 Hello, this is a video covering module 3 probability with our first subtopic being the basic concepts of probability. So when we say probability, we're referring to the likelihood that an event will happen. Well, in order to get to calculating probabilities, there's a few key vocabulary words we should know about. So, first off, an experiment that's a planned activity that's carried out under controlled conditions. So, for instance, an experiment could be you flipping a coin, could be you rolling a dice, could be you pulling a card from a deck of cards. And then the sample space consists of all possible outcomes of an experiment. So when you flip a coin, the coin could land on heads or tails. That's the possible outcomes of an experiment. An event is any outcome of an experiment. So the sample space is all possible outcomes and an event is any outcome of an experiment. So the probability, like I said, is the likelihood an event will occur. So what is the probability when you flip a coin that it will land on heads? What is the probability it will land on tails? What is the probability that it will rain today? Those are all probability calculations. So what I have displayed is a table that shows some probability calculations or some discussion about the probability of children being born and whether they're boys or girls. So in my display, I used B to denote a baby boy and G to denote a baby girl. So if my experiment, my experiment, my task to be performed is to have a single birth. An example of an event for that would be one girl. It's a simple event because only one thing happens, you get at one girl. An example of another event would be just the other child or one child being born. But it's a boy. Those are both simple events. The sample space or all possible outcomes would be boy, girl. You could have a boy. You could have a girl. And you have one child born, it's a boy or girl, probability-wise. Three births. So if my experiment is now looking at three children being born, one possible event out of many would be having one boy and two girls born. So this could be boy, girl, girl. This could be girl, boy, girl. This could be boy, girl, girl. And either of those orders, that's how you can get one boy and two girls. And these are all simple events here once again. All right, so the sample space when you have three children being born is they could all be boys. All three could be girls. You could have two boys and a girl in various orders. You could have one boy and two girls in various orders and so forth. So like the more things that are involved in your experiment, the more complicated the sample space, and also the more complicated your events will be actually. So probability notation, we will often use capital S to denote the sample space. Remember, that's all possible outcomes. We will use P to denote probability, and we will typically use capital letters like A, B, and C to denote specific events. For instance, let event A be the coin lands on heads. So the probability of event A or the probability of the coin lands on heads is 1 out of 2 or 0.5. So putting this all together, if you see P open parentheses, event A close parentheses, that means what is the probability of event A occurring? So some basics for probability is that a probability will always be between 0 and 1, whether it's a fraction or decimal, it should always be between 0 and 1, otherwise you did something incorrectly. And the probability of an impossible event is 0, that means it will not happen, and the probability of event certain to occur is 1. That means it will absolutely happen. And the sum of all the probabilities of all of the outcomes of an experiment is equal to 1. For instance, when you flip a coin one time, probability of landing on heads is 0.5, probability of landing on tails is 0.5, you add those together and it does equal the total of 1. So before we jump into examples, we define an event as unlikely or unusual, sometimes they'll use those two words interchangeably, if its probability is very small. And when we say small, we're talking 0.05 or smaller. So we're talking less than or equal to 0.05. It's when an event is considered unlikely or sometimes unusual is the word they will use. So the empirical probability, whenever I say calculate empirical probability, this is actually when an experiment's being performed real time live action. So if I'm calculating the probability of sum of event A, it's literally going to be the number of times event A occurs divided by the number of times the procedure is repeated. So for instance, if I flip a coin a hundred times and I want to find the probability that I got heads, well you flip the coin a hundred times and then the numerator of your fraction or your probability will be how many heads you got and then on the bottom will be your hundred because you flip the coin a hundred times. So that's the empirical probability. But when we're in statistics class, we rarely talk about the empirical probability just because there's no way I'm going to perform every single experiment I talk about. That's a little bit rough. I'm not going to sit here and flip a coin a hundred times. No thank you. So that's where theoretical probability comes in. So this is the classical approach to probability. And so we assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. So for instance, when you flip a coin one time, there's two different simple events heads or tails. If event A can occur in S of these n ways, so A is the theoretical number of times something occur, then how they calculate the probability of some event A is the number of ways A can occur divided by the number of different simple events. So basically what this means is if you're flipping a coin two times, your sample space, you're flipping a single coin, sorry, you're flipping a coin one time, your sample space is that coin's either heads or tails. So if you wanted to calculate the probability of that coin landing on heads, how many different simple events are there? How many items are there