  Fisher's exact test is often a preferred method to estimate the significance
of statistical dependence. However, in large data sets the test is usually too
worksome to be applied, especially in an exhaustive search (data mining). The
traditional solution is to approximate the significance with the
$\chi^2$-measure, but the accuracy is often unacceptable. As a solution, we
introduce a family of upper bounds, which are fast to calculate and approximate
Fisher's $p$-value accurately. In addition, the new approximations are not
sensitive to the data size, distribution, or smallest expected counts like the
$\chi^2$-based approximation. According to both theoretical and experimental
analysis, the new approximations produce accurate results for all sufficiently
strong dependencies. The basic form of the approximation can fail with weak
dependencies, but the general form of the upper bounds can be adjusted to be
arbitrarily accurate.
