Bonferroni Sometimes Loses

Bonferroni inequalities often provide tight upper and lower bounds for the probability of a union of events.The bounds are especially useful when this probability is difficult to compute exactly.There are situations, however, in which the Bonferroni approach gives very poor results.An example is given in which the upper and lower Bonferroni bounds are far from the probability they seek to approximate and successive bounds do not converge.Even an improved first upper Bonferroni bound may not be close to the probability of the union of events.