in the sample space? There's two. That's the denominator. How many ways can the coin land on heads while looking at the sample space one? So that's why the probability is a half or .5. So it's number of favorable outcomes in the sample space divided by the total number of items in the sample space or total number of simple events. So why theoretical probability is often used is because empirical probability, that's what actually happens when you perform an experiment, theoretical is just that it's theoretical. Well here's the deal. As procedures or experiments are repeated again and again, the empirical probability of an event does approach the theoretical probability. That's why we use theoretical probability for our calculations. So for instance I have here a coin flipping app and I just want us to observe. Just because you flip a coin 10 times, does that mean you're going to get an even split of heads and tails, a 50-50 split, 5 and 5? I don't know, let's check it out. So I'm going to flip a coin 10 times and I'm going to toss that coin and we'll see what happens. Look at this. I flip the coin 10 times and I received one head out of 10 and nine tails out of 10. Definitely not a 50-50 split. What's going on here? Well how about I flip the coin 50 times? Let's see what the breakdown is. So I flip the coin 50 times, look at this. We have tails occurred 42% of the time or 21 out of 50 and heads occurred 58% of the time or 29 out of 50. Let's do 100. We did get closer to that 50-50 split, 0.5, 0.5, 50%, 50%. Now I flip the coin 100 times, I get 45 heads and 55 tails. What about 1000 times? Flip the coin 1000 times and look at this. I got 494 heads, which is 0.494, very close to 0.5, and I got 506 tails. That's 0.506, very close to that 0.5, 0.5 split, 50-50 split. So that's what the law of large numbers is telling you. When an experiment is performed infinitely many times, the empirical probability, the actual probability when you perform the experiment, does approach the theoretical probabilities. So that's why we use theoretical probabilities. So let's do some stuff now. So three children are born. I want us to list a sample space. So when a single birth occurs, V represents boy and we'll let G represent girl. Alright, so I want to list a sample space. So you write S equals open a bracket, that just means this is a set, it's a collection of items in the sample space. So option one, you could get three boys, all three children born are boys. You could get two boys and a girl. You could get boy-girl-boy. You could get girl-boy-boy. Now let's do the outcomes where you would have two girls and a boy. So you could have two girls born and a boy. You could have girl-boy-girl. And you could have boy-girl-girl. And lastly, you could have three girls born. So there's one, two, three, four, five, six, seven, eight items in the sample space here. Eight possible outcomes, eight simple events, so to speak. So assuming that boys and girls are equally likely, find the probability of getting three children of all the same gender. So the probability that all three are the same gender. So out of my possible outcomes, I'll say number of possible outcomes, how many of those were all same gender? Well, if I look, I see I have triple boy and I have triple girl. So two out of eight, which is actually just one out of four, or .25. Your question will specify whether to give a fraction or a decimal. So the probability is .25 or one fourth, that all children will be the same gender. So contingency table, actually, displays the frequency of outcomes as a table with rows and columns. For instance, if I wanted to analyze exam grades, those that got A's, those that got F's, and then of those who studied and who did not, the contingency table breaks that up. For instance, what I have on the screen is I have a 10. Those are the people that made an exam grade of an A who studied. So contingency tables are great for using the calculate probabilities. So an employer drug test its employees. If an employee is randomly selected, what is the probability of the test results in a false negative? So I want to calculate probability of false negative. That's going to equal the number of favorable outcomes or number of false negatives from our table over the total number of tests. Many people were tested, total number of tests or total number of people. So in my table, it looks like how many people were tested, how many test results do we have? Looks like 99. So my denominator is going to be 99. And now we have to figure out what in the world is a false negative, a false negative. So let's look in our negative column, test negative for drugs, subject uses drugs. Those would be the false negatives. It's a false negative of someone test negative, but they actually do use drugs. Someone test negative for drugs, but they actually do use them. So that's 10 out of 99. And if you were to divide those, you would get 0.1. So 10 out of 99 or 0.1, that is the probability of a false negative occurring, a 10 percent chance. Yikes. I have results for a polygraph test. If one of the people was randomly selected, what is the probability the subject was not lying? So not lying is what we're going to focus on. Give your answer as a person rounded to one decimal place. So I want to calculate the probability the subject was not lying. That would be the number not lying over number of people or number of polygraph test results, whatever you want to call it. So what do we have? 10 plus 30 plus 42 plus 13. How many test results do we have total? I believe 95 is appropriate. The number not lying, so that would be the people that did not lie. That would be the 10 and the 30. So that would be 10 plus 30. There's two different groups here. So 40 out of 95 is going to give you 0.421 and they want you to give a percent rounded to one decimal place. So that means move the decimal to the right two spots or multiply by 100 and you get 42.1 percent. So that's an introduction to basic probabilities and how to calculate them. Thanks for watching.