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We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1- q A = - p (8a) Prove that X =1 is an eigenvalue of A (for any choice of p and q). (8b) Set p = 3/4 and q = 1/2. Find the eigenvector of A corresponding to A = 1, such that x + y = 1. (8c) It turns out that the numbers a and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.

We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1- q A = - p (8a) Prove that X =1 is an eigenvalue of A (for any choice of p and q). (8b) Set p = 3/4 and q = 1/2. Find the eigenvector of A corresponding to A = 1, such that x + y = 1. (8c) It turns out that the numbers a and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1- A = (8a) Prove that A = 1 is an eigenvalue of A (for any choice of p and q). (8b) Set p = 3/4 and q = 1/2. Find the eigenvector v = of A corresponding to A= 1, such that x + y = 1. (8c) It turns out that the numbers x and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.
Transcribed Image Text:We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1 – p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 – q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1- A = (8a) Prove that A = 1 is an eigenvalue of A (for any choice of p and q). (8b) Set p = 3/4 and q = 1/2. Find the eigenvector v = of A corresponding to A= 1, such that x + y = 1. (8c) It turns out that the numbers x and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.
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