 this whole bit right here, right? I might not need this stuff because now I've got the numbers that just basically represent it. So sometimes it might be nice just to unclutter things to actually delete this and then click on these little lines so you get the lines and delete them so that you just get the numbers that are representing them. And that can be another format that you can use just as we build our charts. But you get this kind of boring uniform type of distribution. And if we were to write an equation for the uniform ULA, it would be something just like F of X equals C. So for every X we have, we're gonna get C. So it's a uniform distribution. And note that this is a family of distributions because it's possible that if we roll the dice, less numbers, like if we roll the dice only 300 times, then we have a different distribution. It's still a uniform distribution here that is at the 50, right? So then I'm gonna say 1000 and bring it back there. Now, of course in real life, we wouldn't get a uniform distribution because we're using a sample in essence instead of the entire population of infinite rules, right? We're just doing 1000 rules. So we can simulate, okay, what would actually happen and compare a simulation of rolling the dice? So let's say we do a simulation of rolling the dice, rolls, and now we'll actually test this out and see how, and then we can compare to what the expectation is. That's basically what we often do, right? We're gonna say this is the perfect model of the world, our expectations, and then we'll basically see what actually happens in the test and how close is it to the actual expectation? So we're gonna go to the home tab, font group, make this black and white, and then we'll center it and then we're gonna roll our die. Things are gonna get dicey here. Things are getting dicey. So we'll do this by using our random function again, equals rand brackets or rand between, I should say between tab. There's our formula, and I'm gonna say we wanna start at one, comma, and go up to six. So it's gonna give me a random generation of one to six representing a dice rule, so that's just perfect. And then I can copy that down. I'm gonna go down to 1,001, looking at the numbers on the right-hand side because one is already taken up to get 1,000 rolls. So I'm just gonna bring this down to 1,001, and note I can do that quite easily in Excel, even though it's quite a ways, quite a lot of rolls, would be a lot more difficult to do this. And the old days when you actually had to sit at the table and roll the dice a thousand times, right? It'd be tough. That would be a pretty cushy job for somebody to be the dice roller for the experiment, but now we can just kind of simulate it. And so then if we do that, then this is what it's spitting out for the random outcomes. So notice it changes again all the time. So I don't want it to change. I want that to be my generator. And now with the generator, I'm gonna copy this whole column, right-click and copy, and then paste it, but hard-code it so that I'm gonna say right-click and paste just the numbers. So now it's not regenerating every time, and my generator's still over there if I wanna do it again. So let's go ahead and make this one, home tab, font group, black, white. Let's center this thing. And so there are our results. So now let's copy this same table from my expected results and then add the actual results in another column. So to do that, I'm gonna put my cursor in column Q equals, scrolling to the left, and just picking up the dice rolls, and then I'll just copy that down. I'll copy everything that's in that table down, six, and then the total, and then I'll copy it to the right as well, and there's the same information that we pulled in from the table. Let's make us format it. I'm gonna go to the home tab, font group, black, white, wrap it, center it. By the way, you could do it this way too. Let's do it this way, it'll be easier. I could select this entire thing and say I just want the formatting so I can go home tab, format painter. Just give me the formatting and then just put that right here, boom, and it paste the format of it beautifully. Okay, so then let's take our actual rules, actual rules and compare it. And so I'm gonna format paint this one again, home tab, format painter, boom, to get that. And then in order to get the actual rules, what I'd like to do is say, Excel, take everything in this series of numbers and count them if every time you see a one. So we can use our trusty count if function to do that. So this equals count if brackets. I'm gonna go over here, put my cursor in O2, hold down, control shift and down on the keyboard, taking it all the way down to the bottom and then we can have control backspace taking me back up to the top and so I can see the formula. So there it is, closing it up and we have too few arguments. Count if, I need to finish the argument, is a comma, what's the criteria? Second condition, number one. So count if in that range you see a number one. Enter and then I can copy this down. Now notice this time I didn't put a table over here and sometimes the tables are useful and sometimes they're not. So I'm gonna try to go back and forth between using the table or not. If I had a table that I was referring to, I wouldn't need to make the cells absolute references but here since I don't, I could make them absolute by selecting F4 here and F4 so the range of the table doesn't move. Now note, you can also use spills and arrays. So there's actually multiple ways to do the same things these days which is really cool but also confusing. So I'm gonna try to mix in some of that stuff as we go. So in any case, I'll do that. I'm gonna copy this down, put my cursor on the fill handle, drag it down. So then I should have the same range here, right? It's picking up the same numbers this way and it's counting if there's a six in there. So that looks good. Now I can kinda double check by summing it up and I should still get up to 1000. So that's our double check that my range didn't get a skewed or anything like that. I could take the difference then, difference between what we expected and what actually happened. This equals what we expected minus what actually happened. I can copy that down, double clicking on the fill handle and then down here alt equals that's our keystroke for the sum function. There we go with our differences. I can go home tab font group and format this in the same formatting and then let's make all of this bordered and blue and let's put some underlines under here and underline. So there we have it. We can make this a little bit thinner maybe. We can make that a little thinner. I don't really need a space in between these two or this one we can close that up if we want to and so then we could make a histogram based on what the actual results are. So there's a couple ways we can do that. We could take a histogram of the entire column of results or since we've summarized the results in essence into buckets over here we can use our bar charts. So let's use the bar chart. I'm gonna select this column holding down control and select this column because I wanna look at the actual results and then I'm gonna go into the insert tab and we're gonna go into the charts and add a bar chart. So there's the bar chart. I'll drag it to the right. Let's make it a little bit smaller. Bring it on over to the right and so there we have it. So then I'm gonna go into my chart here.