![]() You know that behind one of them is a car (which is most desirable), and that behind the other two is a goat. ![]() Suppose that you're offered the choice of three closed doors. The Monty Hall Problem in probability theory asks the following simple setup. It is named after the host Monty Hall of the show, "Let's Make a Deal." Here's what it was like. Perhaps the most famous and notorious example of this comes from a problem in probability known as the Monty Hall Problem. However, the change from 1/36 to 1/6 reflects a common occurrence in probability theory, that introducing new information can often change the probabilities from what they were before that information was introduced. After all, we already know that when there's only one die to throw, the probability of any given number is 1/6. And, for reasons that we will see soon, the odds of not throwing double sixes on 24 independent throws is given by multiplying this value 24 times: So, the odds that you won't throw double-sixes in a given throw is 35/36. When you throw a pair of fair dice, there are 36 equally likely outcome, one of which is double-sixes. Pascal and Fermat showed how to analyse the problem. Gombaud was betting on the assumption that if you throw a pair of dice 24 times, you'll more likely than not to get two sixes on one of the throws. So, Gombaud asked two of the greatest mathematicians of the time to have a look at the problem, Blaise Pascal and Pierre de Fermat. Unfortunately, the legend goes, Gombaud was horrible at these games and kept losing money, in spite of the fact that he had followed all the local rules of thumb concerning how to win. One of the most famous such stories concerns a French aristocrat named Antoine Gombaud, who liked to gamble on games involving dice throws. And it coincided with the new industry of maritime life insurance that arose in the 16th and 17th centuries. Although this kind of gambling had been played for thousands of years, it became institutionalised in Europe with the advent of casinos. But the real scoundral's mathematics was always the study of probability, which was largely developed to win at games of chance and bet on people dying. The history of mathematics is not an honourable one, having been developed closely with the history of codes and with weapons of war. Today we're going to talk about what probability theory is, with an aim toward trying to understand why it works so well. ![]() Only the player who flips the coin wins or loses the flip no other players are involved.The theory that captures these kinds of non-intuitive facts about uncertain outcomes is probability theory. For all other effects that instruct a player to flip a coin, the player that flips the coin calls “heads” or “tails.” If the call matches the result, the player wins the flip. No player wins or loses a coin flip for this kind of effect. Some effects that instruct a player to flip a coin care only about whether the coin comes up heads or tails. For example, the player may roll an even-sided die and call “odds” or “evens,” or roll an even-sided die and designate that “odds” means “heads” and “evens” means “tails.” If the coin that’s being flipped doesn’t have an obvious “heads” or “tails,” designate one side to be “heads,” and the other side to be “tails.” Other methods of randomization may be substituted for flipping a coin as long as there are two possible outcomes of equal likelihood and all players agree to the substitution. ![]() A coin used in a flip must be a two-sided object with easily distinguished sides and equal likelihood that either side lands face up. From the Comprehensive Rules (April 14, 2023- March of the Machine) ![]()
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