Finding conditional and joint probabilities from a table of data

[user366312]

TL;DR

Finding conditional and joint probabilities from a table of data generated by Markov Chain simulation.

Let,

    alpha <- c(1, 1) / 2
    mat <- matrix(c(1 / 2, 0, 1 / 2, 1), nrow = 2, ncol = 2)

    chainSim <- function(alpha, mat, n)
    {
      out <- numeric(n)
      out[1] <- sample(1:2, 1, prob = alpha)
      for(i in 2:n)
        out[i] <- sample(1:2, 1, prob = mat[out[i - 1], ])
      out
    }

Suppose the following is the result of a 5-step Markov Chain simulation repeated 10 times:

> sim
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    2    1    1    2    2    2    1    1    1     2
[2,]    2    1    2    2    2    2    2    1    1     2
[3,]    2    1    2    2    2    2    2    1    2     2
[4,]    2    2    2    2    2    2    2    1    2     2
[5,]    2    2    2    2    2    2    2    2    2     2
[6,]    2    2    2    2    2    2    2    2    2     2

What would be the values of the following?

1. P(X1=1|X0=1)P(X1=1|X0=1)
2. P(X2=1|X0=1)P(X2=1|X0=1)
3. P(X5=2|X2=1)P(X5=2|X2=1)
4. P(X1=1,X3=1)P(X1=1,X3=1)
5. P(X5=2|X0=1,X2=1)P(X5=2|X0=1,X2=1)
6. E(X2)E(X2)

I tried them as follows:

1. mean(sim[2, sim[1, ] == 1] == 1)
2. mean(sim[3, sim[1, ] == 1] == 1)
3. mean(sim[6, sim[3, ] == 1] == 2)
4. mean(sim[4, ] == 1 && sim[2, ]== 1)
5. ?
6. c(1,2) * mean(sim[2, ])

What would be the solution of (5)?

Am I correct withe the rest?

 

[FactChecker]

Are you getting meaningful probabilities between 0 and 1 from your code?
I guess that I wasn't clear in a similar thread of yours Post #8 of similar thread
I think that your use of 'mean' is wrong here. It will give you the average of a lot of values of 1 and 2, which will be over 1. That can not be a probability. You must count the number of entries, not their values. You can do that by using the 'length' function as I showed in Post #8 of the other thread.

PS. I just noticed that this thread is several days old, so my answer here might already be known and understood by the OP.