Optimal strategy under statistical uncertainty - unsteady markets
If bets are placed with a 50\50 probability, nothing will work. The interaction of these two random processes, flipping the wrong coin and guessing, will result in 50\50.
No one forbids placing any bet within the deposit on either side of the coin with any frequency or skip betting.
If the "coin" is definitely not right, then the betting system will just have to account for the variance of the distribution around the "heavy" side... How this variance is best calculated is up to Mathemat ...
Before you talk nonsense along the lines of "heard the bell", please read the rules. It says that:
1. It is not known in advance which side of the coin is "heavy".
2. No statistical research is allowed.
3. the algorithm only knows the result of the previous coin flip
Question: Under the above conditions, is there a betting system that can be used to extract a positive mathematical expectation?
Yes. Betting on the more frequent side. In any case, the strategy has to take history into account. In this case -- a simple adaptation to it.
2. The side of the coin, which was struck on the previous flip.
A scratchy story. In this case the strategy is to bet on the same side.
Oh, that's how... Yes, I apologise, if you don't know, then of course you can't figure it out...
No one forbids betting on either side of the coin at any frequency or skipping betting.
Do you mean to bet on the same side as the previous toss-up?
Yes, if there is an edge, the correct side will fall out more often.
2. No statistical research is acceptable.
this is really nonsense... how are you going to create your profitable betting system then?
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Many people may have heard of or even be familiar with the contents of J.L. Dub's book Probabilistic Processes. Chapter VII of the book, devoted to martingales, says that one cannot profit from pure martingales (a martingale is a fair game, i.e. with zero mathematical expectation, which, according to the author's proof, remains zero regardless of the strategy used).
Profit, according to G. Doub, is possible only on submartingals - games that have positive mathematical expectation for a player.
Suppose we have a trading system, but we do not know in advance if it is profitable or unprofitable ... (we don't know beforehand if trading signals of this very system give the advantage in the mathematical expectation of profit or if it is more profitable to use the inversion of this very TS for interpretation).
Let's set the problem even simpler, let's assume we have a wrong coin (wrong means one side is falling out more often than the other). We do not know beforehand which side matches more often and with what exact probability, but we know for sure that the coin is wrong.
By conditions, it is necessary to create profitable betting system, which does not allow to calculate statistically the advantage of one of sides of a coin, therefore, its algorithm must be based on knowledge of only two parameters:
1. The number of the next flip.
2. The side of the coin, which was struck in the previous flip.
It is possible to bet on either side of the coin before the next flip. It is possible to skip a particular coin toss, i.e., not bet, i.e., the bet amount is 0. It is possible to increase or decrease bets.
If the side of the coin after the flip is guessed, the player wins in the amount of his bet, if he loses - he suffers a loss in the amount of the bet. (I.e. he bets a certain amount, if he guesses the side of the coin before the flip, he gets back double the bet amount, if he loses, the bet amount goes to the bookmaker).
Question: Under the above conditions, is there a betting system that can be used to extract a positive mathematical expectation?
The gambler only needs to write down the algorithm by which the computer program, the bot, will be created. The gambler bets a certain large amount of money before starting a series of coin flips with which the bot makes bets - the deposit. You can play for as long as the deposit is not zeroed out.