Hi, this is Jeff Spence, your Math 135 instructor for the Community College of Denver, and this is our video lecture over 5.1, which introduces probability. Now, probability is going to be a big tool that we're going to use throughout the rest of the course, and we need to understand what probability means, how it's used, and what information it gives us to make good estimations and predictions. So, first we're going to...
Go right to the rules of probability. There's two main rules that you have to understand, which we're going to carry through for the rest of the course. So that's very important that you remember these rules.
They're pretty simple, though. Understand what the second thing we're going to do is go through the meaning of an experiment, outcome, event, and sample space. So kind of the framework of an experiment and the events themselves. And then we're going to talk about the classical method of assigning probability.
Classical method is... Things like what's the probability of flipping a heads when you flip a coin? So that's one half or 50% or what's the probability of rolling a two when you roll a die? That's one out of six because there's six sided die. So one divided by six We'll also look at the relative frequency method for assigning probability.
We've already done some relative frequency with our data distributions remember relative frequency is the frequency of the event happening divided by the total number of observations. So it gives us a proportion. And any proportion can be converted to a percentage, which also can be interpreted or written as a probability. So once we do the relative frequency method, we'll talk about the law of large numbers and how that applies to probability. So first of all, the most important thing of this video lecture right now is the rules of probability.
So when we say P of A, capital P right there, you see that capital P of A. Anytime we have a capital P, we're talking about the probability of, and then anything in this parenthesis stands for just whatever event we're talking about. So it could be the probability that it rains, or the probability that you flip ahead, or the probability that you roll a 2. The probability of any event, first rule, the probability of any event is always between 0 and 1, or 0 and 100% if you convert it to a percentage.
So in other words, the mathematical statement is that 0 is less than or equal to the probability of any event, which is less than or equal to 1, or 0 to 100%. So the probability of any event could be 0, or it could be 1, or any number in between. The other thing is that the law of total probability, you know, for any experiment, the sum of all the outcomes has to add up to 1. So, you know, when we're talking about rolling a die, you have six different outcomes, they all have a probability of 1 sixth. If you add that all up, you're going to get 1. So in other words, you can't roll a 7 when you roll a die, because that's not a possible outcome.
There are only six possible outcomes, and the probability of all those outcomes adds to 1. Same thing if you did the probability that it's going to rain tomorrow or snow tomorrow. Let's say it's 40%. Then therefore, the probability that it doesn't rain must be 60%, because... the sum of all probabilities of all outcomes has to add to 1. So those are two really, really important rules. And we also want you to be comfortable with certain probability values and what they mean.
So when we say the probability of a... Sorry, we get the probability of something that's equal to 0, we say it's impossible, like rolling a 7 on a die. If it's near 0, we say it's very unlikely.
If the probability is low, like 0.2 or 0.1 or something, we say, or actually, sorry, not low, like 0.05 or less, we say the event is unusual. High, we say the event is not unusual, like 0.8, 0.7, something like that. When we get near one, what's very certain to occur, like 90% and up, we're feeling pretty strong that that event will happen.
And if it's equal to one, then it's 100% chance that it will happen. We're sure that it will happen. NASCAR get out of here get out of here all right now building blocks of probability so every every time we do an experiment We have to set up the possible outcomes that happen from that experiment So when you roll a die, it's one through six you flip a coin. It's head or tails you're talking about the weather It's rain shine Snow whatever the possible outcomes, so we need to set up these kind of parameters and The whole point of probability is, you know, to really kind of deal with uncertainty.
There's a lot of randomness out there in the world, especially with weather or when you roll a die. We really can't predict the future. So what we do is we assign probability, a numerical value, to the chances that something will happen for us to get a little bit more comfortable with this uncertainty.
So the probability of any outcome is defined as the long-term proportion of times the outcome occurs. So for instance, the probability of flipping a heads is one half, but if you flip a coin 10 times, that doesn't guarantee that you're going to get five out of 10 heads, you might get four, you might get three. But eventually, the idea is, in the long run, if you keep flipping, keep flipping, this has to do with the law of large numbers, that eventually, the probability should get very close or equal to 50%.
So now the experiment is any, you know, We conduct experiments and each experiment has different outcomes. And the list of all outcomes is the sample space. So when we're talking about rolling a die, the sample space is 1 through 6. When we're talking about flipping a coin, it's heads or tails.
If we're talking about the number of times that a student has shown up late to class, that can be anywhere from 0 to the number of classes. Let's call it 12 at this point. So the sample space is just the collection of all possible outcomes. Get comfortable with the fact that we might be doing some sample spaces where we count the number of things.
And when we count things, it's always going to start at zero and go up to the number of things that we could possibly count. An event is a collection of those things. So like, you know, with an outcome, when you're talking about rolling a die, that's an outcome would be the individual outcome of 1 or 2 or 3. But when we're talking about an event, we can get a little bit more complicated and talk about maybe the probability of rolling an even number, which is three different outcomes, 2, 4, and 6. So sometimes we'll be looking at probabilities of events, which are collections of different outcomes. Don't worry about that too much. We'll get into that when we see more examples.
