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Suppose that $C >0$ and that $X,Y$ are random variables such that $X>Y$. I am wondering if it is true in general, or under what conditions do we have:

$$ P(X>C) \leq P(Y>C) $$

In other words, if my random variable $X$ is bounded below by $Y$, is the event $\{X>C\}$ a subset of $\{Y>C\}$? Thanks.

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    If $X > Y$ almost surely, and $Y > C$ almost surely, then $X > Y > C$ almost surely. The same transitivity property of inequality applies. – 2017-02-14
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    This was asked very recently. Same class, with same homework? – 2017-02-14
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    @Did, This is a stepping stone to a research problem I have been stuck on. – 2017-02-14

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Yes (although I think you got the direction of the inequality wrong in your question. If $X>Y$ it is true that $P(X> C) \ge P(Y>C)$).

If $X>Y$ then for any outcome where $Y>C,$ we also have $X>C$ (in other words, as events, $\{Y>C\}\subseteq \{X>C\}$). As a result the event $X>C$ occurs at least as often as $Y>C$ so we must have $P(X>C)\ge P(Y>C)$.