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And so there should be no advantage for a corner move over another corner move. Once having a position on the board, all the squares end up being unique in relation to pieces being placed on the board. And we fill out the rest of the board. So there's no way for the other player to somehow also make a path. And then, if you get a relatively high number, you're basically saying, two idiots playing from this move. So for this position, let's say you do it 5, times. And we want to examine what is a good move in the five by five board. So we're not going to do just plausible moves, we're going to do all moves, so if it's 11 by 11, you have to examine positions. So here is a wining path at the end of this game. And that's the insight. So we make every possible move on that five by five board, so we have essentially 25 places to move. White moves at random on the board. So it's a very useful technique. We manufacture a probability by calling double probability. So you might as well go to the end of the board, figure out who won. And we'll assume that white is the player who goes first and we have those 25 positions to evaluate. And these large number of trials are the basis for predicting a future event. The insight is you don't need two chess grandmasters or two hex grandmasters. Of course, you could look it up in the table and you could calculate, it's not that hard mathematically. Because once somebody has made a path from their two sides, they've also created a block. Maybe that means implicitly this is a preferrable move. And at the end of filling out the rest of the board, we know who's won the game. So you can use it heavily in investment. And in this case I use 1. A small board would be much easier to debug, if you write the code, the board size should be a parameter. The rest of the moves should be generated on the board are going to be random. It's int divide. All right, I have to be in the double domain because I want this to be double divide. So it's not truly random obviously to provide a large number of trials. Rand gives you an integer pseudo random number, that's what rand in the basic library does for you. But for the moment, let's forget the optimization because that goes away pretty quickly when there's a position on the board. So here you have a very elementary, only a few operations to fill out the board. Given how efficient you write your algorithm and how fast your computer hardware is. Indeed, people do risk management using Monte Carlo, management of what's the case of getting a year flood or a year hurricane. So you could restricted some that optimization maybe the value. I've actually informally tried that, they have wildly different guesses. I have to watch why do I have to be recall why I need to be in the double domain. It's not a trivial calculation to decide who has won. And that's now going to be some assessment of that decision. How can you turn this integer into a probability? You could do a Monte Carlo to decide in the next years, is an asteroid going to collide with the Earth. And you're going to get some ratio, white wins over 5,, how many trials? I think we had an early stage trying to predict what the odds are of a straight flush in poker for a five handed stud, five card stud. Why is that not a trivial calculation? So probabilistic trials can let us get at things and otherwise we don't have ordinary mathematics work. You can actually get probabilities out of the standard library as well. And we're discovering that these things are getting more likely because we're understanding more now about climate change. Use a small board, make sure everything is working on a small board. And if you run enough trials on five card stud, you've discovered that a straight flush is roughly one in 70, And if you tried to ask most poker players what that number was, they would probably not be familiar with. You're not going to have to do a static evaluation on a leaf note where you can examine what the longest path is. Critically, Monte Carlo is a simulation where we make heavy use of the ability to do reasonable pseudo random number generations. So if I left out this, probability would always return 0.{/INSERTKEYS}{/PARAGRAPH} You're not going to have to know anything else. This white path, white as one here. And you do it again. So it's really only in the first move that you could use some mathematical properties of symmetry to say that this move and that move are the same. {PARAGRAPH}{INSERTKEYS}無料 のコースのお試し 字幕 So what does Monte Carlo bring to the table? And there should be no advantage of making a move on the upper north side versus the lower south side. So what about Monte Carlo and hex? So we could stop earlier whenever this would, here you show that there's still some moves to be made, there's still some empty places. Okay, take a second and let's think about using random numbers again. That's the character of the hex game. Sometimes white's going to win, sometimes black's going to win. So it's a very trivial calculation to fill out the board randomly. We're going to make the next 24 moves by flipping a coin. And indeed, when you go to write your code and hopefully I've said this already, don't use the bigger boards right off the bat. You're going to do this quite simply, your evaluation function is merely run your Monte Carlo as many times as you can. No possible moves, no examination of alpha beta, no nothing. That's going to be how you evaluate that board. So black moves next and black moves at random on the board. That's the answer. Instead, the character of the position will be revealed by having two idiots play from that position. So here's a way to do it. So it can be used to measure real world events, it can be used to predict odds making. But with very little computational experience, you can readily, you don't need to know to know the probabilistic stuff. This should be a review. You'd have to know some facts and figures about the solar system. That's what you expect. And then by examining Dijkstra's once and only once, the big calculation, you get the result. Turns out you might as well fill out the board because once somebody has won, there is no way to change that result. We've seen us doing a money color trial on dice games, on poker. Who have sophisticated ways to seek out bridges, blocking strategies, checking strategies in whatever game or Go masters in the Go game, territorial special patterns. Here's our hex board, we're showing a five by five, so it's a relatively small hex board. You readily get abilities to estimate all sorts of things. So it's not going to be hard to scale on it. And that's a sophisticated calculation to decide at each move who has won. You'd have to know some probabilities. I'll explain it now, it's worth explaining now and repeating later. One idiot seems to do a lot better than the other idiot. And the one that wins more often intrinsically is playing from a better position. And then you can probably make an estimate that hopefully would be that very, very small likelihood that we're going to have that kind of catastrophic event. Because that involves essentially a Dijkstra like algorithm, we've talked about that before. But it will be a lot easier to investigate the quality of the moves whether everything is working in their program. Filling out the rest of the board doesn't matter. So here's a five by five board. Now you could get fancy and you could assume that really some of these moves are quite similar to each other. But I'm going to explain today why it's not worth bothering to stop an examine at each move whether somebody has won. So we make all those moves and now, here's the unexpected finding by these people examining Go.