 Statistics and Excel. Binomial distribution coin flip random number generation. Got data? Let's get stuck into it with statistics and Excel. You're not required to, but if you have access to OneNote, we're in the icon left hand side. OneNote presentation 1560 binomial distribution coin flip random number generation tab. We've also been uploading our transcripts so that you can go to the view tab, immersive reader tool, change the language if you so choose, be able to read or listen to the transcripts in multiple different languages, and use the timestamps to tie into the video presentation. In prior presentations, we've been thinking about how we can represent different data sets, both with mathematical tools, such as using calculations of the average or mean the median quartiles and with pictorial representations like the box and whiskers and histogram. The histogram being the primary tool we use to envision the spread of data, and we can describe the spread of data on a histogram using terms like it's skewed to the left, it's skewed to the right. We're now looking at smooth curves that have formulas related to them that sometimes can represent data in real life, at least approximating that data set. And if we can do that, then the curves give us more predictive power into the future because we have a mathematical formula related to them. Prior presentations, we looked at the uniform distribution, Poisson distribution. We're now looking at the binomial distributions, which you will recall are those that we need to break out into whatever we're looking at have basically two outcomes related to them that we can define as basically success or failure. Let's look at a coin flip type of scenario to apply our concepts here. So we're going to say that the probability of a coin flip is going to be 50 50 because we're going to imagine it's a fair coin flip. Now let's just imagine if we were to plot out our binomial distribution with just x as zero, and then we'll increase the number of flips. And we'll see it's because that'll give us more of an intuitive feel of what is going on. So if we just simply have x being zero, meaning we don't actually flip the coin at all, then that means that the likelihood of us getting zero successes. And if we define success in terms of a coin flip, we can define it as either heads being a success, or sales being a success and the other being a fail, we will define it here as heads being a success. Well, if there's no flips, then of course, then there's going to be 100% chance that we have no successes. We're using the binomial dot dist dot range. And we'll talk more about the binomial dot dist range versus the binomial dot dist in future presentations. But the range is the latest and most flexible one of the two binomial functions. So we have the trials and the trials are going to be zero. In this case, the probability per trial 50%. And then the numbers we're looking at are going to be then the zero. So let's take a look at a scenario this time where we have the probability of success is still 50% because it's a coin flip situation, but we flip the coin one time. Well, if we only flip the coin one time and we're defining success as heads, then as you would expect, the likelihood of having zero heads out of two flips is 50%. The likelihood of having one success out of two flips is 50%. So that means, and again, we're doing this with our binomial dot dist range. And this is a spill array. So we just once again took the trials, the probability of the 50%. And then the numbers being this range, and it spilled out these two ranges will do that. We have done this in Excel if you want to check it out in Excel as well. Let's add add it to two. So now you've got the number of rounds is two. So if we have two numbers of rounds, what's the probability that we get zero successes zero heads out of two rounds or two flips 25%. What's the likelihood that I get one success defined as a heads out of two flips, we have the 50%. And then what's the likelihood that we get two successes out of two flips, two heads, in other words, 25%. You'll notice that the total adds up in all these cases to 100%, which is kind of our check figure. What if we bring it up to three then. So now we've got the likelihood that we get zero successes out of three flips, zero heads, that is 12.5% likelihood that we get one success out of three flips, 37.5 likelihood that we get two successes, 37.5, three successes, three heads, 12.5. And if we can also ask questions, of course, what's the likelihood that I get either zero or one successes. And that would be a cumulative type concept 12.5 plus 37.5. We're coming to the 50 if I added those correctly. Let's do one more. If we bring it up to four rounds, each individual flip being at 50%, we have the 6.25% that we get in zero successes, zero heads, we get 25% that we get one success exactly one success one heads out of four. We have the 37.5% that we get two heads out of four. We have the 25% that we get the three heads out of four and then the 6.25 that we get four successes or heads out of four. Now, we did this with a coin flipping scenario, but you can imagine just like we talked about in the prior presentation, other scenarios, any so many different scenarios actually, there's a lot of applicability here. If you can break out the outcomes to success or fail, we talked about sales calls in a prior presentation. Now let's think about a situation where the probability of success is the 50% and the number of rounds is 12. So we have a coin flipping situation, success is heads, we're now flipping it 12 times. So now we have zero if we have zero out of 12 successes, that's very unlikely that we don't flip any heads, one head or a success out of 12.29 and so on and so forth. If we were to plot this out, then we get something that