(Extreme values) An ordinary six-sided die is rolled five times. Let the results be X₁, X2, X3, X4, X5, in that order. Let the greatest result be G. For example, if the results are 1, 4, 3, 2, 2, then G = 4. (a) What are the probabilities for the events {G ≤ k}, for each of the integers k = 1, 2, 3, 4, 5, 6? It would be possible to list all 65 = 7776 possible result sequences, and collect the cases where, say, G≤ 3, but we really do not want to do that by hand. Instead, use this hint. What is the condition that each individual roll has to fulfill so that G < 3? What is the probability that this condition is fulfilled for all five rolls? Do the same for each k. (b) Using the results from the previous item, and the additive rule of probability, determine the probabilities for the events {G = k}, for all k = 1, 2, 3, 4, 5, 6. Now present the distribution of G as a table. Explain the rough shape of the distribution in words (or use a picture). (c) Show that G and X₁ are dependent (i.e. not independent), e.g. by showing that equation (4.3.8) [Ross section 4.3.1] is not fulfilled. Hint: Consider, for example, the events {G = 1} and {X₁ = 1}.
(Extreme values) An ordinary six-sided die is rolled five times. Let the results be X₁, X2, X3, X4, X5, in that order. Let the greatest result be G. For example, if the results are 1, 4, 3, 2, 2, then G = 4. (a) What are the probabilities for the events {G ≤ k}, for each of the integers k = 1, 2, 3, 4, 5, 6? It would be possible to list all 65 = 7776 possible result sequences, and collect the cases where, say, G≤ 3, but we really do not want to do that by hand. Instead, use this hint. What is the condition that each individual roll has to fulfill so that G < 3? What is the probability that this condition is fulfilled for all five rolls? Do the same for each k. (b) Using the results from the previous item, and the additive rule of probability, determine the probabilities for the events {G = k}, for all k = 1, 2, 3, 4, 5, 6. Now present the distribution of G as a table. Explain the rough shape of the distribution in words (or use a picture). (c) Show that G and X₁ are dependent (i.e. not independent), e.g. by showing that equation (4.3.8) [Ross section 4.3.1] is not fulfilled. Hint: Consider, for example, the events {G = 1} and {X₁ = 1}.
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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![(Extreme values) An ordinary six-sided die is rolled five times. Let the results be X₁,
X2, X3, X4, X5, in that order. Let the greatest result be G. For example, if the results are
1, 4, 3, 2, 2, then G = 4.
(a) What are the probabilities for the events {G ≤ k}, for each of the integers k = 1, 2, 3, 4, 5, 6?
It would be possible to list all 65 = 7776 possible result sequences, and collect the cases where,
say, G≤ 3, but we really do not want to do that by hand. Instead, use this hint. What is the
condition that each individual roll has to fulfill so that G < 3? What is the probability that this
condition is fulfilled for all five rolls? Do the same for each k.
(b) Using the results from the previous item, and the additive rule of probability, determine
the probabilities for the events {G = k}, for all k = 1, 2, 3, 4, 5, 6. Now present the
distribution of G as a table. Explain the rough shape of the distribution in words (or use
a picture).
(c) Show that G and X₁ are dependent (i.e. not independent), e.g. by showing that equation
(4.3.8) [Ross section 4.3.1] is not fulfilled.
Hint: Consider, for example, the events {G = 1} and {X₁ = 1}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3802b3a4-f93d-4b7b-94b3-88a2c6183309%2Fa9f5bdb9-27d2-4716-a03e-d68849cb0a07%2Fng31l3f.jpeg&w=3840&q=75)
Transcribed Image Text:(Extreme values) An ordinary six-sided die is rolled five times. Let the results be X₁,
X2, X3, X4, X5, in that order. Let the greatest result be G. For example, if the results are
1, 4, 3, 2, 2, then G = 4.
(a) What are the probabilities for the events {G ≤ k}, for each of the integers k = 1, 2, 3, 4, 5, 6?
It would be possible to list all 65 = 7776 possible result sequences, and collect the cases where,
say, G≤ 3, but we really do not want to do that by hand. Instead, use this hint. What is the
condition that each individual roll has to fulfill so that G < 3? What is the probability that this
condition is fulfilled for all five rolls? Do the same for each k.
(b) Using the results from the previous item, and the additive rule of probability, determine
the probabilities for the events {G = k}, for all k = 1, 2, 3, 4, 5, 6. Now present the
distribution of G as a table. Explain the rough shape of the distribution in words (or use
a picture).
(c) Show that G and X₁ are dependent (i.e. not independent), e.g. by showing that equation
(4.3.8) [Ross section 4.3.1] is not fulfilled.
Hint: Consider, for example, the events {G = 1} and {X₁ = 1}.
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