Chapter 22

Non-Transitive Dice and Other Paradoxes

Katie Kephart

           

            This chapter talks about transitivity. An example of transitivity would be if A is heavier than B and B is heavier than C than A is heavier than C. A statistician whose name is Bradley Efron designed three sets of four dice that violate transitivity. It does violate this because the fact of being more likely to win is not transitive between pairs of dice. It also talks about with these dice the greatest possible advantage that can be done with four dice is 2/3. The fundamental principle in calculating probability with dice came from classical probability theory in the 18^th century. This principle is called the “principle of indifference’. A definition of it would be if you have no reason to believe that any number of mutually exclusive events is more likely to occur than any other, a probability of 1/n is given to each. There are some cases where the principle of indifference leads to a logical contradiction. An example of this could be a cube with a random size. You know the cubes edge is not less than one foot or more than three feet. In estimating the probability that the edge is between one and two feet compared to two and three feet you could not say each probability is ½. On the other hand if the cubes edge was picked at random the principle of indifference does apply. Also Carnap gives a paradox where of the principle of indifference where you know that every ball in an urn is blue, red or yellow, but you don’t know how many of each color is in the urn. In wanting to know what the probability will be for red you say it is ¼, but if you ask the probability the first ball will be red you get ½ which contradicts the other estimates. Another paradox could be the probability of plant or animal life on Mars to assume that 1/2+1/2=1 which would be wrong. Pascal’s wager is another example of a paradox of the principle of indifference.

            From the article written by Rudolf Carnap on Statistical and Inductive probability the author should have mentioned statistical probability versus inductive probability. It talks about the principle of indifference but does not talk about the main difference between statistical and inductive probability. The statistical probability being mathematical statistics where as inductive probability has a hypothesis with respect to a body of evidence. It measures the strength of support by the degree of confirmation. It is purely based on logic. On the other hand one thing I the other omitted that is good would be that the author didn’t include all of the details about the different types of probability.

            From the article the paradox of non-transitive dice by Richard P. Savage something that could have been included would be the example of the non-transitivity paradox with the voting paradox where a majority of voters may prefer candidate A to candidate B, B to C. and C to A. Although it’s good the author didn’t include all of the different formulas in the article.

In the book The Emergence of Probability by Ian Hacking something that could have been included was the history of probability in Europe before the mid seventeenth century. Also it is good the author didn’t include the growth of economics from the book and the theology of the period.

In conclusion the chapter on Non-Transitive dice and Other Paradoxes is about transitivity and the principle of indifference. It is also about probability of dice and what the probability would be based on the principle of indifference or not. It is also about the different people throughout history like Pascal and Rudolf Carnap. Also some things that could be added to chapter would be the main difference between statistical and inductive probability, the voting paradox, and the history of probability in Europe before the mid seventeenth century. Some things that are good the author omitted could be not including all of the different types of probability, not including all of the formulas, and the theology of the period.