Consider the Markov chain with state space S = {1, 2}, transition matrix

and initial distribution α = (1/2, 1/2).

1. Simulate 5 steps of the Markov chain (that is, simulate X₀, X₁, . . . , X₅). Repeat the simulation 100 times. Use the results of your simulations to solve the following problems.

• Estimate P(X₁ = 1|X₀ = 1). Compare your result with the exact probability.

# returns Xn 
func2 <- function(alpha1, mat1, n1) 
{
  xn <- alpha1 %*% matrixpower(mat1, n1+1)
  
  return (xn)
}

alpha <- c(0.5, 0.5)
mat <- matrix(c(0.5, 0.5, 0, 1), nrow=2, ncol=2)
n <- 10


for (variable in 1:100) 
{
   print(func2(alpha, mat, n))
}

I have a few questions here:

1. Is my code correct here?
2. What is the difference if I run this code once or 100 times (as is said in the problem-statement)?
3. How can I find the conditional probability from here on?

Consider the Markov chain with state space S = {1, 2}, transition matrix

and initial distribution α = (1/2, 1/2).

1. Simulate 5 steps of the Markov chain (that is, simulate X₀, X₁, . . . , X₅). Repeat the simulation 100 times. Use the results of your simulations to solve the following problems.

• Estimate P(X₁ = 1|X₀ = 1). Compare your result with the exact probability.

My solution:

# returns Xn 
func2 <- function(alpha1, mat1, n1) 
{
  xn <- alpha1 %*% matrixpower(mat1, n1+1)
  
  return (xn)
}

alpha <- c(0.5, 0.5)
mat <- matrix(c(0.5, 0.5, 0, 1), nrow=2, ncol=2)
n <- 10


for (variable in 1:100) 
{
   print(func2(alpha, mat, n))
}

What is the difference if I run this code once versus 100 times (as is said in the problem-statement)?

How can I find the conditional probability from here on?