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I'm having a little trouble obtaining an intuition wrt hitting times in random walks.
Let $X_n$ be the position of a simple $1$-dimensional random walk on $\mathbb{Z}$ and $T_k$ the first hitting time of $k\in\mathbb{Z}$.

Why do we have $\mathbb{P}(X_n\geq k)\leq \mathbb{P}(T_k\leq n)?$

I understand that $\mathbb{P}(X_n\geq k)$ is the probability that our RW is at a position greater or equal to $k$ after $n$ steps and $\mathbb{P}(T_k\leq n)$ is the probability that $k$ will first be reached before $n$ steps, but why is the latter more probable than the former?

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    $\begingroup$ Because the latter (event) contains the former (event). $\endgroup$
    – Did
    Commented Oct 14, 2017 at 14:11

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I guess that k>0 and that you are starting from zero, right?

For to be $X_n\ge k$ it is needed that $T_k\le n$ and not the opposite.

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