Generalizing the result by Y. L. Tong, a chain of inequalities for probabilities in some types of multivariate distributions is proved. These inequalities embrace a large number of interesting special cases. Nine illustrations are given: cases of multivariate equicorrelated normal, $t,\chi^2$, Poisson, exponential distributions, normal and rank statistics for comparing many treatments with one control, order statistics used in estimating quantiles, and characteristic roots of covariance matrices in certain multiple sampling.
 Z. Šidák: Rectangular confidence regions for the means of multivariate normal distributions
. J. Amer. Statist. Assoc. 62 (1967), 626-633. MR 0216666
 Z. Šidák: On probabilities of rectangles in multivariate Student distributions: their dependence on correlations
. Ann. Math. Statist. 42 (1971), 169-175. DOI 10.1214/aoms/1177693504
| MR 0278354