Chance constraint ($P\{\text{constraint holds}\}\ge 1-\epsilon$)

A constraint imposed with a specified probability when data or demands are random. Distributional assumptions can turn it into a deterministic condition on decision variables. The choice of strict versus non-strict event matters when the distribution has atoms, as in the deterministic boundary in an upper-tail chance constraint.

A deterministic demand has probability one of meeting or exceeding its mean. Thus replacing a positive-variance Gaussian chance constraint by its naive zero-variance limit with a non-strict capacity inequality loses the boundary condition. The example $X=0$, $C=0$, $\epsilon =e^{-1}$ satisfies $m+\varphi \sqrt{v}\le C$ for finite $\varphi$, but violates the chance requirement.

For $X\sim N(m,v)$ with $v>0$ and $0<\epsilon <1$, $P(X\ge C)\le \epsilon$ is equivalent to the displayed chance constraint, because the standard normal distribution function is continuous and strictly increasing. The coefficient is a standard normal quantile, which can be negative. If $v=0$, equality is an atom and the exact condition becomes $C>m$ instead.