So there's really three ways of assigning probability. We're really only going to deal with the classical and relative frequency method. The subjective method is mentioned at the end, and it's something that you need to be aware of and understand that, yes, while it exists, it's not something we're going to deal with in intro to statistics, but it could be a decent way of assigning probability in some cases.
Now, the classical method can be used for rolling die, flipping coins, pulling cards, selecting markers out of a... group of markers and if you worried about what color they were because when we have that we can set up this fraction here where we have the number of outcomes in the sample space so once again if I'm rolling a die the number of outcomes in that sample space is six and if I want to know the probability of rolling an even number it would be three three even numbers out of six so one half here are some examples so let's say there's a deck of cards Alright, and this is the sample space of a deck of cards, all 52 cards. And we wanted to figure out the probability of getting an ace. Well, there are four aces, that's the number of ways that you can get an ace, divided by the 52 possible cards. So 4 divided by 52 reduces to 1 out of 13. Or you could give this as a decimal or a percentage.
So it's just, when we use the classical method, We know the number of outcomes in the sample space, 52, and then we just count up the number of ways that the event can occur. Well, how many different ways can you select an ace? There are four different ways, so 4 divided by 52. Another example is when you roll two die.
Here's the sample space of rolling two die. Let's say we wanted to find the probability of getting a sum equal to 4. So which die up here in this sample space? could you find where if you added up the two die, you get a four?
Well, there they are right there. So therefore, there are three different outcomes in which that could happen. Out of the 36 different outcomes, there are six by six here, 36 different outcomes.
Three divided by 36 is one divided by 12. Or you could give that as a percent or a decimal. So that's the relative frequency method. Another, or sorry, excuse me, the classical method.
This is the classical method, looking at a sample space. And notice in the classical method, we're not actually rolling die or selecting cards. We're just kind of theoretically looking at the possibility of it happening.
So we're computing probabilities without actually conducting an experiment. Another way to kind of do this when you have multiple experiments in a row, is with a tree diagram. And this is something I'll probably put up in class in notes or when I explain things. But this is an example of if you wanted to flip two coins and get the probability of obtaining one head and one tails.
Well, when you flip two coins, turns out that there's four different outcomes, two times two. Because there's two outcomes on the first flip, two outcomes on the second flip, and that gives us four different outcomes in order here. And the probability of getting one head and one tail is actually 2 out of the 4 outcomes, 2 out of 4. So once again, we're not conducting this experiment, we're just doing it theoretically. That's another name for our classical probability. The other probability method which is used pretty frequently is the relative frequency method.
So actually conducting an experiment, collecting data, and computing the frequency of your event divided by the total number of data values or observations that you made. So usually this happens in the real world because we can't necessarily use the classical method for everything. It's used a lot for cards and other things where we can look at the sample space and figure out that probability. But a lot of times we need to be able to run the experiment itself to get some data and get some numbers. So it's still a fraction of the number of ways the event occurs over the total.
Let me give you an example. Let's say I wanted to figure out the probability that a student in your class is from Colorado. So I'm going to randomly select the students, figure out what the probability that they're from Colorado is. Well, in order to do that, one way I could do it is just, you know, do the survey on the second day like I did and ask how many people are from Colorado.
Well, let's say that 12 out of the 28 folks that were there that day are from Colorado, which is 0.4286 or 42.86%. So then I would say. Since there are 12 out of 28, if I close my eyes and pick a name out of the hat, the probability that that person would be from Colorado is 42.86% because it was 12 out of 28. Or if I wanted to actually start conducting an experiment where I flip a coin and I want to know what's the probability of flipping a tail, so maybe I decide to run the experiment 100 times, I flip a coin 100 times, and I get 56 tails.
So then I would say the probability of flipping a tail with this particular coin is 56 percent. But I know that the probability of getting from the classical method is, it should be 50 percent. And the idea is, is that this diagram is showing that if you keep, this is showing flipping, sorry, rolling a die, but the idea is, imagine if I kept flipping a coin over and over and over again, 100 times, 200 times, 300 times, 400 times, 500 times, that eventually that 56% would continue to get closer and closer to 50%. So that's what the law of large numbers says, that if you conduct an experiment many, many times, we're talking about over 100, over 200, you know, many hundreds of times, the probability, the relative frequency probability, your actual data will get close to the classical probability. So in other words, the more you conduct an experiment, the better that probability that you get from that experiment is.
So subjective method is the last one. This is like when people put odds on the weather or football games or sporting events. And, you know, there's not a way to numerically assess everything because there's so many variables that we can't control.
So that's when we use subjective probability. And it's used a lot in the real world, but it's not something we're going to deal with in Intro to Stats. It's just something you have to be aware of that... there is probabilities of other events that we can't control, or with a lot of variables that we can't control.
But the big thing I want you to be comfortable with is, number one, the rules of probability. Going all the way back to the rules of probability. Probability of any event is between 0 and 1. You have to know that.
And then the law of total probability, the sum of all the probabilities must equal 1. Be careful with these interpretations. Sorry, B. Understand these. interpretations of certain probabilities, but the probability of any event needs to be between 0 and 1. From there we're going to do some basic classical probabilities and basic relative frequency probabilities. And after this we're going to go straight into 6.1, which talks about distributions of outcomes and probabilities.
So we call those probability distributions, which we use for the rest of the course.