So, let's start our little adventure. Now most classes in statistics are going to roll some dice some other time and we're going to be no different. Now a normal die contains six sides. And if it's a fair die it has an equal likelihood of landing with any of its six sides facing up. So there really are six outcomes, and I can wonder what the chances of me are, rolling a two. Well because there's six different outcomes, and we have the six faces rolling on one, well, there's just one in six chance of throwing that. So that gives me a probability of 0.1667 if I wanted to round it off, or 16.67%. The same goes for rolling a two. It's one out of six, there's one out of six chance of the die landing with that face up. Same for three, etc. Now on the screen, you can see beautiful little bar graphs. So to throw a one, to roll a one, there's one way to get a one. To roll a three, there's one way to get that. But we can represent this slightly differently, look at this. Now we just do the heights as if they are the probabilities. And I can now ask you to do your very first, very rudimentary p-value. I can ask you what the probability is of you rolling a two. And you see it is the geometrical area of that little bar. It has got a height of 0.1667 and it's got a width of 1. Now, that width of 1 is because this is discrete data points. We've spoken about those. And it's not because the difference between one and two is just one, I could make very funny dice. Dice, a pair of dice that have one, one and a half, two, two and a half, three, three and a half dots. And the difference between those would be half. But it's discrete, we see it as a single entity. We can't divide that entity any further. So the width of our little bars are one, and the area of that little bar is 0.1667, and that is the probability. Now we can ramp things up a bit. See the two bars are colored in now because I can ask you what the probability is of getting a value of five or six. We can add those and you can work out for me what that probability was. Now that's one die, let's roll a pair of dice, and we sum up the values that land face up. Now you can well imagine I can get anything from a two to a 12. Now, how many ways would there be to roll a two? Well, there's just one way to sum to two. You can have one and one on each die. Doesn't matter which one comes first, there's one and one, it's exactly the same thing. But, I can ask you how many times can you get a 12? Well, same story, you're going to throw a six and a six. A six is going to appear on both die. So there's one way each for two and one each way for two and 12. But what about a three? Well, there are two ways to get a three. The first die can land on one and the second one on a two or the other way around. What about a three? Well, what about a four? Well, there's three ways. There'd be a one and a three, a three and a one, or a two and a two. And here you see the table that contains all the results. And it's intuitive to understand that the probability of rolling a seven is much higher than is rolling a two or a 12, and you can see actually to get a seven there are six ways. And if you count all the probabilities up, there's about 36 ways in which we can get certain outcomes. So, I can now ask you what was our probability of ending up with a 12? Oh, this was one way to get this. So one divided by 36, that's very unlikely. It will give us, can say a p-value of 0.00278. But rolling a seven, that's going to be much more likely. That's going to be six in 36, so 0.16667, or 16.67%. Now the other thing I want you to notice is that if we add all the probabilities, we end up with a one, 100 percent. Nothing else is possible. It contains all the probabilities, so the sum of all our little areas they have to be 100 percent. Now look at this, we've drawn this nice little histogram. It shows us if we look in the middle seven, there's six ways to do that. And if we look at twos and 12 this is one way to do that. But once again we can represent it differently. And that is just with its probabilities again. I can ask you what was the probability of rolling an 11 or 12, well you just add those geometrical areas, the surface are of those two little rectangles. And you get a p-value of probability of rolling an 11 or a 12. So that's a geometrical area gives us a p-value of probability. Now, these were discrete values. So, we chose the width of the bar to be one. They stand on their own, and you've got to ask yourself, what are we going to do when we talk about continuous data points? That's up next.
