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A set of dice is called intransitive (or non-transitive) if it contains three dice, A, B, C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time (reproduced from Wikipedia: Intransitive Dice) Consider the following set of dice: is not transitive on its elements. • Die A has sides {2,2,4, 4, 9,9}. • Die B has sides {1, 1,6, 6, 8, 8}. • Die C has sides {3,3,5, 5, 7,7}. You can readily verify that the probability of rolling die A higher than B is 5/9, and similarly for B> C and A (see Wikinedia article above)

A set of dice is called intransitive (or non-transitive) if it contains three dice, A, B, C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time (reproduced from Wikipedia: Intransitive Dice) Consider the following set of dice: is not transitive on its elements. • Die A has sides {2,2,4, 4, 9,9}. • Die B has sides {1, 1,6, 6, 8, 8}. • Die C has sides {3,3,5, 5, 7,7}. You can readily verify that the probability of rolling die A higher than B is 5/9, and similarly for B> C and A (see Wikinedia article above)

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.6: Counting Principles
Problem 74E: Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is...
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A set of dice is called intransitive (or non-transitive) if it contains three dice, A, B, C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation a higher number than Y more than half the time (reproduced from Wikipedia: Intransitive Dice) Consider the following set of dice: X rolls is not transitive on its elements. - • Die A has sides {2,2,4, 4, 9,9}. • Die B has sides {1, 1,6, 6, 8, 8}. • Die C has sides {3,3,5, 5, 7, 7}. You can readily verify that the probability of rolling die A higher than B is 5/9, and similarly for B > C and C > A (see Wikipedia article above). For this question, assume you roll all three dice at once and are interested in which one gives the highest number overall. Define the event HA="die A is the highest overall", and similarly for HB, Hc. (a) Are the events HA, HB, Hc a) mutually exclusive, or b) a partition? Justify your answer. (b) Find the probabilities P(H A) , P(HB), P(Hc); are they all equal?
Transcribed Image Text:A set of dice is called intransitive (or non-transitive) if it contains three dice, A, B, C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation a higher number than Y more than half the time (reproduced from Wikipedia: Intransitive Dice) Consider the following set of dice: X rolls is not transitive on its elements. - • Die A has sides {2,2,4, 4, 9,9}. • Die B has sides {1, 1,6, 6, 8, 8}. • Die C has sides {3,3,5, 5, 7, 7}. You can readily verify that the probability of rolling die A higher than B is 5/9, and similarly for B > C and C > A (see Wikipedia article above). For this question, assume you roll all three dice at once and are interested in which one gives the highest number overall. Define the event HA="die A is the highest overall", and similarly for HB, Hc. (a) Are the events HA, HB, Hc a) mutually exclusive, or b) a partition? Justify your answer. (b) Find the probabilities P(H A) , P(HB), P(Hc); are they all equal?
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