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prob.part19.39_40 - 1.2 DISCRETE PROBABILITY DISTRIBUTIONS...

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1.2. DISCRETE PROBABILITY DISTRIBUTIONS 31 be 0, 1, or 2, and hence we should assign equal probabilities for these three possible outcomes. 16 Cardano chose the correct sample space for his dice problems and calculated the correct probabilities for a variety of events. Cardano’s mathematical work is interspersed with a lot of advice to the potential gambler in short paragraphs, entitled, for example: “Who Should Play and When,” “Why Gambling Was Condemned by Aristotle,” “Do Those Who Teach Also Play Well?” and so forth. In a paragraph entitled “The Fundamental Principle of Gam- bling,” Cardano writes: The most fundamental principle of all in gambling is simply equal con- ditions, e.g., of opponents, of bystanders, of money, of situation, of the dice box, and of the die itself. To the extent to which you depart from that equality, if it is in your opponent’s favor, you are a fool, and if in your own, you are unjust. 17 Cardano did make mistakes, and if he realized it later he did not go back and change his error. For example, for an event that is favorable in three out of four cases, Cardano assigned the correct odds 3 : 1 that the event will occur. But then he assigned odds by squaring these numbers (i.e., 9 : 1) for the event to happen twice in a row. Later, by considering the case where the odds are 1 : 1, he realized that this cannot be correct and was led to the correct result that when f out of n outcomes are favorable, the odds for a favorable outcome twice in a row are f 2 : n 2 - f 2 . Ore
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