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Chapter 24: Problem 631

In the brother-sister mating problem, two individuals are mated, and from among their direct descendants two individuals of opposite sex are selected at random. These are mated, and the process continues indefinitely. a) What is the transition matrix for this experiment? b) What are the absorbing states?Business Applications

Short Answer

Expert verified
The transition matrix for the brother-sister mating problem is: \[ P = \begin{bmatrix} 1 & 0 & 0 \\ 1/4 & 1/2 & 1/4 \\ 0 & 0 & 1 \end{bmatrix} \] The absorbing states are State A (the two descendants have the same parent) and State C (the two descendants have no common grandparents or parents).

Step by step solution

01

Identify possible states

There are three possible states in the brother-sister mating problem: 1. State A: The two descendants have the same parent. 2. State B: The two descendants have the same grandparents but not the same parent. 3. State C: The two descendants have no common grandparents or parents.
02

Determine the transition probabilities

We need to find the probability of moving from one state to another: 1. From state A to A: The probability of selecting two siblings as direct descendants is 1 (since they must have the same parent). 2. From state A to B: The probability of selecting two cousins as direct descendants is 0 (since they share grandparents but not a parent). 3. From state A to C: The probability of selecting two individuals with no common grandparents or parents is 0 (since they share a parent). 4. From state B to A: The probability of selecting two siblings as direct descendants is 1/4 (as there's a 1 in 4 chance they share a parent). 5. From state B to B: The probability of selecting two cousins as direct descendants is 1/2 (as there's a 2 in 4 chance they share grandparents but not parents). 6. From state B to C: The probability of selecting two individuals with no common grandparents or parents is 1/4 (as there's a 1 in 4 chance they don't share any grandparents). 7. From state C to A, C to B, and C to C: There's no way to move back into state A or B from state C, so these probabilities are 0. The probability of staying in state C is 1.
03

Create the transition matrix

Now that we have the transition probabilities, we can create the transition matrix P: \[ P = \begin{bmatrix} 1 & 0 & 0 \\ 1/4 & 1/2 & 1/4 \\ 0 & 0 & 1 \end{bmatrix} \]
04

Identify the absorbing states

An absorbing state is a state that, once entered, cannot be left. In this transition matrix, states A and C have a probability of 1 to remain in the same state, making them absorbing states. So the absorbing states are: - State A: The two descendants have the same parent. - State C: The two descendants have no common grandparents or parents.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Matrix
A **transition matrix** is a fundamental tool used in the study of Markov chains. It represents the probabilities of moving from one state to another. Each entry in the matrix corresponds to the probability of transitioning from one specific state to another.

In this exercise, we consider the brother-sister mating problem with states A, B, and C. The transition probabilities are laid out in a matrix form, reflecting the likelihood of each possible transition. For instance:
  • State A has a 100% probability of staying the same (1), while there is 0% chance of transitioning to states B or C (0).
  • State B can transition to state A with a probability of 0.25, remain in state B with 0.5, or transition to state C with 0.25.
  • State C will always remain in state C, with a probability of 1.
These probabilities are aligned as rows in the matrix:\[ P = \begin{bmatrix} 1 & 0 & 0 \ 1/4 & 1/2 & 1/4 \ 0 & 0 & 1 \ \end{bmatrix}\]Each row sums up to 1, confirming that the total probability across potential transitions is complete.
Absorbing States
**Absorbing states** in a Markov chain are states that, once entered, cannot be exited. When a process reaches an absorbing state, it will remain there indefinitely.

In the given transition matrix, states where the probability of staying is 1 are absorbing states. According to the matrix:
  • State A is absorbing, because if the process enters this state, it will remain there ( 1 ).
  • State C is also absorbing for the same reason ( 1 ).
These states are crucial in probabilistic analysis, as they represent end points in a process, where transitions cease. This concept helps us understand long-term behaviors of processes modeled by the matrix.
Probability Distribution
A **probability distribution** in the context of Markov chains, such as in the brother-sister mating problem, shows how likely each state is to occur over time as the process iterates.

Initially, the probability distribution might be uniform, with each state having equal likelihood. As transitions occur according to the transition matrix, the distribution will evolve, eventually converging to a stable pattern called the steady state. This state depends on the initial distribution and the structure of the transition matrix.
  • The presence of absorbing states can greatly influence the final distribution. If one starts in a non-absorbing state, the probability distribution will gradually shift towards the absorbing states.
  • For example, starting in state B, as transitions occur, there's a possibility of moving towards absorbing states A or C, due to their probabilities.
  • Understanding these distributions helps in predicting which states are likely to be occupied in the long term.
Analyzing how these distributions evolve provides insight into the long-term trends and expected outcomes of processes defined by Markov chains.

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Most popular questions from this chapter

According to a psychological study, the personality traits emotional stability and sociability are related. Specifically, the study indicates that the relationship between these two traits can be represented by the matrix \(\mathrm{S}=\left[\mathrm{x}_{\mathrm{i}}\right]\) where:$$ \begin{array}{|l|l|l|l|} \cline { 2 - 4 } \multicolumn{1}{l|} {} & \begin{array}{l} \text { High } \\ \text { sociability } \end{array} & \begin{array}{l} \text { Average } \\ \text { sociability } \end{array} & \begin{array}{l} \text { Low } \\ \text { sociability } \end{array} \\ \hline \text { High emotional stability } & \mid 0.5 & 0.3 & 0.2 \mid \\ \hline \text { Average emotional stability } & 10.4 & 0.3 & 0.3 \mid=\mathrm{S} \\ \hline \text { Low emotional stability } & 10.1 & 0.4 & 0.5 \mid \\ \hline \end{array} $$and for \(i=1,2,3\) and \(j=1,2,3, x_{1 j}\) represents the probility that a person with rating i in emotional stability will have rating \(j\) in sociability. In a group of 1000 randomly selected individuals, 300 are rated high in emotional stability, 500 are rated average in emotional stability, and 200 are rated low in emotional stability. If the entries in \(\mathrm{S}\) are correct, how many of these individuals can be expected to fall into each of the sociability rating categories?

Assume that rabbits do no reproduce during the first month of their lives but that beginning with the second month each pair of rabbits has one pair of offspring per month. Assuming that none of the rabbits die and beginning with one pair of newborn rabbits, how many pairs of rabbits are alive after \(n\) months?

Which of the following matrices can be interpreted as perfect communications matrices? For those that are, find the two stage and three stage communication lines that are feedback. \(\begin{array}{lllllllll} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & & \mathrm{A} & \mathrm{B} & \mathrm{C} \\\ \mathrm{A} & 0 & 1 & 1 & 2 & \mathrm{~A} & 0 & 1 & 1 \\ \mathrm{~B} & 1 & 0 & 1 & 0 & \mathrm{~B} & 1 & 0 & 1 \\ \mathrm{C} & 1 & 1 & 0 & 1 & \mathrm{C} & 1 & 1 & 0 \\ \mathrm{D} & 2 & 0 & 1 & 0 & & & & \end{array}\)

Give examples of the following concepts a) Graph b) Digraph c) Matrix of a digraph d) Incidence matrix.

Bob, Ted, Carol and Alice are throwing a ball to one another. Alice always throws it to Ted; Bob is equally likely to throw it to anybody else; Carol throws it to the boys with equal frequency; Ted throws it to Carol twice as often as to Alice and never throws it to Bob. Construct a stochastic matrix to answer the following question: What is the probability that the ball will go from a) Bob to Carol in two throws b) Carol to Alice in three throws?
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