Scan statistics have been extensively used in many areas of science to analyze the occurrence of observed clusters of events in time or space. Since the scan statistics are based on the highly dependent consecutive subsequences of observed data, accurate probability inequalities for their distributions are of great value. We derive accurate second-order Bonferroni-type inequalities for the distribution of linear and circular scan statistics. Our approach is based on the scanning window representation of scan statistics. Both the one-dimensional continuous and discrete cases are investigated. Based on the numerical results presented in this article, it is evident that the Bonferroni-type inequalities are tight. For all practical purposes, they determine the scan statistics probabilities used in testing. Based on the probability inequalities for the distribution of scan statistics accurate inequalities for the expected size of a possible cluster are derived. Numerical results presented in this article indicate that these inequalities are tight; thus, providing valuable information on the expected size of the largest cluster of events.