Okay, to close the course out, let's talk a little bit about golf. And everything we're going to talk about here is based on a fantastic book. And if you have any interest in golf, you've gotta buy this book. Every Shot Counts by my friend and colleague, Mark Brody. Who's a professor at Columbia University Business School, teaches MBAs. And he is an avid golfer, great at math and a great teacher. He's won many teaching awards and he is just fascinated by the math of golf. And he brought Moneyball to golf trying to understand, like we did in baseball, understand fielding, trying to understand if you putted well, were your tee shots good? So, how do you look at this? Well luckily the PGA has like every shot from just about every tournament, PGA shot data, Mark got this data and he did a great analysis. So let's just start with putting. Okay, my golf career was not that good. I shot 98 for nine holes. The last time I played and then I looked in my garage the next day and my clubs were gone. Probably somebody I played with stole my clubs and that was the end of my golf career. Miniature golf, I'm okay. All right, so let's suppose you have 20 putts on a round of golf. And on another round I told you, you had 25 putts. You would probably think if you don't think too hard but hopefully after this course you'll understand that you should think a little harder about this. Most people who don't think about it to hard would say well, 20 putts, that was a better round of putting than 25 putts. Well clearly, it's like fielding. It depends on where the putts were from. If I took 20 putts and all my putts were 1 foot, that's pretty bad. If I took 20 putts and all my putts were 10 feet or longer, that's fantastic. Okay, and so basically, what you need to know is basically like in fielding, what's the chance somebody will field a ball based on where it's hit? You need to know on putting, what's the chance a pro golfer will hole out from a particular range or how many strokes will he need? So an average, from let's say eight feet, takes one and a half strokes From 40 feet, 2.06 strokes, from 67 feet, 2.25 strokes and Mark has all the stuff. Now it does matter if the course was a hard putting course, a course that's hard to putt on. Think Pebble Beach was the hardest course to putt on, I think Mark said in the book. And you can analyze that and you can adjust for the course sort of like park factors but we won't get into that. But, Let's suppose you have four eight-foot putts. And let's suppose it took you one putt three times, and you two-putted once. Okay, you have four holes, you're eight feet from the hole, sorry. To start. Okay, did you do a good job putting or not? Well, it's really simple, each of those four holes starting eight feet away, expected putts would be 4 times 1 and a half, that's 6. You used five putts so the concept is strokes gained. So the putting there gained one stroke, which she took five putts when a typical PGA golfer would have taken six putts. Where this is so useful is you can break down every part of your game. The tee shots, the approach, and the putting. Okay, we're just going to scratch the surface here, give you some easy problems to work on. Okay, so here's a Tiger Woods round. Here's how far away he was on each hole when he putted, and here's how many putts he used. So, did he have a good putting game or not? Well, he used one putt from four feet away, and that should've taken 1.13. So that one, he gained 0.13 strokes. But here was not so good. He was 13 feet away, and it should have taken him 1.71 putts, and it took him 2, so he lost 0.29 strokes. And you add that all up, his putting gained minus 1 stroke. Or lost one stroke. Okay, and so when Mark was doing the analysis, I forget what you were. But the best PGA putter was Luke Donald. Now Jordan, who won the Masters, basically is good at every part of the game. I mean, every part of his game is really, really good. But I think one lesson from Mark's book is most people think putting's the most important part of golf. But you look at where the great golfers in the PGA Tour differentiate themselves from the other golfers, it's on the approach shots much more than the putts. And this is true for amateur golfers too. So spending time practicing those approach shots, although I think it's hard to practice those, will probably pay off in your score. He also makes very interesting points like you should really try and drive the ball as far as you can, don't worry so much about the accuracy. If you can gain 10 or 20 yards and be a little bit less accurate, that's going to help your score also. But if you really golf and you really care about your score, every shot counts will improve you and certainly improve your understanding of the game. And I know it's available at Barnes and Noble. I've seen at the Houston Barnes and Noble in River Oaks like 20 copies on the shelf. because I think everybody there golfs. Okay, so let's talk about what happens if you want to figure out how many strokes were gained on every shot, starting from the tee shot. So Mark looked at based on how far you are from the hole, up to 600 yards. You're on the tee, you're in the fairway, you're in the sand or some place where you need a recovery shot, where basically, you've got a really obstructed view or something like that. So for instance, if you start 300 yards away, you'll need usually 3.71 strokes from the tee to hole out, little bit less than four. And if you were 300 yards away in the sand for an extra 4.04 strokes, and if you were like the trees were bugging you and stuff like that, you would need 4.2 strokes. So let's look at a hole here, a sequence of four shots where you hit a par, you got a par 4. Its a 400 yard par 4. Its going to be an unusual sequence of shots but you can break down the shots gained or lossed on each shot and you'll see it'll come out. There's basically no shots gained or lossed on that hole net because you got a par 4. But let's suppose the first tee shot was horrible. It was only 120 yards. That would be like one of my tee shots on a good day and I would end up 280 yards from the hole. Then I'm 280 on the fairway. So I'm 280 yards from the hole on the fairway and the next shot puts me in the sand 60 yards away. Then I make a pretty good shot, I chipped from the sand, then I end up 20 feet out. And then amazingly enough, I make the putt so I got a par 4. So how do we break this down? So we start out 400 yards away. We needed 3.99 strokes. And we ended up by 280 yards away, we need 3.65. So basically, we gained 0.34 strokes on that shot, but we used up one stroke. So, you take the starting strokes needed minus the ending strokes minus one. That was a horrible shot, it lost 0.66 strokes. All right, so I'm 280 yards away. I need 3.65 strokes. I hit it in the sand 60 yards away. I need 3.15 strokes. So that shot sort of gained me 0.5 less strokes, but I used up a stroke. So I gain minus 0.5. Okay, now I'm in the sand 60 yards away, I need 3.15 strokes. And I go to the basically 20 feet away on the green, now I need 1.87 strokes. That shot helped me by 1.28 strokes. I used up one stroke. That stroke gave me 0.28 strokes. Okay, now I'm 20 feet away, I need 1.87 strokes on average to hole out from there. And I made the putt. So I gained 1.87 strokes and gave up one stroke so I gained a net of 0.87. And when you add that together, you get minus 0.01 stroke, or pretty much a wash. So taking these basic concepts and working with these, Mark can really tell you a lot about golf. He can also tell you where should you aim a putt. Usually you should aim a putt beyond the hole. Basically you don't want to leave it short. And so how do you determine how far beyond the hole to aim the putt? Well, it depends. Is the putt uphill? Is the putt downhill? It depends on the STIMP ratio, which is sort of, so uphill or downhill, you measure by the slope, by the angle of the incline. And then the STIMP ratio is how fast the green is. So Mark has this all figured out. And, believe me, buy his book and you will learn a ton about golf using basic math. And, I can't say enough good things about the analysis Mark did. I mean, there's work to be done, I'm sure. But, he did a fantastic job. And, if you want to see ratings of golfers, my friend, Jeff Sagarin, has ratings in Golf Digest that you may want to check out. Okay, and there is fantasy sports for golf, as I mentioned, on DraftKings. Okay, well that seems to be our last video, so I hope you've enjoyed the class. Let us know.