 Hello there, in this video we will discuss module for discrete probability distributions. Our first stop is going to be an introduction to these types of distributions. So when I say discrete probability distributions, we are talking about looking at all of the possible outcomes of an experiment and their corresponding probabilities. For instance, if I wanted to flip a coin five times, what is the probability you get zero heads, one head, two heads, three heads, four heads, five heads, and listing each of those probabilities? Yes, it is a lot of calculating. So first, a random variable is a variable which represents the outcomes of a statistical experiment. So we represent random variables with capital letters like x, y, and z, and then specific values these random variables can equal are represented with lowercase letters. So don't let the word random variable scare you. It just means basically the outcomes of a statistical experiment. So let's say let x be a random variable representing the number of heads you get when you flip a coin five times. That's all it's saying. A discrete random variable is, if you recall what discrete data is or discrete data are, it's a random variable whose outcomes are countable. So remember, these are those nice pretty whole numbers, zero, one, two, three, four, and so forth. A continuous random variable is a random variable whose outcomes are measured. So we said in particular this is decimals. And then you get to choose how accurate you want to be, how many decimal places you want to go to in a nutshell. That's what we said continuous was. So this module, we focused solely on discrete. Continuous is going to come at a later time. So we're focusing only on discrete random variables, nice whole number values. So identify each of the following random variables as discrete, continuous, or neither. So it's like a little game. So A, the amount of rainfall which occurs in Jacksonville, Florida on a certain day. So amount of rainfall, typically it's measured in inches. Can I take on decimal values to as many places of accuracy or precision as I would like? The answer is yes. Amount of rainfall is continuous. It is a measurement. What about the number of sandwiches made by a deli on a certain day of the week? The number of sandwiches is whole number values. So we're talking discrete about the gender of a person. Well, that's either male or female. It is not a numeric value that gender takes on. Therefore, it is neither. About the time it takes for someone to run a mile. So we're talking time. Well, when you measure time, you can have decimal values and you can go to as many decimal places as you want as far as the precision goes. So what happens here is you say, okay, it took me 1.5 minutes to run that course. It took me 1.51 minutes. It took me 1.523 minutes and so forth. You can be as specific as you want. Therefore, time is continuous. We're talking about minutes and seconds. So a probability distribution gives the various probabilities of outcomes of an experiment. So like I said, with the coin flipping, you flip a coin five times. What's the probability of getting anywhere from zero to five heads? That's what you would have to list as your outcomes for the experiment. You would have to calculate each of those probabilities. So here are the requirements for a probability distribution. There is a random variable X. Remember that is the experiment. You're performing what you're looking at, your corresponding probabilities and their outcomes. The sum of all probabilities must be one. So all of the probabilities total of every outcome will add up to one. And each probability, each individual probability of each outcome must be between zero and one. So we will mainly focus on requirements two or three here. Those are the very important ones you have to check. So I have an example of a probability distribution. Let X be the number of threes when a die is rolled three times. So the possible number of threes you can get when you roll a die three times is zero, one, two or three. You calculate each of these individual probabilities. So there's four probabilities to calculate. The reason why this is a probability distribution is because all probabilities are between zero and one. And notice the probabilities add up to one. That's very important. Very important. The probabilities must add up to one. What about a non-example? Let X be the number of field goals a football player makes in three kicks. Sure, you could football player can get anywhere from zero to three field goals. And we have probabilities between zero and one. However, if you were to add up those probabilities, feel free to try it if you want. The probabilities do not add up to one. So do not add to one. So this is not a probability distribution. So someone didn't make it correctly. That's a non-example. We can look at probability distribution and calculate their mean. It's the long-term average when an experiment is repeated many times. And standard deviation is a number that measures how far the outcomes of a statistical experiment are from the mean. So it's just like standard deviation you learned about previously, except now we're talking about with probability distributions. So formulas, the mean of a probability distribution that would be mu is equal to we literally take every data value multiplied by its probability. In other words, we multiply rows together and then add up all of those products. Multiply rows together from the distribution and add up all of those products. Or all of those, yeah, all those products. The variance or sigma squared formula is a bit more complicated. Literally, you subtract the mean from every data value, you square that result, then you multiply by the probability, then you add everything together. Sounds to me like we're going to use a technology shortcut there. There is an actual mathematical shortcut or a mathematical shortcut you can use for variance. And that's multiplying every data value squared by its probability and subtracting the overall mean or expected value of the distribution square. After you add up all the data value squared times probability. Standard