In a proof I'm working through the following is stated:
$h$ is a random variable taking values in the closed interval $[0,1]$. For some $\varepsilon > 0$, we know that $E[h] \geq 1-\varepsilon$. Therefore, we have $P(h<{2\over 3}) < 3\varepsilon$.
I don't see how one can conclude that $P(h<{2\over 3}) < 3\varepsilon$ holds. Is it possible to see this with just the given information, or is there something missing?
$$1-\epsilon \leq Eh =EhI_{h<\frac 2 3} +EhI_{h \geq\frac 2 3}\leq \frac 2 3 P(h<\frac 2 3)+P(h \geq\frac 2 3)$$ $$=\frac 2 3 P(h<\frac 2 3)+1-P(h <\frac 2 3).$$ From this you can easily get $P(h <\frac 2 3) \leq 3\epsilon$.