All right, so i created just a small Excel file that you can take a look at and let's just fire this up so that we see what we're dealing with. So what we've got here is the Excel worksheet that I've built. And you can download this separately from the course website. But let's just zoom in a little bit already and we can just take a look at what this looks like. So the two parameters corresponding to the binomial distribution n, in this case, how many offers do we make? And the value P, that's our probability of an offer being accepted. And what I've put in here is how likely am I to observe this specific number of offers being accepted. All right, so the if statement is in there, because we're only going to be able to observe a particular number of acceptances if I make at least that many offers. But the bulk of what we want to focus on is this probability associated with the binomial distribution. So let's say if I make seven offers, pretty small chances that I get zero acceptances, 36% chance that I get six acceptances, 20% chance I get seven acceptances. What I've highlighted in yellow is the sum of getting five, six, or seven acceptances because that was our target. Well, is seven offers the best number if what I'm trying to achieve is the highest number possible falling in five to seven. Well let's see what happens if we lower the number of offers. If I only make six offers, I only have a 65.5% chance of getting between five and seven acceptances. Again we go up to 85% if I make seven offers. If I make eight offers, it's going to drop down to a 77.6% chance. So in this particular example it says, you know what, if your goal is to get between five and seven, our best bet is to make those seven offers. That's where we maximize our chances of falling in our target range. All right, another distribution that we might choose to work with is the exponential distribution. So very common if we're thinking about customer lifetimes or time between purchases. Also gets used in the case of product lifetimes, how long until a product failure. And you can see the different possible shapes that we're observing with the exponential distribution. So, depending on the failure rate or this rate parameter lambda, you see in our legend, In the legend here, the higher the parameter, the faster it's going to fall off. So if we look at the lowest level of this parameter that's in green, it seems like it's a very drawn out distribution. If we look at the highest value of this parameter, in this case our colors aren't matching exactly, so I'm just going to trace this line. But notice how quickly it falls off and then less likelihood of having one of these long observations. So, if we're dealing with durations, we might focus on using an exponential distribution. And just like the normal distribution, just like the binomial distribution, there are commands built into Excel to be able to calculate the probabilities associated with an exponential distribution. Just a couple of caveats in terms of using these distributions to characterize uncertainty and when we incorporate them into our decision making. It's very important that we get as close a fit as we can to the actual data. The tails are going to be particularly important for those very high and very low outcomes. But the intent of the probability distribution is to give us that approximation. If we're too far off it's not mimicking the data, and it's going to put all of our conclusions, it's going to call them into questions. Right, so the next exercise, what we're going be looking at is using that binomial distribution and using that as a means of saying how much variation do we have around our expected outcomes? We use that to characterize the variable that we don't necessarily have perfect knowledge of. And based on the extent of uncertainty that we're having, we can see how that's going to have an impact on the decisions that we ultimately make. All right, so we're going to take a look at this in terms of the airline industry immediately dealing with the number of no shows for a flight and should we sell more tickets? If we take advantage of the fact that people don't show up, we're going to be able to sell more tickets, generate revenue that way. But there is a chance as we sell more tickets that the flight goes into an over sold condition and it costs the airline. So, what's the appropriate number of seats for us to be showing? Well, it's going to depend on my best guess for how many no-shows we have. So we're going to assume that we know the likelihood that people don't show up for the flight. We're going to use a binomial distribution to characterize that and use that as our best guest to say, how many seats should we be selling to maximize profit? In our next session, what we're going to look at is products breaking down, product failures, and needing to be serviced. Well, if we know that products are going to break down and need to be serviced should consumers buy warranty coverage for them and should retailers offer that coverage? Well, how do we go about pricing that? Depends on how likely the products are to break down, how much it's going to cost to repair, what could consumers do on their own. So that will be the next application that we take a look at. But if we've got resource allocation decisions, maybe I don't know how many people are going to show up at a restaurant on a given night. How many people do I need to have working? Am I going to have enough tables? Do I need to order new equipment to meet capacity? These are all case where there is uncertainty in our decision making. There's some factor that we don't have perfect knowledge of so that's where the probability distribution is ultimately going to get used.