Understanding an upper bound in the analysis of Karger's algorithm
$$(1-P(n))^{O(\log^{2} n)} \le (1-\frac{c}{\ln n})^{\frac{3}{c} \ln^{2} n}$$

First, to get a tighter bound, you should choose $\frac{4}{c}$ or larger, instead of $\frac{2}{c}$ or smaller. 

In particular, the probability for $\frac{2}{c}$ is
$$\textrm{Pr}[\textrm{miss any min-cut}] \le O(1).$$
This does not make sense for a probability value.

Second, the larger the chosen coefficient is, the higher the time complexity of the algorithm is (to achieve a tighter bound). That is, there is a trade-off.