Suppose you see, weather forecast and the weatherman says

Suppose you see, weather forecast and the weatherman says "There is a fifty percent chance of rain tomorrow." What do you do? Take an umbrella or not? The rain just as likely as not - rain. If so, what helped the forecast? Weatherman could equally say "rain or tomorrow, or tomorrow will not rain." Ridiculous, is not it? Yet there is forecasters are taking this method and declare any impairment percent chance of rain. Can you think of a better way to measure pain same old song the Quality "of such forecasts? I want to convince you because - and immediately afterwards, to convince you that - and do it using minimax theorem from game theory. If so, how can nevertheless good measure forecast as "fifty percent chance of rain"? If it is a single forecast, nothing pain same old song to do - whether it rained or not, the weatherman was wrong. Even if he says "one percent probability of rain" raining, the weatherman went wrong - he had said that rain may fall. So, as always a probability, have to do a long time. Suppose we follow the predictions of the weatherman and test a hundred different days when he predicted the probability of a fifty percent pain same old song chance of rain - and see fifty-three of them had rain and more is not - it's pretty impressive, because it means that in this case did about fifty percent of the time it was raining. However, if ninety-four days, these days there was no rain, so the weatherman's forecast is quite bad. Could put it more formally, but I guess there's no need for it - the main thing we talk about diagnosis differences between the probability of "real" rain days fifty percent, and the probability H"hazoih. "
This probability should weigh along with other predictions of the weatherman - it probably is also sometimes predict rain probability of ten percent, and rain probability of thirty percent and so on. Of course you have to limit his resolution not to allow him to give predictions as "the probability pain same old song of thirty-seven point eight five three percent chance of rain" then it could reach a point where each forecast, pain same old song we have no more than one day for which it was given. So it can be assumed he was talking about predictions percentages are double ten - this ensures we will not need too many days to collect such amount of information that will allow us to give the forecaster judgment is not bad at all (that would allow us to distinguish clearly between him and someone who just throws random projections).
For me personally this test method sounds very good, and certainly intuitive and clear - making the score I'm going to describe interesting, because it illustrates how our intuition can mislead us in such cases. pain same old song In short, what I'm going to show (with waves, of course - here's pain same old song the article describes it formally) is that you can make "predictions" according pain same old song to the test method I suggested would be considered pain same old song excellent (formal - at the level of accuracy required, we can achieve the level of accuracy, given enough days trial ), without knowing anything about the weather, or even leave the room. I'm a little Varma sense that my argument pain same old song will not constructive - I see that you can make such a forecast without me to explain exactly how; But in this case you can also give a constructive proof that I did well because it will slip too many technical details, and also because I think the non-constructive proof has great beauty (who is interested in the technical details can read the article presented by the subject).
As mentioned, I'm going to rely strongly on the minimax theorem, I mentioned not long ago; He has already been described blog before, but I want to display it again, with an emphasis on a point probably not stressed before. "Playing field" is our so-called "sum games - zero." These games involve two players - call them Alice and Bob - so that the interval of Alice is the loss of Bob. "The Game", for that matter, is any situation in which both Alice and Lviv have any choice about the way they work - through the selected action is called "strategy" - and as a result of the two strategies chosen obtained score of the game, for which the matter will be just a real number that says how Alice Bob earned and lost ("profit" of minus 2 Alice says Alice and Bob lost two earned two, so we are limiting the generality here.) Need to clarify some of the interest strategy - Strategy is a kind of "owner's manual" means the player how to act in every possible situation in the game. Thus we can reduce the complexity of games like chess following description: "Alice and Bob choose a strategy independent pain same old song of each other, and the profit pain same old song is determined according to both of these elections and nothing more."
This is one of the places where formal writing helps to understand what it is. If \ (x \) denotes any strategy of Alice from the group strategies \ (X \) and \ (y \) any strategy of Bob Strategies Group \ (Y \), so \ (\ pi: X \ times Y \ to \ mathbb {R} \) is "profit function" Alice: \ (\ pi \ left (x, y \ right) \) is the space of Alice from the game if it is taken at \ (x \) and Bob takes on \ (y \) (and recall, this number also describes the loss of Bob. could also be Bob's profit. then is the loss of Alice. did not confusing at all).
Take for example the playing partner - or - except that I talked to him recently. Strategy in this game is a total number of fingers being spent. If we spend between 0 to 5 fingers, winning a game marked by profit-sized one, and Alice is a player of "couple", then \ (\ pi \) function \ (\ pi \ left \ {0,1, \ dots, 5 \ right \} ^ {2} \ to \ mathbb {R} \) are examples of \ (\ pi \ left (2,2 \ right) = 1 \) and \ (\ pi \ left (3, 2 \ right) = -1 \). When talking about chess situation is much more complex in terms of how that strategy seems that every strategy pain same old song contains instructions of the form "If you get to a position of such a board, follow pain same old song this ..." (which is even further pain same old song complicated pain same old song because the manner in reaching a position pain same old song is important that example lies In