not represent a probability? In your module, I think they expect you to use probability in a decimal format and you will use proportions in a percentage format. So which one does not represent a probability in this instance, it will be option number one, which is in a percentage. So proportion will be percentage probabilities will be decimal. Okay, question number three. If 10,000 students who wrote a university admission test 7,000 passed, it means they obtained 50% or more. So it means they got greater than or equals to 50%. So it includes 50% and 400 only obtained exactly 50%. So out of this 7,400 got 50, so what is the probability of a randomly selected student will fail a test? No, this is just to confuse you. Read the question carefully and make a decision from there and say which statement is relevant to answer this question and which statement is not. So let's start with that. We have our sample space and we have our event A and we also have our joint event. I'm not gonna say it's joint event because this is exactly, I can call them B as of exactly. So B is equals to 5,400 and it is also included in this. So now what is the probability of a randomly selected student would fail? What does fail mean? Fail means the student would have received less than 50%. So if I have greater than or equals to 50%, then I've got my entire of those ones. So the rest means they failed. So 70, 7,000 passed, the rest from 10,000 failed. So our B fail, I'm gonna use this was for this one, and this was for this one, the B there. And I'm gonna use A complement with this one. So A complement is 10,000 minus 7,000 is 3,000. So 3,000 of the students failed. So in order to calculate the probability of a fail, we're going to say 3,000 divided by 10,000. And the answer is, the answer is 0.3. And it's none of those ones that are there because they're looking for the probability of those who have failed the test and to fail the test, they've received less than 50. Unless if they say they obtained 50% or more would mean it does not include the 50. Let me just see if I don't include from the 3,000. If I subtract also the 400 to get to 26,000 divided by 10,000 but this is very wrong, it's 26% so it's not. So I think one of the options here is written wrong. Maybe probably I'm gonna assume that this one was 0.3. Okay, let's look at exercise four. A probability of an event occurring which depends on something else occurring such as passing a test when you do not understand your course can be described as what is the probability of an event happening given that another event has happened. Is it one conditional probability to independent events, the real mutually exclusive event or multiplicative? Independent event describes the event here. We're talking about the probability. Mutually exclusive event describes the event. It's not a probability multiplicative probability. We've never spoken about such thing. We know that this should be multiplicative law, not probability so the only one that is outstanding is just the conditional probability and we remember conditional probability is probability given that another event has happened. The formula, we write it that way, right? And that is the definition of a conditional probability. Let's look at exercise five. Yo, you must let me know if I'm going too fast. Let me just ask you to answer this one so that then it doesn't mean like I'm giving you all the answers. A ball is drawn at random from a box containing six balls. White, four, six red balls. Four white balls, five blue balls. What is the probability that it is red? So remember to always calculate your sample space N and your sample space N will be a total of all outcomes. All outcomes is all of the balls. So there are six plus four plus five, which is equals to 15, that is your sample space. So calculate the probability of red. That will be outcome satisfying red divide by N. How many balls are red? Divide by the total sample space. What is the answer? Are you still there? So the answer would be observation satisfying red. They have six of them. The sample space is 15. And what is the answer? Six divide by 15 is 0.4 and we can just add the zero zeros because they put your answers in three decimal. You can just add the other zeros because it doesn't change the number. Let's look at another one. The probability that Jamie will pass his research methodology exam is 0.5. Find the probability that they will fail. The probability of fail is given by one minus the probability of pass. So they have given you the probability on red. So they told you that this is a probability. Always remember, one, two, three, 100, 16, 20, 10,000. These are what we call events. If it's 0.1, 0.30, 0.5, 0.78, these are what we call probabilities with an exception of one. So one can either be an event and can also be a probability. Always remember, if they give you the event, you calculate the probability. If they give you the probability, you just substitute. So this will be one minus 0.53. And the answer is 0.47, 0.47, which is option