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Hoeffding's Inequality gives, for independent Bernoulli random variables, that $\mathbb{P}\{|\sum_{i=1}^n X_i| > t\} \leq 2\text{exp}(-\frac{t^2}{2}) $. I'm wondering if this is also a lower bound for $\mathbb{P}\{|\sum_{i=1}^n X_i| \leq t\} \geq 2\text{exp}(-\frac{t^2}{2})$. I think that this is true intuitively but cannot quite show it.

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What you propose is not true. The event in your second equation is the opposite event, therefore what you can say is that its probability is $\ge 1-2e^{-t^2/2}$

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