All right, well let's take another potential example, how would we go about calculating this? All right, we've got our same distribution, mean of 500 for demand, standard deviation of 100. We currently have enough seats to meet demand of 450. If we think about adding another flight to this route, now we're going to have a capacity for 600 people, all right? Well how likely is it, that demand's going to fall between 450 and 600? So how do we go about calculating this? Well, one thing we could do is say, well let's calculate the z-score for 600 and standardize it. Let's calculate the z-score based on demand of 450, standardize that. So we've got our z-scores for the two, we look up the probabilities associated with both of these values, and we can get, what's the probability that demand falls in this range. Another way of doing this, want to just jump into Excel, is to use this particular set of commands. So here's, what's the probability that demand is less than 600. So that's saying that demand is anywhere below 600, and what I want to do is subtract from that, the chance that demand is below 450. So, if I kind of take this piece out, this is the range that I want to focus on. So, the probability of demand being below 600, minus the chance of demand being below 450, that's going to tell me how likely is it, that demand falls between those two values. And it turns out, that it's about a 53% chance of happening. Now, that's not telling us exactly what the level of demand is going to be. Might be 451, might be 559, 599, we don't know. And this is not enough information for us to make the decision of, should we add another flight to this route. You know, think about what's going to go into that decision. Well, what's it cost us to operate that flight? How much revenue are we going to generate? What's the cost associated with not offering enough seats, versus offering too many seats? You know, these are some of the issues that airlines, they have to grapple with, when they're allocating their resources. And that's what our next exercise is going to be based on, is an over booking problem, another situation that airlines have to deal with. Another example that airlines might have to contend with, would be a case where they say, I want to make sure I have enough seats for passengers, a particular percentage of the time. So, for example, in this case, what we're looking at is saying, I'm only willing to run a 10% chance of not having enough seats. Now that's something that the airlines might look at, what about in the context of a retailer? You know, a new product coming on the shelves, or it's the holiday season, the hot product. I want to make sure I have enough of my product available to meet demand 90%, 95% of the time. You know, there's a lot of costs involved, if I don't have that product in stock, I potentially put a customer's business at risk. So, how much do I have to plan to have available? So in this case, saying, I'm willing to be short on seats 10% of the time, well, what does that mean for the amount of seats that I need to have available? I don't want to make that many mistakes, so I want to make sure that 90% is covered here. Well, if I know that, if I know that I want to make sure that I've got seats available at least to cover 90%. That's where this norm inverse function's going to come into play. So we know the probability, 0.9, we have our mean of 500, our standard deviation of 100. Tells me, I need to have seats available of at least 628.15. Can't have 0.15 seats available, let's bounce it up, call it 629 seats available. If I only want to have a 10% chance of not having enough seats. Right, so using characteristics of the normal distribution, as well as using these distribution and inverse commands within Excel, we can perform these calculations, to tell us how many seats we need to have available, how likely is it that we can expect business to fall within these particular ranges? Again, we could look this up on our z tables, to find out what's the appropriate value that will correspond to this. In these case, we'd say, well let's find 90% in our z table, and then reverse engineer what the appropriate number of seats would be. So, we've talked about being able to characterize the extent of uncertainty, being able to characterize the variance, if you will, that we're observing. The amount of dispersions, the range and frequency of possible outcomes. The other piece that's remaining for us is, well, how do we go about making predictions. And if we're talking about sales forecasting, there are a number of factors that we might take into account. We might consider seasonality, we might consider competitive actions, we might consider our marketing mix. And all of those factors, they're going to be assumed to influence that best guess, that prediction for what the level of demand is going to be. Everything that we're talking about, in this case, the normal distribution, is the variation, or the fluctuations, around that best guess. All right, so we've talked a fair amount already about the normal distribution. And it's great if your data actually looks like a normal distribution, it's an appropriate distribution to use. We've got the commands built into Excel, we can visualize it easily with a bell curve. We have a good sense for one standard deviation, two standard deviation, but, does your data actually look like a bell curve? That's not always going to be the case, so here are just some examples of places where other distributions might be necessary. If we think about the number of employment offers made, versus the number of employment offers that are accepted. If we think about people making brand choices. These are choices, these are not following the normal distribution, they're not continuous outcomes. If we look at how technology or new products diffuse over time, that diffusion curve does not look like a normal distribution. It might start off slow and then speed up, until when it hits the mass market. If we look at product failure times, or if we look at customer lifetimes, these tend to be skewed. You have a lot of customers who have short lifetimes, and very few customers, potentially, who have long relationships with companies. Well, if that's what the data looks like, the normal distribution isn't going to be appropriate. And, we can actually make very serious mistakes from a financial standpoint, if we don't choose the appropriate distribution. The normal distribution, 99.7% of that data falls within three standard deviations. Well, what if the tails are a lot fatter? What if there's more of the mass out in those extreme observations? Well, those extreme observations could be what we would consider catastrophic losses and blockbuster successes. Well, if we use a normal distribution, we might under weigh the likelihood of those outcomes occurring. So just to give you some examples using different distributions, suppose we're dealing with a company that's looking to hire new employees. And they want to get between 5 and 7 new employees in a given year. Well, how many offers do they have to make in order to do that? So, let's say in the past, 80% of the people they made offers to, accept those offers. If I'm looking to net between 5 and 7 employees, all right, well I've got to make at least five offers, that's the easy part. But if I want to maximize my chances of falling in the range of 5 to 7, what do I consider a success? I would consider it successful if I got five, six, or seven new hires. So I need to figure out, how many offers do I make, in order to fall within that range? Well this is a case where the binomial distribution is going to be appropriate for us. The binomial distribution is used when we have, for a fixed number of trials or, in this case, employment offers, how many successes do I observe, how many yeses? How many people said yes to that employment offer? Again, we have formulas that we can use if we want to calculate the mean, the variants, and the standard deviation. You'll notice these differ considerably, compared to what the equations look like for the normal distribution. Again, it's a different distribution, it's a different distribution that we could use to fit our data, so it really depends on the context, in terms of, which distribution is going to be appropriate. Now, if i'm dealing with very large numbers of trials, the binomial distribution will actually end up looking like a normal distribution. So for large values of n, we can approximate the binomial distribution with a normal distribution. But for small values of n, we want to be careful, and let's take a look to see what this distribution's actually going to look like. All right, so within Excel, these commands are going to look pretty similar to what we saw before, in terms of their syntax. Instead of NORM.DIST, now we have BINOM.DIST. So, if I want to know the probability of my outcome being below a particular value k, that's where BINOM.DIST is going to be used. I input n, the number of trials, I input p, the probability, and i input that true statement, to find out how likely is it, that the number of successes falls below the value k. If I want to know, how likely is it that I get exactly k successes, that's where changing that true statement to false is going to come into play for us. And just like we had the inverse statement for the normal distribution, we have the critical value statement for the binomial distribution. So, what is that cut-off probability, if you will. So in this case, if I know that I'm looking to fall above a particular probability, a cut-off probability, I'm going to look for, what's the smallest possible value to make that happen.