four. Which is option four. Okay, now we go into interesting stuff. If a coin is flipped three times, the sample space of all possible outcomes would be. So now, in this instance, you can use a decision tree because they say a coin is flipped three times. So we're going to create three, three ways. So we start at the beginning and we branch out. So it was a head and a tail. And at that point, we taught this coin. So this is the first time, right? The second time, it creates a head. Let me write on there. I'm gonna do it like that. So we toss one time, two times, it lands on a head and a tail. So those will be our three. A head and a tail. A head and a tail. A head and a tail, even though they are not that visible. So this one is head, head, head. Head, head, head. This one, it's head, head, tail. Head, head, tail. And this one and that one, I'm gonna do all of them like that. I'm gonna have all the outcomes. So this one is head, tail and a head. Head, tail and a head. And this one is head, tail, tail, head, tail, tail. And this one is tail. Oh, I forgot to put head and tail on this. Tail, head, head, tail, head, head. And this is a tail, head, tail. Tail, head, tail. And this one is tail, tail, head. Tail, tail, head. And this is tail, tail, tail. Tail, tail, tail. And those are the three possible outcomes. So let's see. Head, head, head. Head, head, tail. Head, head, head, head, head, head. Head, tail, head, head, tail, head. Head, tail, tail, head, tail, tail. Head, tail, tail, head, head, tail, head, head. Tail, head, tail, tail, head, tail. Tail, tail, head, tail, tail, head. And tail, tail, tail, tail, tail. You see how easy it is to find your outcomes. If I look at the other ones, they could also be, but they might be repeating themselves. Head, head, head, head, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail, tail. It's repeating. This one is repeating. So it cannot be that one. So the others, they are missing other things. So you can see. So let's do the last one, or this is the second last. Then you flipped four unbalanced or well-balanced coin. What are the odds that two of the coins will lead, will lend heads up? Also, you can create a decision tree on this one so that then it helps. So it means I have to draw a big one now. So we start at the beginning and it lands on a head and a tail and going to split two. That's one, two, three, and this will be the last one. It should do the same with this one. It has to split twice and I'm going to run out of space. It has to split twice and it also has to split again. And the same on this one. It has to split and it has to split and it has to split again. Okay, so we always start with the head. So it's head, head, tail, and on this one, head, tail, and on this one, head, tail, and we do the same. Head, tail, head, tail, head, tail, head, tail, head, tail, head, tail, head, tail, head, tail. And I'm going to also do the same, head, tail, head, tail, head. Okay, so I've created all those scenarios. So I'm doing it this way so that I know that I don't skip anyone. I include, I'm going to run out of space. Yeah, so the first one is head, head, head, head, that line. Head, head, head, head. This one, head, head, head, tail. Head, head, head, tail. The next one is head, head, tail, head, head, head, tail, head. The next one, head, head, tail, tail. The next one, head, tail, didn't put the head, tail. The next one is head, tail, head, head. The next one, head, tail, head, tail, head, tail, head, tail. The next one, head, tail, tail, head. Head, tail, tail, head. head, tail, tail, head, tail, tail, head, tail, tail. Tail, sorry. That's another tail. The next one, head, tail, head, head, tail. Tail, head, head, head, tail, head, tail. Tail hat, head tail, tail, head, tail, tail head, tail, tail, head, tail, head, tail, Tail, tail, head, head, tail, tail, head, tail, tail, tail, head, tail, tail, tail, tail, tail. So I've got all scenarios. Now, what are the odds of that two coins will land on a head? So we need to find where there are two heads. Let's change the paint color. I know that we, oh, sorry, that is the last one. I just want to go in here and change my pointer color. So we want to find where we've got only two heads in there. One, head, head, head, there are three, two, two, there are three, two, two, two, and those are the ones. So one, two, three, four, five, six, six, two heads coin. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, is our total outcome. So six divided by 16, which is the probability. If we're going to call this A, I always label it my events A, A, A, which is 0, 3, 7, 5, which is option three. That's how easy it is, or how tricky it can be to answer questions on probabilities. And that concludes today's session. And like always, we from Pambili Analytics, we're trying to bridge the gap in terms of numeracy and statistical literacy and data literacy skills. We offer a range of suite of services with skills development as one of our flagship. 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