Interperetation of tau in marokov chains

Question

I'd like to ask you a question about the meaning of certain equation in my exercise.

This concerns Markov Chains I have:

$\tau =inf\{n>=1:X_n\in\{3,5\}\}$

and I have to calculate $P(\tau=1)$

What exactly is tau? Is it the step in which process achieves state 3 or state 5? Meaning that $P(\tau=1)$ is the probability of achieving 3 or 5 in step 1?

Thank's for help.

Answer 1

$\tau$ is simply the first occurance of a certain state. If the state is say, "The machine breaks down", it may be of interst to know what the probablity of this happening at various times is, since the first occurance of this state has special significance.