Have you ever thought about this: can probability paradoxes be applied to gambling? After all, if there are situations in which probabilities become, so to speak, anomalous, then surely there is a temptation to apply these anomalies in gambling? I mean such well-known probability paradoxes as the Monty Hall paradox and the Bertrand Box paradox. In all of these (and some other) paradoxes, there are disputes about which probabilities we choose. Usually, there are disputes between the odds of 2/3 and ½. The Monty Hall paradox is a well-known paradox. Some commentators believe that if the player changes his initial choice, his chances increase from 50% to 66%. Some commentators are sure that the chances remain at 50% in any case.
The Bertrand Box paradox is a slightly less well-known paradox, but the debate about probabilities is similar.
I can tell you a little about this paradox. There are three boxes, each containing 2 coins. The first contains 2 gold coins, the second contains one gold coin and one silver coin, and the third contains 2 silver coins. We choose a random box and take a coin out of it. Let's say our coin is gold. What is the probability that another coin from the same box is also gold? Some people think that this probability is 2/3 and explain this by the fact that there were 3 coins in the boxes in total, and now there are 2 left, which means the probability is 2/3. Personally, I think that there is no paradox here and the probability of taking out the second gold coin is ½. The logic is simple:
If we took out a gold coin, then it is definitely not the third box, which contains 2 silver coins. This means that it is either the first box (which contains 2 gold coins), or the second (which contains one gold and one silver coin). If we opened the first box, then the second coin must be gold, if we opened the second box, then the second coin must be silver.
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In this regard, I would like to ask you the following questions:
1. What do you think is the true probability of events in the Monty Hall paradox and the Bertrand Box paradox?
2. If there are anomalies in the calculation of probabilities, can these anomalies be used to develop strategies in gambling? In other words, can this be used to create a gaming system in which your understanding of these and other probability paradoxes would become an advantage over other players or over the bookmaker?