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