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 higherthan C more than half the time, but it is not true that A rolls higher than C more thanhalf the time. In other words, a set of dice is intransitive if the binary relationa higher number than Y more than half the time(reproduced from Wikipedia: Intransitive Dice)Consider the following set of dice:X rollsis 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, andsimilarly 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 onegives 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?

There are three dice, which have the following faces: Die A= {2,2,4,4,9,9} Die B= {1,1,6,6,8,8} Die…

Answered: A set of dice is called intransitive…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider a game with n ∈ N participants which are somehow ordered (P1, P2, ..., Pn). The game starts with the first numbered player (P1) tosses a fair coin until the first “Tail” appears. By x1 we denote the number of flips made by P1. Player P1 is eliminated from the game if x1 < y. In this case, the second player tosses a fair coin until the first “Tail” appears. Similarly, by x2 we denote the number of flips made by P2. Player P2 is eliminated from the game if x1 + x2 < y. The game proceeds this way until either the total number of coin flips attains y, or all players are eliminated. All those players who are not eliminated at the end of the game are winners. Find the expected number of tosses for a player i (E(xi)) assuming there is no limit on the total number of flips (y).
A company has hired a team of computer programmers. Let U be the set of all team members where U = {Amy, Bob, Cam, Deb, Eli, Flo, Gil, Hal}. %3D Let J be the set of members on the Java team where J = {Amy, Flo, Gil, Hal}. Let P be the set of members on the Python team where P = {Bob, Deb, Gil, Hal}. Let S be the set of members on the Swift team where S = {Eli, Flo, Hal}. %3D List the team members in the set (J U S) N P'. O A. (Flo} . (Amy, Eli, Flo} OC {Amy, Cam, Eli, Flo} D. {Amy, Eli, Flo, Hal} O E. {Amy, Cam, Eli, Flo, Hal) F. none of these Reset Selection
14. Event A and Event B are dependent. 21 a) P(B)=; p(A and B) 200 P(B|A) = b) P(B|A) = ; P(A and B) P(B) = 200
12 A panel of Judges A and B graded seven debtors and independently awarded the following marks : - Debator 1 3 4 7 31 Marks by A Marks by B 40 34 28 30 44 38 32 39 26 30 38 34 28 An eighth debator was awarded 36 marks by Judge A while Judge B was not present. If Judge B were also present, how many marks would you expect him to award to the eighth debator assuming that the same degree of relationship exists in their judgement. Stic
Which one of the following statements is False (not true)? Events A and B are independent if and only if P(A and B) = P(A) x P(B). Events A and B are independent if and only if P(A) = P(A given B). O(A) x O(not A) = 1 Events A and B are independent if and only if A and B are mutually exclusive; that is disjoint. If events A and B are mutually exclusive then, P (A or B) = P(A) + P(B).
Afat
5. Show that the conditional independence of A and B given C neither implies, nor is implied by, the independence of A and B. For which events C is it the case that, for all A and B, the events A and B are independent if and only if they are conditionally independent given C?
The events driving a sedan and driving in an attentive style are approximately independent because P(attentive | sedan) ≈ P(attentive). The events driving a sedan and driving in an attentive style are not independent because P(sedan | attentive) ≈ P(sedan). The events driving a sedan and driving in an attentive style are approximately independent because P(sedan | attentive) ≈ P(sedan). The events driving a sedan and driving in an attentive style are not independent because P(attentive | sedan) ≈ P(attentive).
3. A researcher decides to investigate the relationship between an exposure (X) and an outcome (Y). In order to do so, the researcher decides to collect information on a relevant variable Z that may play a role in the relationship between X and Y. Which of the following hypothetical scenarios describes interaction by the variable Z? A) Z is associated with an increase in both X and Y B) Z is associated with only the exposure C) Z is associated with only the outcome D) When Z is present, there is an increase in Y, and when Z is absent, there is a decrease in Y

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