# sub-gaussian tail bound when $t=0$

Let $$X$$ be a sub-gaussian random variable, and $$K_i>0$$ are parameters.

Its tail satisfy $$P(|X|\geq t)\leq 2\text{exp}(-t^2/K_1^2)$$ where $$t\geq 0$$.

I am confused on the case that $$t=0$$. If $$t=0$$, then the tail

$$P(|X|\geq 0)\leq 2\text{exp}(0)=2$$, which is not a meaningful bound. Am I making some obvious mistakes? or Is it possible to know the probability of a sub-gaussian $$X$$ is non-negative?

The bound $$P[|X|\geq t]\leq 2\exp(-t^2/K^2) \quad \forall t \geq 0$$ implies the tighter bound $$P[|X|\geq t]\leq \min[1, 2\exp(-t^2/K^2)] \quad \forall t \geq 0$$ However, the former bound does not have a min[,] and is often easier to work with.
We need the factor of 2 to enable random variables with bounded support to be subGaussian. Specifically if we use $$X=c$$ with prob 1, for some $$c\neq 0$$, then it can be shown this is subGaussian, but there is no $$K>0$$ for which $$P[|X|\geq t] \leq \exp(-t^2/K^2)$$ for all $$t\geq 0$$.
It can be shown that $$X \sim Uniform[a,b]$$ is subGaussian for any finite $$a, so we can have $$P[X\geq t]$$ be any value in $$[0,1]$$.