Geometric minimum law (${\min }_i{X}_i\sim \mathrm{Geom}\left(1-{\prod }_i(1-p_i)\right)$)

For independent positive-support geometric random variables with success probabilities $p_i$, all must exceed $k$ for their minimum to exceed $k$. The resulting survival function is ${\prod }_i(1-p_i{)}^k$. This is the positive-support geometric distribution with the displayed success probability. Independence is needed for multiplying the survival probabilities.