A mouse roams between the living room and the kitchen of a house. However, a cat also lives in the house, so the mouse is at great risk. For simplicity, we'll assume that the mouse can be at these 3 different places: in the living room, in the kitchen, or in the cat. Here are the transition probabilities: If the mouse is in the living room, then one hour later the probability of being in the living room is 0.42 and the probability of being in the kitchen is 0.52. So the probability is 0.06 that the mouse will have been eaten by the cat. If the mouse is in the kitchen, then one hour later the probability is 0.7 that the mouse will be in the kitchen, 0.23 that the mouse will be in the living room, and 0.07 that the mouse will have been eaten. Treat this as a 3-state Markov Chain with "in the cat" being an absorbing state (literally). Since absorbing states should be listed first, set this up with "in the cat" as state #1. The order of states 2 and 3 won't matter, since the other two states are non- absorbing states. Use the same order for row and column labels. Compute the matrix N- (I - Q). Recall from the videos that this matrix can be used to find expected values. Use your N matrix will be a single entry of the matrix N.) answer the following questions. (Hint: Your answer for (a) will be the sum of a row of the matrix N. Your answer for (b) (a) If the mouse is in the kitchen, what is the expected value for the number hours that elapse before the mouse is eaten by the cat? (Give your answer correct to 2 decimal places.) hours (b) If the mouse is in the kitchen, and if you check every hour to see if the mouse is in the kitchen, what is the expected value for the number of times you will find the mouse in the kitchen before the mouse finally gets eaten by the cat? (Give your answer correct to 2 decimal places.) hours
A mouse roams between the living room and the kitchen of a house. However, a cat also lives in the house, so the mouse is at great risk. For simplicity, we'll assume that the mouse can be at these 3 different places: in the living room, in the kitchen, or in the cat. Here are the transition probabilities: If the mouse is in the living room, then one hour later the probability of being in the living room is 0.42 and the probability of being in the kitchen is 0.52. So the probability is 0.06 that the mouse will have been eaten by the cat. If the mouse is in the kitchen, then one hour later the probability is 0.7 that the mouse will be in the kitchen, 0.23 that the mouse will be in the living room, and 0.07 that the mouse will have been eaten. Treat this as a 3-state Markov Chain with "in the cat" being an absorbing state (literally). Since absorbing states should be listed first, set this up with "in the cat" as state #1. The order of states 2 and 3 won't matter, since the other two states are non- absorbing states. Use the same order for row and column labels. Compute the matrix N- (I - Q). Recall from the videos that this matrix can be used to find expected values. Use your N matrix will be a single entry of the matrix N.) answer the following questions. (Hint: Your answer for (a) will be the sum of a row of the matrix N. Your answer for (b) (a) If the mouse is in the kitchen, what is the expected value for the number hours that elapse before the mouse is eaten by the cat? (Give your answer correct to 2 decimal places.) hours (b) If the mouse is in the kitchen, and if you check every hour to see if the mouse is in the kitchen, what is the expected value for the number of times you will find the mouse in the kitchen before the mouse finally gets eaten by the cat? (Give your answer correct to 2 decimal places.) hours
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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Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
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Transcribed Image Text:A mouse roams between the living room and the kitchen of a house. However, a cat also lives in the house, so the mouse is at great risk. For simplicity, we'll assume that the mouse can be at these 3 different places: in the living room, in the kitchen, or
in the cat.
Here are the transition probabilities:
If the mouse is in the living room, then one hour later the probability of being in the living room is 0.42 and the probability of being in the kitchen is 0.52. So the probability is 0.06 that the mouse will have been eaten by the cat. If the mouse is in
the kitchen, then one hour later the probability is 0.7 that the mouse will be in the kitchen, 0.23 that the mouse will be in the living room, and 0.07 that the mouse will have been eaten.
Treat this as a 3-state Markov Chain with "in the cat" being an absorbing state (literally). Since absorbing states should be listed first, set this up with "in the cat" as state #1. The order of states 2 and 3 won't matter, since the other two states are non-
absorbing states. Use the same order for row and column labels.
Compute the matrix N =
(I - Q)1. Recall from the videos that this matrix can be used to find expected values. Use your N matrix to answer the following questions. (Hint: Your answer for (a) will be the sum of a row of the matrix N. Your answer for (b)
will be a single entry of the matrix N.)
(a) If the mouse is in the kitchen, what is the expected value for the number of hours that will elapse before the mouse is eaten by the cat? (Give your answer correct to 2 decimal places.)
hours
(b) If the mouse is in the kitchen, and if you check every hour to see if the mouse is in the kitchen, what is the expected value for the number of times you will find the mouse in the kitchen before the mouse finally gets eaten by the cat? (Give
your answer correct to 2 decimal places.)
hours
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