The following is Hoeffding’s Inequality:
$P(|\nu - \mu| > \epsilon) \leq 2e^{-2\epsilon^2N}$
The left side looks like a probability notation but the right side becomes $2$ when the exponential part becomes very close to zero, then I would get $2$ for the right side. What does the number $2$ mean?
The Hoeffding's inequality tells that how likely $\nu$ is more than $\epsilon$ away from $\mu$. More specifically, it upper bounds the probability that $\nu$ is more than $\epsilon$ away from $\mu$. As you can see, the term in the RHS decreases as $\epsilon$ gets larger; that is, $\nu$ is more likely to be near $\mu$ and is less likely to be very distant from $\mu$.
The number $2$ here is just an upper bound. It states that the probability that $\nu$ is more than $\epsilon$ away from $\mu$ is not greater than $2$ when $\epsilon$ is very small. Actually, the upper bound $2$ here is quite loose here and it does not provide any insight. However, when $\epsilon$ gets larger, the upper bound can be much smaller than $1$, and in this case, we can use it to estimate the probability rather than computing the probability exactly.