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In the context of Markov chains, the transition probability matrix plays a fundamental role in predicting the behavior of a stochastic process over time. This matrix, usually denoted as P, contains all the probabilities of moving from one state to another in one time step. Each entry P_ij represents the probability of transitioning from state i to state j.

For the problem involving genetic types, the transition matrix is defined using binomial probabilities to account for the random selection of genes from one generation to the next. The matrix is constructed such that the sum of the probabilities in any given row is equal to 1, validating the concept that a transition from one state must result in some subsequent state. Thus, the P_ij entries account for all possible outcomes of gene type distributions in the next generation.

The expected value in a Markov chain provides us with a way to forecast the average outcome over many iterations of the process. For our genetic example, the expected value of X_n, where X_n is the number of type 1 genes in the nth generation, can be calculated using the transition probabilities and the states' values. To add clarity to the existing solution, the formula for the expected value is given by:
\[E\left[X_{n}\right] = \sum_{j=0}^{m} j \cdot P_{ij}\]The equation sums up the product of each state's value (the number of type 1 genes) and the probability of reaching that state from the current state i. Even though we're dealing with a random process, the expected value remains constant over successive generations because the probabilities don't change—this is a property of stationary Markov chains, where the system's dynamics are stable over time.

Within Markov chains, absorbing states are special because once entered, the process cannot leave them—there is no escape. The problem at hand sets the scene for an absorbing state when all genes become type 1, denoted as state m. This state is 'sticky' because once all genes are of type 1, no transitions to other states are possible, reflecting the real-world scenario where certain conditions become permanent.

Identifying absorbing states is crucial for understanding the long-term behavior of the system. In such states, the transition probability to itself is 1, and to all other states is 0. This is significant because it allows us to predict ultimate outcomes and their probabilities, which—when calculated—can be of immense value in various fields such as genetics, finance, and even sports.

The stationary distribution of a Markov chain, a key concept within probability theory, is a distribution of probabilities that remains unchanged as the process evolves over time. In other words, it represents an equilibrium state of the stochastic process.

For a Markov chain to have a stationary distribution, it must be regular, meaning it is possible to go from any state to any other state in a finite number of steps. This stationary distribution can be found by solving a system of linear equations that come from the transition probability matrix. When a Markov chain has a stationary distribution, the expected number of genes of each type remains stable across generations—even as the actual number fluctuates randomly from generation to generation—reflecting the process's underlying stability.

The fundamental matrix, often denoted by N, is a concept tightly linked with absorbing Markov chains. It tells us about the expected number of times the chain will visit a transient state before being absorbed. In our example involving genes, the fundamental matrix can help understand how many generations, on average, it will take before all genes are of type 1 under different starting conditions.

To find the fundamental matrix, we subtract the transition matrix of transient states, denoted by Q, from an identity matrix of the same dimension, taking the inverse of the result:
\[N = (I - Q)^{-1}\]N is then used to determine many important quantities, such as the mean first passage times—the average time to be absorbed starting from a transient state—and different long-term behavior probabilities. It can be especially insightful when evaluating processes that have clear end conditions or 'goals,' providing a measure of the expected time to reach such goals.