A laboratory animal may eat any one of three foods each day....

A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal chooses one food on one trial, it will choose the same food on the next trial with a probability of $50 \%,$ and it will choose the other foods on the next trial with equal probabilities of 25$\% .$ a. What is the stochastic matrix for this situation? b. If the animal chooses food $\# 1$ on an initial trial, what is the probability that it will choose food $\# 2$ on the second trial after the initial trial?

A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal chooses one food on one trial, it will choose the same food on the next trial with a probability of $50 \%,$ and it will choose the other foods on the next trial with equal probabilities of 25$\% .$
a. What is the stochastic matrix for this situation?
b. If the animal chooses food $\# 1$ on an initial trial, what is the probability that it will choose food $\# 2$ on the second trial after the initial trial?

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Key Concepts

Markov Chain

A Markov chain is a stochastic process that satisfies the memoryless property, meaning that the probability of transitioning to any future state depends solely on the current state and not on the sequence of events that preceded it. This technique is widely used in various fields to model random processes that evolve over time.

Stochastic Matrix

A stochastic matrix is a square matrix used to represent the transition probabilities in a Markov chain. Each entry in the matrix represents the probability of moving from one state to another in one time step, and the sum of probabilities in each row is 1. This matrix simplifies the analysis of multistep transitions and long-term behavior of the process.

Transition Probability

Transition probabilities are the chances of moving from one state to another in a stochastic process. In the context of a Markov chain, these probabilities are fixed and represented by the entries in the stochastic matrix. They form the building blocks for predicting future states and analyzing the dynamics of the system.

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