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1. You have two dice, one with 4 sides (labelled 1, 2, 3, 4) and one with 8 sides (labelled 1,2,...,8). Each die is fair in the sense that each side is equally likely to be rolled whenever the die is rolled.¹ You pick one of the dice at random, choosing the 4-sided die with probability p and otherwise choosing the 8-sided die. You then roll the chosen die twice and record the numbers rolled (in order). (a) Are the events "the first number is 1" and "the second number is 1" independent? (b) Are the events "the first number is even" and "the second number is even" indepen- dent? If there are unstated assumptions in the question, state clearly what reasonable extra assumptions you are making.

1. You have two dice, one with 4 sides (labelled 1, 2, 3, 4) and one with 8 sides (labelled 1,2,...,8). Each die is fair in the sense that each side is equally likely to be rolled whenever the die is rolled.¹ You pick one of the dice at random, choosing the 4-sided die with probability p and otherwise choosing the 8-sided die. You then roll the chosen die twice and record the numbers rolled (in order). (a) Are the events "the first number is 1" and "the second number is 1" independent? (b) Are the events "the first number is even" and "the second number is even" indepen- dent? If there are unstated assumptions in the question, state clearly what reasonable extra assumptions you are making.

A First Course in Probability (10th Edition)
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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1. You have two dice, one with 4 sides (labelled 1, 2, 3, 4) and one with 8 sides (labelled 1,2,,8). Each die is fair in the sense that each side is equally likely to be rolled whenever the die is rolled. You pick one of the dice at random, choosing the 4-sided die with probability p and otherwise choosing the 8-sided die. You then roll the chosen die twice and record the numbers rolled (in order). (a) Are the events "the first number is 1" and "the second number is 1" independent? (b) Are the events "the first number is even" and "the second number is even" indepen- dent? If there are unstated assumptions in the question, state clearly what reasonable extra assumptions you are making.
Transcribed Image Text:1. You have two dice, one with 4 sides (labelled 1, 2, 3, 4) and one with 8 sides (labelled 1,2,,8). Each die is fair in the sense that each side is equally likely to be rolled whenever the die is rolled. You pick one of the dice at random, choosing the 4-sided die with probability p and otherwise choosing the 8-sided die. You then roll the chosen die twice and record the numbers rolled (in order). (a) Are the events "the first number is 1" and "the second number is 1" independent? (b) Are the events "the first number is even" and "the second number is even" indepen- dent? If there are unstated assumptions in the question, state clearly what reasonable extra assumptions you are making.
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