I am new to Markov Processes, and while learning the discrete Markov chain with the following matrix $$P=begin{bmatrix}frac{1}{3} & frac{2}{3} & 0 & 0 \ frac{1}{2} & frac{1}{2} & 0 & 0 \

frac{1}{4} & 0 & frac{1}{4} & frac{1}{2} \0 & 0 & 0 & 1end{bmatrix},$$

I was told that $f_{34}(n)=left(frac14right)^{n-1}frac12$, where $f_{ij}(n)$ denotes the probability that starting in state $i$, we visit $j$ for the first time at time $n$. I can also verify this by hand.

I’ve tried the following input in Mathematica:

```
p = {{1/3, 2/3, 0, 0}, {1/2, 1/2, 0, 0}, {1/4, 0, 1/4, 1/2}, {0, 0, 0, 1}};
P = DiscreteMarkovProcess(3, p);
f34 = FirstPassageTimeDistribution(P, 4);
PDF(f34, n)
```

But the result is $$left{begin{array}{ll} 3 times 4^{-n} & n>0 \0 & text { True }end{array}right.$$

I’m confused since $f_{34}(1)=p_{34}=frac12$, how can it be $frac34$ as given? I’m using the version 12.1.1.0, could anyone give me a hint where I went wrong? Thanks!