 It looks like this and it looks somewhat, you know, like a bell shaped curve in this case because as we looked at in prior presentations, the percent of success is at that 50%. And this looks kind of like what you would expect in a coin flipping scenario, right? Because you're saying, okay, that means if I flipped it 12 times, you would kind of think that six would, you know, the data would be, would be formatting around six and basically is tapering off, uh, given those conditions. And if I look down here, we can do a similar kind of chart with a line type of chart as opposed to with the bar chart or a histogram type of chart. All right. Now let's actually approximate this. Let's say that we simulate these, uh, this, this situation in Excel. We have some tools instead of using just the equals random function that we did when we did a normal, like a coin flip in prior presentations to represent a heads or a tails, we're going to represent the outcomes for the sets of flips that we have. So we're, we're going to say that we flip them 12 times. So when we run these tests, I'm going to say that we're flipping the coin 12 times each round and see how many heads we were defining as successes that we get every 12 times we flip. So in the first 12 times we flip the coin, we've got five successes and the second 12 times we flip the coin, seven successes or seven heads out of 12, the third time, seven successes out of 12, fourth time, four successes or heads out of 12 and so on. So in the past, you know, the teachers union and the colleges, you know, they had people that just sat in there and flipped coins all day and it was like a union job and whatnot, but even they couldn't even hold on to that. Right. They, they moved over to the, to, to, to write for Hollywood and whatnot. But, uh, uh, but now, now we have this number generator and you could check out how to do that in Excel to kind of simulate these models. So now we're going to imagine a real life scenario or in practice, we actually did this test, right? And see how closely it matches what we did theoretically with a bi-nome.dist, which would be the smooth curve. So if we break this out into our bends, we're going to say, all right, how many times are we going to get zero? How many times are we going to get one, uh, success out of 12 flips? How many times did we get two successes out of 12 flips? We're using our frequency, which is taking our outcomes here of 12 flip outcomes. I can't remember exactly how many, we did it a thousand times, a thousand 12 time flips. And we're saying out of a thousand 12 time flips, how many times did we get zero out of 12, two out of a thousand 12 times flips? We never got one exactly one success or head. We got, how many times did we get two successes or heads out of 12 when we flipped it a thousand times, 13 times and so on. Now, again, you might ask, why don't I use the count if function? I could do that. I could say, Excel, I would like you to count all the items in this column if there's a zero. However, because we use this random number generator, sometimes it kind of throws out that count if or throws it off. Therefore, this frequency, uh, usually picks up all the numbers. It's an array function. So it's kind of a fancier function. So if you want to check that out in Excel, we will do it there. But if I sum up the columns, it should come out to a thousand. Why? Because we ran a thousand 12 flip tests. So that's our check figure. Then we can also look at the percent of the total. So for example, how many times, what's the percent of times that we've, we got four successes or heads out of the thousand trials that we did? Well, we have one, one, nine divided divided by a thousand. And that's going to give us the 11.9. Now remember that if we actually run the test, I can make a graph about the frequency and I can make a similar graph about the percentages. When I look at the Poisson distribution data that we have over here, I didn't actually run the tests. What we got instead is simply the frequency. So over here, if I got the frequency, I would predict then the number of flips if I ran a thousand tests by doing it this way, right? I would say, okay, how many times would I get a five if I ran the test a thousand times 1000 times, uh, times the point 1934 or so on, right? And over here, what we did is we actually ran the test and got the frequency and then we, and then we, we divided by the total, total to get the, to get the percent of the total that outcome to that 5%. If we compare the two, I'm subtracting this number from what we got over here to see how close it lines up to a binomial distribution. And we can see it's not perfect, but it looks pretty close to the binomial, uh, distribution. So as we would basically expect, and if it, and if it is, if it was a real life situation like phone calls or something like that, then we would be likely to be able to use the binomial formula to make predictions about outcomes on sales calls. All right. So if I was doing my own sales calls, I can kind of figure out what would be expected and see if my data is lining up to that. Or if you're supervising someone, you might have some expectations that are designed in a similar fashion. So if I graph these two things together, meaning this is the percent of the total of the actual data in blue, and this is the actual smooth curve from the function that we graphed, but, or basically we graphed this data that we generated from the function or formula, we could see they're pretty smooth. So we have a similar situation that we've seen in prior presentations. We're saying, uh, we could say whatever scenario we're looking at, does it fit the conditions to be a binomial distribution? If it does, then maybe we can use the function. If we actually plot data outcome and say, does the data outcome look like it ties out to what we would get from using the binomial distribution? Again, good evidence that we could use the binomial distribution. If we plot the data that we actually got versus what we would get with the smooth binomial distribution curve and those two things line up, again, we would say, okay, now maybe I can, I'm convinced that I can use the binomial distribution curve to make predictions about whatever we're talking about, in this case coin flips or in future presentations, we'll talk about the sales calls and what not, whatever we're talking about might be able to be predicted with this smooth curve or formula.