What is the sample path of a stochastic process

Question

Assume $\Omega $={head, tail}, let T=$\mathbb N$ and $X_t$ $t\in T$ be a collection of i.i.d random variables following Bernoulli distribution. Since a stochastic process is a function of two variables. When $\omega$ is fixed, we get a sample path, so when fix $\omega$={head}, Is the sample path of $X_t(\omega)$ be the constant 1? But a realization of coin tossing can't be always head . So where is my fault?

Answer 1

The error is that $\Omega=$ {heads,tails}$^T$, where {heads,tails}$^T$ is the set of all functions from $T$ to {heads,tails}. So a typical $\omega$ is an infinite sequences whose entries are either heads or tails.