Okay, we all love to bet or pick play in NCAA Tournament Pool. And there's lots of things called props bets where you can bet on just about anything that moves in the NCAA tournament. So, let's play. Take two popular props bets and use our simulation worksheet to analyze the chance of these happening. What's a probably number one seed loses the first game? Okay, and this has never happened. I think the closest that came to happening was when Princeton almost beat Georgetown. I believe they had Patrick Ewing and Princeton was leading the whole game, but then they didn't win. And then, we'll figure what is the chance that basically no number one seed will make the final four. Okay, and most people would say, okay, that basically, it's more likely or sorry, what's the chance of all the number one seeds make the final four or none of them make the final four? Most people would say that basically, it's more likely that all the one seeds would make the final four than a one seed looses the first game. And they would actually be wrong on that. Okay, we'll fill this stuff in later. But let's go through calculating these two probabilities. What's the probability a one seed loses the first game? Well, we'll use something called the law of complements. Well, either no number one seed loses the first game or at least one loses the first game. So, probably at least one number one seed loses the first game would equal one minus the chance no number one seed loses the first game. But probably no number one seed loses the first game is the chance they number one seeds all win their first game. So, what I put in here were the ratings of the teams, now number one seeds, Kentucky, Wisconsin, Villanova, and Duke from the 2015 tournament, and when there's a playing game, I simply average the rating of the possible opponents. It's more likely they would play the better team, but it really doesn't matter. Okay, too much, so Kentucky would play either Hampton or Manhattan. Duke would play North Florida or Robert Marks. Okay, now, what's the chance that basically, so I took the difference in these ratings, and then, what's the chance that each of the one seeds will win. Let's do this from scratch here. The chance the one seed will win is one minus the chance they'll lose. So, the chance they'll lose is the chance they win by zero or less. We really should count the chance of overtime. So, chance to win by zero or less is put in at zero, and then, the mean would be the difference and the standard deviation is 11. You need the word true there, I keep forgetting, to let your true probabilities come shining through in the words of Cyndi Lauper. Okay, so now, Ctrl+C, I can copy that here. So, Wisconsin's got a 97% chance to win, going over to 96 and Duke a 98. So, the chance that all the one seeds win there, I should multiply these probabilities. because they're independent, you take the probability the first event will happen times the second event, times the third event, times the fourth event. So, you take product of probably Kentucky wins their game, that's probably all the one seeds win it. And one minus that is the chance that a one seed loses. So, the chance all the one seeds win is 92%. Okay, and one minus that, there's about an 8% chance that a one seed loses. Okay. And, even though it's never happened, there's a reasonable chance that would happen. So, Las Vegas is not going to give you Incredibly great odds, betting that 16 beats a one. You might think, you'd get 100 to one odds, but you won't, because there's a reasonable chance of that happening even though it's never happened. You just do the math on it. Okay. Now, let's look at the number of number one seeds that make the final four. How can we analyze that? Well, the winner of the Midwest region is in cell D16. The winner of the west region is in D29. The winner of the east region is in D47, D42. And the winner of the south region is in D55. Now, the code number for the one seed, Kentucky is code number 1, Wisconsin, 17, Villanova, 33, and Duke, 49. Okay, now, how can we figure out how many one seeds made the final four? Okay, we simply figure out, are these two numbers equal? That's who won in our simulation, and that's the code number for one seed. And if those are equal, if those are equal, start from scratch here, if this equals that, we have a one seed making the final four. So, Kentucky didn't make it, I'm glad of that. Okay. Here we had only two. Kentucky made it and Duke made it, which actually did happen. So, add this up and you get the number of one seeds to make the final four. Okay. So, we should run that, let's say, 5000 times. And we could figure with a cabinet the fraction of times that no one seed makes the final four, one one seed makes the final four, and basically all four one seeds make the final four. Okay. So, we know this one's 8%. Now, what's the chance all the one seeds make the final four? It's going to be a little bit smaller. We'll see. So, how many one seats make the final four? So, we'll run this 5000 times, home, fill series, column input, columns 1 through 5000. And the output cell was how many of the one seeds make the final four? So I go data, what if analysis, again a blank column input cell we know from our study simulation means the RAND's keep recalculate. So, I have to hit F9 to make these change. Okay, so I should count if, how many times I see in zero through four here, we need a dollar sign because I didn't need and we start up. I would say count if how many of these guys hit F4 oops, I should divide it by 5,000. Okay. So, there's about a 6% chance, sorry, a 4% chance that all four teams make the final four. So, there is, indeed, a larger chance that a one seed loses its first game than all the one seeds make the final four, and I would think if you asked 100 PhD's in math who were college basketball fans, which is bigger, and you gave them ten seconds to answer, they would get it wrong. because they just, since it never happened that a one seed lost its first game, people would say, hey, it just can't happen. But it's really unlikely for all one seeds to make the final four, I mean why? Even if they each have a 90% chance to win each game, which they don't, when it gets after the first round, you would need to flip a coin within 90% chance of heads, and get 16 heads, four for each team. And okay, and that you'd get 18% then. And it's way less than, let's suppose it's an 80% chance that the one seed wins each game. That may be a better approximation. Then you'd get two or 3% which seems to be a closer approximation. So, armed with this type of technique, we can analyze a lot of these great interesting NCAA props bets. And so, I hope you're ready for NCAA 2016.