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