deviation, remember that's just square root of the variance. So we'll use technology for that in our next upcoming examples. But for this one, let's actually calculate the mean by hand. So we're going to let X be our random variable and it's the number of girls and two births. So you can have anywhere from zero, one and two girls born. So the probabilities are given in the right hand column to find the expected value, literally multiply across on your rows. Multiply every data value by its probability. The expected value is the sum of these products. Notice that your probabilities add up to one. That means the probability distribution was made correctly. So when you multiply across your rows, zero times point 25 is zero. One times point five is point five. Two times point two five is also point five. So remember we're literally multiplying across the rows. I'll put the asterisk there to represent multiplication. And then you add up these products to get one. You could get two. You could get three. You could get anything for the expected value. One is the answer. So the mean or expected value, the two words are used interchangeably. The mean mu is equal to one. So this is mu, the Greek letter mu. The mean is equal to one. And what this means is that if you go to infinitely many people that are giving birth to two children, the average number of girls born would be one. Which makes sense because typically there's a 50-50 chance. What is the probability of exactly one girl being born? So what is the probability that a random variable takes on the value of one? That the number of girls being born is exactly one. So we look at the one row in the table and what is the probability? The probability is actually point five. So that's how you interpret the table. What is the probability of one or more girls being born? So what is the probability that the number of girls born is greater than or equal to one? So that means what are all of the categories that are greater than or equal to one? Well, we have one itself and two. Those are the only data values or outcomes that are greater than or equal to one. So we add together the two probabilities for those corresponding outcomes. You have point five or point five zero plus point two five. And the answer is point 75. That is the probability of one or more girls being born. So that's just kind of to get your feet wet with regard to probability distributions. We will actually use Google Sheets and I'm going to show you how to use Google Sheets to calculate the mean and even the variance or standard deviation of a probability distribution. A discrete probability distribution. So let X be the number of threes when a die is rolled three times. We're going to find the mean. We're going to find the standard deviation. So we're going to find mu and we're going to find sigma. We'll do that using Google Sheets. So for Google Sheets, we're literally going to go to our Google Sheets document and we're going to go to the two variable stats tab. Notice you have a column A and B that both have data. This is where you will actually type your probability distribution. So starting in cell A2, you'll type zero one two three. So probabilities point five seven nine point three four seven point five six nine point zero zero five. And you'll notice that you'll see this bar in the top right corner that's calculating and it's going to calculate many times. As long as that bar is up there, we have not calculated our answer just yet. So here's what's going to happen now is we need to find the mean and standard deviation and actually the mean and standard deviation are way off to the right hand side. So make sure you get plenty of time for the spreadsheet to calculate. The gray bar on top must be gone for a long period of time and that means your calculation is done. So here we go over to the right. Stats is frequency table or expected value. You see here you have a mean, you have a standard deviation. Those are the two values that you want for your answer. So over here in column R that's where you're looking. The mean is point five and the population standard deviation to three decimal places is point six four seven. So column R is where we look. Make sure you get plenty of time for your for your spreadsheet to calculate. So we said the mean is point five and we said the standard deviation is point six four seven. Much easier than doing a bunch of calculations by hand. What is the probability of exactly one three being rolled? So what is the probability that the number of threes are random variables is representing the number of threes rolled? What is the probability that x is equal to one? Remember look at your table. Look at our row that has one. The outcome of one has a probability of three point three four seven. There's your answer. Point three four seven. What's the probability of at least one three being rolled? So the probability that the number of threes rolled is greater than or equal to one. So this means one two and three add up those three probabilities. Add up point three four seven plus point zero six nine plus point zero zero five. You will add up these three probabilities and what's actually going to happen is you will get point four two one. Point four two one is the answer for part C. And another way to do this another way to do this would be to say hey I know that all of the data or all the probabilities and distribution add up to one. Well if you take out the only outcome that's not included if you take away point five seven nine it'll tell you the sum of the other three outcomes. This is kind of like the compliment rule. So you say okay all the probabilities out of the one let me take out the value that is supposed to be excluded zero which has a probability of point five seven nine and that would still give you point four two one. So there's many ways you can attack this sort of question. So that was some information about probability distributions that are discrete and we learned how to calculate their mean standard deviation and how to read them and what the requirements must be to have a probability distribution. So that's all I have for now. Thank you for watching.